From 4aef5c959dcc3a76cdf2f594faf7519a5457cf1b Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Fri, 13 Jan 2023 16:12:48 +0100 Subject: New appeal letter for PRL. --- appeal.tex | 88 ++++++++++++++++++++++++++++++++++++++----------------- frsb_kac-rice.bib | 44 +++++++++++++++++++++++++++- 2 files changed, 104 insertions(+), 28 deletions(-) diff --git a/appeal.tex b/appeal.tex index 7ae78f4..ae37548 100644 --- a/appeal.tex +++ b/appeal.tex @@ -8,12 +8,14 @@ urlcolor=purple, linkcolor=black, citecolor=black, - filecolor=black + filecolor=black, ]{hyperref} % ref and cite links with pretty colors \usepackage{xcolor} \usepackage[style=phys]{biblatex} -\addbibresource{bezout.bib} +\renewcommand{\thefootnote}{\fnsymbol{footnote}} + +\addbibresource{frsb_kac-rice.bib} \signature{ \vspace{-6\medskipamount} @@ -30,36 +32,68 @@ \begin{document} \begin{letter}{ - Editorial Office\\ + Agnese I.~Curatolo, Ph.D.\\ Physical Review Letters\\ 1 Research Road\\ Ridge, NY 11961 } -\opening{To the editors of Physical Review,} - -We wish to appeal the decision on our manuscript \emph{How to count in -hierarchical landscapes: A ‘full’ solution to mean-field complexity}, which was -rejected without being sent to referees. - -The problem of characterizing the geometry of complex energy and cost -landscapes is long-standing. Until this work, the correct calculation of the -complexity has only been made for a small minority of systems, those with -so-called replica symmetry. We show explicitly how such calculations can be -made for the vast majority of cases. - -Landscape complexity even for the simple models we consider is relevant to a -broad spectrum of physics disciplines. These models appear explicitly in modern -research of machine learning, like tensor denoising, and understanding how -complexity, dynamics, and equilibrium interplay in them provides powerful analogies -and insights into emergent phenomena in more complicated contexts, from -realistic machine learning models to the behavior of structural glasses. -Already in this work, we identify the surprising result that the purported -algorithmic threshold for optimization on mean-field cost functions lies -\emph{far above} the geometric threshold traditionally understood as the dynamic limit. - -We urge you to allow this paper to go to referees and allow it to -be judged by other scientists at the forefront of these fields. +\opening{Dear Dr.~Curatolo,} + +We wish to appeal the decision concerning our manuscript, +and expect to take this to the highest level possible in Physical Review. +Our motivation is scientific, but also one of principle: {\em there should +not be an implicit bias in favor of partial or purely numerical solutions, +and against the actual closure of +a long standing problem, on the basis that the latter is harder to read.} + +Neither of the referees criticize the content of the manuscript. They both +find the article somewhat technical, +but it should be born in mind that: +\begin{enumerate} + \item PRL has been publishing articles on precisely this problem in the + last 30 years.\footfullcite{Fyodorov_2004_Complexity, Bray_2007_Statistics, Fyodorov_2012_Critical, Wainrib_2013_Topological} + \item These works were often limited by the fact that general landscapes (for + which an annealed solution is not exact) were inaccessible. It is perhaps + true that the final solution of an open problem may often be more technical + than the previous ones. +\end{enumerate} + +The other criticisms are: + +Referee A remarks that a full connection with dynamics +is not presented in our paper. This is true, and such a connection does not +exist at present. We sincerely hope to +find it, and if we succeed we firmly believe the resulting article will also be +worthy of PRL. + +Referee B states that +\begin{enumerate} + \item ``the topic has been studied extensively in the last thirty years and + more'' as a shortcoming, but fails to realize that the solution to the full + problem has been open, in spite of all these efforts. + \item The meaning of the related remark: ``the only novelty with respect to + previous work is that the results are obtained at zero temperature'' + completely escapes us. Firstly because temperature is not an issue here -- + it suffices to replace energy by TAP free-energy to introduce it, with + essentially no change. Again, the novelty of the article is that generic + models, most notably the Sherrington Kirkpatrick which motivated the first + (pioneering but failed) attempt 42 years ago, are now finally accessible. + \item ``instead the analysis of the static landscape, to which the present + paper is a variation, failed to deliver answers to these questions up to + now.'' + One could say that development of higher fidelity $q$-bits has failed to + deliver the factorization of large integers up to now, but this is hardly a + barrier to publishing substantial progress about their development. +\end{enumerate} + +Again, PRL has devoted a sustained attention to these problems. The referee +goes on to remark that what is truly important ``involves dynamics and +activated processes.'' But one of the reasons that they have not been fully +understood, as the referee claims correctly, is that these take place +between states whose geometrical arrangement were not known (except for the +pure $p$-spin case, or in general, equilibrium), and our work is, we believe, a +true breakthrough in this direction. \closing{Sincerely,} diff --git a/frsb_kac-rice.bib b/frsb_kac-rice.bib index f8d4265..157c14f 100644 --- a/frsb_kac-rice.bib +++ b/frsb_kac-rice.bib @@ -168,7 +168,7 @@ @article{Bray_2007_Statistics, author = {Bray, Alan J. and Dean, David S.}, - title = {Statistics of Critical Points of Gaussian Fields on Large-Dimensional Spaces}, + title = {Statistics of Critical Points of {Gaussian} Fields on Large-Dimensional Spaces}, journal = {Physical Review Letters}, publisher = {American Physical Society (APS)}, year = {2007}, @@ -885,3 +885,45 @@ complexity}, } +@article{Fyodorov_2012_Critical, + author = {Fyodorov, Yan V. and Nadal, Celine}, + title = {Critical Behavior of the Number of Minima of a Random Landscape at the Glass Transition Point and the {Tracy}-{Widom} Distribution}, + journal = {Physical Review Letters}, + publisher = {American Physical Society (APS)}, + year = {2012}, + month = {10}, + number = {16}, + volume = {109}, + pages = {167203}, + url = {https://doi.org/10.1103%2Fphysrevlett.109.167203}, + doi = {10.1103/physrevlett.109.167203} +} + +@article{Fyodorov_2004_Complexity, + author = {Fyodorov, Yan V.}, + title = {Complexity of Random Energy Landscapes, Glass Transition, and Absolute Value of the Spectral Determinant of Random Matrices}, + journal = {Physical Review Letters}, + publisher = {American Physical Society (APS)}, + year = {2004}, + month = {6}, + number = {24}, + volume = {92}, + pages = {240601}, + url = {https://doi.org/10.1103%2Fphysrevlett.92.240601}, + doi = {10.1103/physrevlett.92.240601} +} + +@article{Wainrib_2013_Topological, + author = {Wainrib, Gilles and Touboul, Jonathan}, + title = {Topological and Dynamical Complexity of Random Neural Networks}, + journal = {Physical Review Letters}, + publisher = {American Physical Society (APS)}, + year = {2013}, + month = {3}, + number = {11}, + volume = {110}, + pages = {118101}, + url = {https://doi.org/10.1103%2Fphysrevlett.110.118101}, + doi = {10.1103/physrevlett.110.118101} +} + -- cgit v1.2.3-70-g09d2 From e42d569001f3f20dc7b3c49cd159b710cc654e68 Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Fri, 20 Jan 2023 19:10:31 +0100 Subject: New response. --- response.txt | 102 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 102 insertions(+) create mode 100644 response.txt diff --git a/response.txt b/response.txt new file mode 100644 index 0000000..e9337dd --- /dev/null +++ b/response.txt @@ -0,0 +1,102 @@ + +Because neither referee addressed the scientific content of our paper, +nor substantively addressed its presentation, we have not submitted a +revised manuscript. Below, we respond to the referees' comments. + +Report of Referee A -- LY17256/Kent-Dobias +> The authors consider spin glass models with mixed p-spin interactions +> on the N-Sphere and calculate the number of stationary points, the +> logarithm of which yields the complexity. The disorder average of this +> logarithm is computed with the replica trick, and for different model +> variants different replica symmetry breaking (RSB) solutions are +> obtained. A new feature of the solutions, in contrast to previous +> replica symmetric calculations, is that RSB must occur in parts of the +> energy-stability phase diagram. +> +> The paper is clearly written although the content is rather technical +> and probably only accessible to experts in mean field spin glass +> models and the different RSB schemes developed in this field. In +> connection with the well-studied p=3 spin glass model it is briefly +> mentioned that the complexity and its transitions as a function of +> energy and/or stability is relevant for the equilibrium and the +> dynamical behavior of this model – but such a connection has not been +> made here. +> +> Therefore, I feel that the results presented here are only interesting +> for group of experts and I do not assess the finding that the +> complexity of mixed p-spin glass models shows RSB as a major +> breakthrough in the field. Therefore, the manuscript is not suitable +> for publication in Phys. Rev. Lett., and the publication of the +> accompanying longer paper, submitted to PRE, is sufficient to +> disseminate the results summarized in this manuscript. + +Referee A says that our paper is clearly written but technical, and +that its topic of "the different RSB schemes" are not suitable for a +broad audience. This is surprising to the authors, since a quick +search on Google Scholar reveals several recent PRLs with heavy use of +RSB schemes. + +Referee A correctly points out that one new feature of the solutions +outlined in our manuscript is that RSB must occur in parts of the +phase diagram for these models. However, they neglect another feature: +that they are the solutions of the *quenched* complexity, which has +not been correctly calculated until now. We agree with the referee +that "the complexity of the mixed p-spin glass models" is not a major +breakthrough in and of itself, but believe that the technique for +computing the quenched complexity is a major breakthrough. Just +because this new technique is demonstrated on the simplest toy models +should not discount its importance and potential. + +Referee A states that a connection between the complexity and the +equilibrium and dynamical behavior is not made in our paper. Until +recently, this connection was taken for granted, and the demonstration +that the standard correspondence does not hold in the mixed p-spin +spherical models was exciting enough news to be published in PRX 10, +031045 (2020). It is true that our work doesn't solve the problem that +paper opened, but it does deepen it by showing definitively that the +use of RSB and the quenched complexity are not sufficient to +reestablish a landscape–dynamics connection. We disagree with the +referee's implicit assertion that only clean resolutions, and not the +compelling deepening of problems, are worthy of a broad audience. + +Report of Referee B -- LY17256/Kent-Dobias +> The paper presents a computation of the complexity in spherical +> spin-glass models. Neither the techniques nor the results are +> sufficiently new and relevant to justify publication on PRL. This is +> not surprising given that the topic has been studied extensively in +> the last thirty years and more, the only novelty with respect to +> previous work is that the results are obtained at zero temperature but +> this is definitively not enough. Essential open problems in the field +> involves dynamics and activated processes and some results have +> appeared recently, instead the analysis of the static landscape, to +> which the present paper is a variation, failed to deliver answers to +> these questions up to now. +> +> I recommend that the paper is not published in PRL while the companion +> paper is worth publishing on PRE. + +We disagree with the statement of Referee B that "the only novelty +with respect to previous work is that the results are obtained at zero +temperature". For a system where the quenched and annealed +complexities differ, there has not been a correct calculation of the +quenched complexity at finite temperature. (and, besides our work, +only once or twice at zero temperature, e.g., PRX 9, 011003 (2019).) +Rejecting a paper based on a severe misconception of its contents or +of the state of the field is not appropriate. + +We agree with Referee B's assessment of "[e]ssential open problems in +the field," and agree that our work does not deliver answers. However, +delivering answers for essential open problems is not the acceptance +criterion of PRL. These are + + - Open a new research area, or a new avenue within an established area. + - Solve, or make essential steps towards solving, a critical problem. + - Introduce techniques or methods with significant impact. + - Be of unusual intrinsic interest to PRL's broad audience. + +We believe our manuscript makes essential steps toward solving the +critical problem of connecting analysis of the static landscape to +dynamics. We believe that its essential step is through the +introduction of a new technique, calculation of the quenched +complexity, which we believe will have significant impact as it is +applied to more complicated models. -- cgit v1.2.3-70-g09d2 From 84dff2e9f7ed4da29f02ab90105bb85f9a9f6fce Mon Sep 17 00:00:00 2001 From: "kurchan.jorge" Date: Sun, 22 Jan 2023 15:21:21 +0000 Subject: Update on Overleaf. --- frsb_kac-rice.bib | 9 ++ frsb_kac-rice.tex | 79 ++++++++++------ frsb_kac-rice_letter.tex | 124 +++++++++++++++---------- response.txt | 236 ++++++++++++++++++++++++++++++++++------------- 4 files changed, 310 insertions(+), 138 deletions(-) diff --git a/frsb_kac-rice.bib b/frsb_kac-rice.bib index 157c14f..37ffb2d 100644 --- a/frsb_kac-rice.bib +++ b/frsb_kac-rice.bib @@ -927,3 +927,12 @@ complexity}, doi = {10.1103/physrevlett.110.118101} } +@article{ros2021dynamical, + title={Dynamical instantons and activated processes in mean-field glass models}, + author={Ros, Valentina and Biroli, Giulio and Cammarota, Chiara}, + journal={SciPost Physics}, + volume={10}, + number={1}, + pages={002}, + year={2021} +} diff --git a/frsb_kac-rice.tex b/frsb_kac-rice.tex index 875d5a9..3d59148 100644 --- a/frsb_kac-rice.tex +++ b/frsb_kac-rice.tex @@ -36,8 +36,8 @@ \begin{abstract} We derive the general solution for counting the stationary points of mean-field complex landscapes. It incorporates Parisi's solution - for the ground state, as it should. Using this solution, we count the - stationary points of two models: one with multi-step replica symmetry + for the ground state, as it should. Using this solution, we count + {\color{red} and discuss the distribution of the stability indices } of stationary points of two {\color{red} representative} models: one with multi-step replica symmetry breaking, and one with full replica symmetry breaking. \end{abstract} @@ -52,33 +52,56 @@ Sherrington--Kirkpatrick model, in a paper remarkable for being one of the first applications of a replica symmetry breaking (RSB) scheme. As was clear when the actual ground-state of the model was computed by Parisi with a different scheme, the Bray--Moore result was not exact, and the problem has -been open ever since \cite{Parisi_1979_Infinite}. To date, the program of -computing the number of stationary points---minima, saddle points, and -maxima---of mean-field complex landscapes has been only carried out for a small subset of -models, including most notably the (pure) $p$-spin model ($p>2$) -\cite{Rieger_1992_The, Crisanti_1995_Thouless-Anderson-Palmer, Cavagna_1997_An, Cavagna_1998_Stationary} and for similar -energy functions inspired by molecular biology, evolution, and machine learning -\cite{Maillard_2020_Landscape, Ros_2019_Complex, Altieri_2021_Properties}. In -a parallel development, it has evolved into an active field of probability +been open ever since \cite{Parisi_1979_Infinite}. +Many other interesting aspects of the problem have been treated, and the subject has +evolved into an active field of probability theory \cite{Auffinger_2012_Random, Auffinger_2013_Complexity, -BenArous_2019_Geometry}. +BenArous_2019_Geometry} and has been applied to +energy functions inspired by molecular biology, evolution, and machine learning +\cite{Maillard_2020_Landscape, Ros_2019_Complex, Altieri_2021_Properties}. + +To date, however, the program of +computing the statistics of stationary points---minima, saddle points, and +maxima---of mean-field complex landscapes has been only carried out in an exact form for a small subset of +models, including most notably the (pure) $p$-spin model ($p>2$) +\cite{Rieger_1992_The, Crisanti_1995_Thouless-Anderson-Palmer, Cavagna_1997_An, Cavagna_1998_Stationary}. + +{\color{red} Not having a full, exact (`quenched') solution of the generic problem is not +primarily a matter of {\em accuracy} of the actual numbers involved. +In the same spirit (but in a geometrically distinct way) as in the case of the equilibrium properties of glasses, +much more basic structural questions are omitted in the approximate `annealed' solution. What is lost is the nature, at any given +energy (or free energy) level, of the stationary points in a generic energy function: at low energies are they basically all minima, with an exponentially small number of saddles, or +-- as we show here -- do they consist of a mixture of saddles whose index -- the number of unstable directions -- is a smoothly distributed number? Also, in an energy +level where almost all saddle points are unstable, are there still a few stable ones? + +These questions need to be answered for the understanding of the relevance of more complex objects such as +barrier crossing (which barriers?) \cite{Ros_2021_Dynamical}, or the fate of long-time dynamics +(which are the target states?). + + + + + In this paper we present what we argue is the general replica ansatz for the number of stationary points of generic mean-field models, which we expect to include the Sherrington--Kirkpatrick model. It reproduces the Parisi result in -the limit of small temperature for the lowest states, as it should. +the limit of small temperature for the lowest states, as it should. For this kind of situation +it clarifies the structure of lowest saddles: there is a continuous distribution of them, +with stability characterized by a continuous distribution of indices. +} -To understand the importance of this computation, consider the following -situation. When one solves the problem of spheres in large dimensions, one -finds that there is a transition at a given temperature to a one-step replica symmetry +From the point of view of glassy systems, consider the following +situation. Generically, we now know \cite{Charbonneau_2014_Fractal} + that there is a transition at a given temperature to a one-step replica symmetry breaking (1RSB) phase at a Kauzmann temperature, and, at a lower temperature, another transition to a full RSB (FRSB) phase (see \cite{Gross_1985_Mean-field, -Gardner_1985_Spin}, the so-called `Gardner' phase -\cite{Charbonneau_2014_Fractal}). Now, this transition involves the lowest -equilibrium states. Because they are obviously unreachable at any reasonable -timescale, a common question is: what is the signature of the Gardner -transition line for higher than equilibrium energy-densities? This is a -question whose answers are significant to interpreting the results of myriad +Gardner_1985_Spin}, the so-called `Gardner' phase. +Now, this transition involves the lowest +equilibrium states which are obviously unreachable at any reasonable +timescale. We should rather ask the question of what is the signature of the Gardner +transition line for states with higher energy-densities: the answer +will then be significant to interpreting the results of myriad experiments and simulations \cite{Xiao_2022_Probing, Hicks_2018_Gardner, Liao_2019_Hierarchical, Dennis_2020_Jamming, Charbonneau_2015_Numerical, Li_2021_Determining, Seguin_2016_Experimental, Geirhos_2018_Johari-Goldstein, @@ -89,11 +112,9 @@ transition are high energy (or low density) states reachable dynamically?' One approach to answering such questions makes use of `state following,' which tracks metastable thermodynamic configurations to their zero temperature limit \cite{Rainone_2015_Following, Biroli_2016_Breakdown, -Rainone_2016_Following, Biroli_2018_Liu-Nagel, Urbani_2017_Shear}. In the -present paper we give a purely geometric appoarch: we consider the local energy -minima at a given energy and study their number and other properties: the -solution involves a replica-symmetry breaking scheme that is well-defined, and -corresponds directly to the topological characteristics of those minima. +Rainone_2016_Following, Biroli_2018_Liu-Nagel, Urbani_2017_Shear}. +The present paper we provide a purely geometric approach, since we shall address the local energy +minima at any given energy and study their number and stability properties. Perhaps the most interesting application of this computation is in the context @@ -126,6 +147,12 @@ $3+16$ model with a 2RSB ground state and a 1RSB complexity, and a $2+4$ with a FRSB ground state and a FRSB complexity. Finally \S\ref{sec:interpretation} provides some interpretation of our results. +{\color{red} A final remark is in order here: for simplicity we have concentrated on the energy, rather +than the {\em free-energy} landscape. Clearly, in the presence of thermal fluctuations, the latter is +more appropriate. However, from the technical point of view, this makes no fundamental difference, it suffices +to apply the same computation to the Thouless-Andreson-Palmer \cite{Crisanti_1995_Thouless-Anderson-Palmer} (TAP) free energy, instead of the energy. No new +complications arise.} + \section{The model} \label{sec:model} diff --git a/frsb_kac-rice_letter.tex b/frsb_kac-rice_letter.tex index 4fb22e7..27025e7 100644 --- a/frsb_kac-rice_letter.tex +++ b/frsb_kac-rice_letter.tex @@ -29,11 +29,12 @@ Complexity is a measure of the number of stationary points in complex landscapes. We derive a general solution for the complexity of mean-field complex landscapes. It incorporates Parisi's solution for the ground state, - as it should. Using this solution, we count the stationary points of two - models: one with multi-step replica symmetry breaking, and one with full - replica symmetry breaking. These examples demonstrate the consistency of the - solution and reveal that the signature of replica symmetry breaking at high - energy densities is found in high-index saddles, not minima. + as it should. Using this solution, we count the stationary points of two representative + models. Including + replica symmetry breaking uncovers s the full organization of saddles in terms of their energies and stabilities encountered in generic models. + %These examples demonstrate the consistency of the + %solution and reveal that the signature of replica symmetry breaking at high + %energy densities is found in high-index saddles, not minima. \end{abstract} \maketitle @@ -45,48 +46,80 @@ size of the system \cite{Maillard_2020_Landscape, Ros_2019_Complex, Altieri_2021_Properties}. Though they are often called `rough landscapes' to evoke the intuitive image of many minima in something like a mountain range, the metaphor to topographical landscapes is strained by the reality that these -complex landscapes also exist in very high dimensions: think of the dimensions -of phase space for $N$ particles, or the number of parameters in a neural -network. - -The \emph{complexity} of a function is the average of the logarithm of the -number of its minima, maxima, and saddle points (collectively stationary -points), under conditions fixing the value of the energy or the index of the -stationary point -\cite{Bray_1980_Metastable}. -Since in complex landscapes this -number grows exponentially with system size, their complexity is an extensive -quantity. Understanding the complexity offers an understanding about the -geometry and topology of the landscape, which can provide insight into -dynamical behavior. - -When complex systems are fully connected, i.e., each degree of freedom -interacts directly with every other, they are often described by a hierarchical -structure of the type first proposed by Parisi, the \emph{replica symmetry -breaking} (RSB) \cite{Parisi_1979_Infinite}. This family of structures is rich, spanning uniform -\emph{replica symmetry} (RS), an integer $k$ levels of hierarchical nested -structure ($k$RSB), a full continuum of nested structure (full RSB or FRSB), -and arbitrary combinations thereof. Though these rich structures are understood -in the equilibrium properties of fully connected models, the complexity has -only been computed in RS cases. +complex landscapes exist in very high dimensions. +Many interesting versions of the problem have been treated, and the subject has +evolved into an active field of probability +theory \cite{Auffinger_2012_Random, Auffinger_2013_Complexity, +BenArous_2019_Geometry} and has been applied to +energy functions inspired by molecular biology, evolution, and machine learning +\cite{Maillard_2020_Landscape, Ros_2019_Complex, Altieri_2021_Properties}. + + + + + +The computation of the number of metastable states in this setting +was pioneered forty years ago by Bray and Moore +\cite{Bray_1980_Metastable}, who proposed the first calculation for the +Sherrington--Kirkpatrick model, in one of the +first applications of any replica symmetry breaking (RSB) scheme. As was clear +from the later results by Parisi \cite{Parisi_1979_Infinite}, their result was not exact, and the problem has +been open ever since. To date the program of +computing the statistics of stationary points---minima, saddle points, and +maxima---of mean-field complex landscapes has been only carried out in an exact form for a relatively small subset of +models, including most notably the (pure) $p$-spin model ($p>2$) +\cite{Rieger_1992_The, Crisanti_1995_Thouless-Anderson-Palmer, Cavagna_1997_An, Cavagna_1998_Stationary}. + +{\color{red} +Having a full, exact (`quenched') solution of the generic problem is not +primarily a matter of {\em accuracy}. +Very basic structural questions are omitted in the approximate `annealed' solution. What is lost is the nature, at any given +energy (or free energy) level, of the stationary points in a generic energy function: at low energies are they basically all minima, with an exponentially small number of saddles, or +-- as we show here -- do they consist of a mixture of saddles whose index -- the number of unstable directions -- is a smoothly distributed number? +These questions need to be answered for the understanding of the relevance of more complex objects such as +barrier crossing (which barriers?) \cite{Ros_2019_Complexity, Ros_2021_Dynamical}, or the fate of long-time dynamics +(which end in what kind of target states?). + + + + + + +In this paper we present what we argue is the general replica ansatz for the +number of stationary points of generic mean-field models, which we expect to +include the Sherrington--Kirkpatrick model. This allows us +to clarify the rich structure of all the saddles, and in particular the lowest ones. The interpretation of a Parisi ansatz itself, in this context must be recast in a way that makes sense for the order parameters involved. + +} + +{\color{blue} + For simplicity we have concentrated here on the energy, rather +than {\em free-energy} landscape, although the latter is sometimes +more appropriate. From the technical point of view, this makes no fundamental difference, it suffices +to apply the same computation to the Thouless-Andreson-Palmer \cite{Crisanti_1995_Thouless-Anderson-Palmer} (TAP) free energy, instead of the energy. We do not expect new features or technical +complications arise. + +} + + In this paper and its longer companion, we share the first results for the complexity with nontrivial hierarchy \cite{Kent-Dobias_2022_How}. Using a general form for the solution detailed in a companion article, we describe the structure of landscapes with a 1RSB complexity and a full RSB complexity -\footnote{The Thouless--Anderson--Palmer (TAP) complexity is the complexity of - a kind of mean-field free energy. Because of some deep thermodynamic - relationships between the TAP complexity and the equilibrium free energy, the -TAP complexity can be computed with extensions of the equilibrium method. As a -result, the TAP complexity has been previously computed for nontrivial -hierarchical structure.}. - -We study the mixed $p$-spin spherical models, with Hamiltonian +%\footnote{The Thouless--Anderson--Palmer (TAP) complexity is the complexity of + % a kind of mean-field free energy. Because of some deep thermodynamic + % relationships between the TAP complexity and the equilibrium free energy, the +%TAP complexity can be computed with extensions of the equilibrium method. As a +%result, the TAP complexity has been previously computed for nontrivial +%hierarchical structure.}. + +For definiteness, we consider the standard example of the mixed $p$-spin spherical models, with Hamiltonian \begin{equation} \label{eq:hamiltonian} H(\mathbf s)=-\sum_p\frac1{p!}\sum_{i_1\cdots i_p}^NJ^{(p)}_{i_1\cdots i_p}s_{i_1}\cdots s_{i_p} \end{equation} -is defined for vectors $\mathbf s\in\mathbb R^N$ confined to the $N-1$ sphere -$S^{N-1}=\{\mathbf s\mid\|\mathbf s\|^2=N\}$. The coupling coefficients $J$ are taken at random, with + $\mathbf s\in\mathbb R^N$ confined to the $N-1$ sphere +$\{|\mathbf s\|^2=N\}$. The coupling coefficients $J$ are taken at random, with zero mean and variance $\overline{(J^{(p)})^2}=a_pp!/2N^{p-1}$ chosen so that the energy is typically extensive. The overbar will always denote an average over the coefficients $J$. The factors $a_p$ in the variances are freely chosen @@ -95,19 +128,16 @@ models have $a_p=1$ for some $p$ and all others zero. The complexity of the $p$-spin models has been extensively studied by physicists and mathematicians. Among physicists, the bulk of work has been on - the so-called `TAP' complexity, -which counts minima in the mean-field Thouless--Anderson--Palmer () free energy \cite{Rieger_1992_The, + the so-called `TAP' complexity of pure models \cite{Rieger_1992_The, Crisanti_1995_Thouless-Anderson-Palmer, Cavagna_1997_An, Cavagna_1997_Structure, Cavagna_1998_Stationary, Cavagna_2005_Cavity, -Giardina_2005_Supersymmetry}. The landscape complexity has been proven for pure -and mixed models without RSB \cite{Auffinger_2012_Random, -Auffinger_2013_Complexity, BenArous_2019_Geometry}. The mixed models been -treated without RSB \cite{Folena_2020_Rethinking}. And the methods of +Giardina_2005_Supersymmetry}, and more recently mixed models \cite{Folena_2020_Rethinking} without RSB \cite{Auffinger_2012_Random, +Auffinger_2013_Complexity, BenArous_2019_Geometry}. And the methods of complexity have been used to study many geometric properties of the pure models, from the relative position of stationary points to one another to shape and prevalence of instantons \cite{Ros_2019_Complexity, Ros_2021_Dynamical}. -The variance of the couplings implies that the covariance of the energy with +{\color{green} {\bf eliminate?} The variance of the couplings implies that the covariance of the energy with itself depends on only the dot product (or overlap) between two configurations. In particular, one finds \begin{equation} \label{eq:covariance} @@ -120,7 +150,7 @@ where $f$ is defined by the series One needn't start with a Hamiltonian like \eqref{eq:hamiltonian}, defined as a series: instead, the covariance rule \eqref{eq:covariance} can be specified for arbitrary, non-polynomial $f$, as in -the `toy model' of M\'ezard and Parisi \cite{Mezard_1992_Manifolds}. In fact, defined this way the mixed spherical model encompasses all isotropic Gaussian fields on the sphere. +the `toy model' of M\'ezard and Parisi \cite{Mezard_1992_Manifolds}. In fact, defined this way the mixed spherical model encompasses all isotropic Gaussian fields on the sphere.} The family of spherical models thus defined is quite rich, and by varying the covariance $f$ nearly any hierarchical structure can be found in diff --git a/response.txt b/response.txt index e9337dd..058a6f8 100644 --- a/response.txt +++ b/response.txt @@ -1,40 +1,119 @@ +\documentclass[a4paper]{letter} -Because neither referee addressed the scientific content of our paper, -nor substantively addressed its presentation, we have not submitted a -revised manuscript. Below, we respond to the referees' comments. - -Report of Referee A -- LY17256/Kent-Dobias -> The authors consider spin glass models with mixed p-spin interactions -> on the N-Sphere and calculate the number of stationary points, the -> logarithm of which yields the complexity. The disorder average of this -> logarithm is computed with the replica trick, and for different model -> variants different replica symmetry breaking (RSB) solutions are -> obtained. A new feature of the solutions, in contrast to previous -> replica symmetric calculations, is that RSB must occur in parts of the -> energy-stability phase diagram. -> -> The paper is clearly written although the content is rather technical -> and probably only accessible to experts in mean field spin glass -> models and the different RSB schemes developed in this field. In -> connection with the well-studied p=3 spin glass model it is briefly -> mentioned that the complexity and its transitions as a function of -> energy and/or stability is relevant for the equilibrium and the -> dynamical behavior of this model – but such a connection has not been -> made here. -> -> Therefore, I feel that the results presented here are only interesting -> for group of experts and I do not assess the finding that the -> complexity of mixed p-spin glass models shows RSB as a major -> breakthrough in the field. Therefore, the manuscript is not suitable -> for publication in Phys. Rev. Lett., and the publication of the -> accompanying longer paper, submitted to PRE, is sufficient to -> disseminate the results summarized in this manuscript. - -Referee A says that our paper is clearly written but technical, and +\usepackage[utf8]{inputenc} % why not type "Bézout" with unicode? +\usepackage[T1]{fontenc} % vector fonts plz +\usepackage{newtxtext,newtxmath} % Times for PR +\usepackage[ + colorlinks=true, + urlcolor=purple, + linkcolor=black, + citecolor=black, + filecolor=black, +]{hyperref} % ref and cite links with pretty colors +\usepackage{xcolor} +\usepackage[style=phys]{biblatex} + +\renewcommand{\thefootnote}{\fnsymbol{footnote}} + +\addbibresource{frsb_kac-rice.bib} + +\signature{ + \vspace{-6\medskipamount} + \smallskip + Jaron Kent-Dobias \& Jorge Kurchan +} + +\address{ + Laboratoire de Physique\\ + Ecole Normale Sup\'erieure\\ + 24 rue Lhomond\\ + 75005 Paris +} + +\begin{document} +\begin{letter}{ + Agnese I.~Curatolo, Ph.D.\\ + Physical Review Letters\\ + 1 Research Road\\ + Ridge, NY 11961 +} + +\opening{Dear Dr.~Curatolo,} + + +Because neither referee addressed (or criticized) the scientific content of our paper, +nor substantively addressed its presentation, we have only modified +the manuscripts in order to highlight the importance of having a full solution. + + +We have thus added the paragraph: + +{\color{red} +Having a full, exact (`quenched') solution of the generic problem is not +primarily a matter of {\em accuracy}. +Very basic structural questions are omitted in the approximate `annealed' solution. What is lost is the nature, at any given +energy (or free energy) level, of the stationary points in a generic energy function: at low energies are they basically all minima, with an exponentially small number of saddles, or +-- as we show here -- do they consist of a mixture of saddles whose index -- the number of unstable directions -- is a smoothly distributed number? +These questions need to be answered for the understanding of the relevance of more complex objects such as +barrier crossing (which barriers?) \cite{Ros_2019_Complexity, Ros_2021_Dynamical}, or the fate of long-time dynamics +(which end in what kind of target states?). + +} + +Both referees find that our paper is clearly written but technical, and that its topic of "the different RSB schemes" are not suitable for a broad audience. This is surprising to the authors, since a quick search on Google Scholar reveals several recent PRLs with heavy use of -RSB schemes. +RSB schemes. + +We would also like to submit to the referees that it is somewhat +incongruous that the solution to a problem that had remained open for 42 years -- during which it was always present in articles in PRL and PRX -- is rejected +because it demands of the readers a slightly longer attention span. + + + + +\begin{enumerate} + \item PRL has been publishing articles on precisely this problem in the + last 30 years.\footfullcite{Fyodorov_2004_Complexity, Bray_2007_Statistics, Fyodorov_2012_Critical, Wainrib_2013_Topological} + \item These works were often limited by the fact that general landscapes (for + which an annealed solution is not exact) were inaccessible. It is perhaps + true that the final solution of an open problem may often be more technical + than the previous ones. +\end{enumerate} + + + +Below, we respond to the referees' comments. + +{\it Report of Referee A -- LY17256/Kent-Dobias + The authors consider spin glass models with mixed p-spin interactions + on the N-Sphere and calculate the number of stationary points, the + logarithm of which yields the complexity. The disorder average of this + logarithm is computed with the replica trick, and for different model + variants different replica symmetry breaking (RSB) solutions are + obtained. A new feature of the solutions, in contrast to previous + replica symmetric calculations, is that RSB must occur in parts of the + energy-stability phase diagram. + + The paper is clearly written although the content is rather technical + and probably only accessible to experts in mean field spin glass + models and the different RSB schemes developed in this field. In + connection with the well-studied p=3 spin glass model it is briefly + mentioned that the complexity and its transitions as a function of + energy and/or stability is relevant for the equilibrium and the + dynamical behavior of this model – but such a connection has not been + made here. + + Therefore, I feel that the results presented here are only interesting + for group of experts and I do not assess the finding that the + complexity of mixed p-spin glass models shows RSB as a major + breakthrough in the field. Therefore, the manuscript is not suitable + for publication in Phys. Rev. Lett., and the publication of the + accompanying longer paper, submitted to PRE, is sufficient to + disseminate the results summarized in this manuscript.} + + Referee A correctly points out that one new feature of the solutions outlined in our manuscript is that RSB must occur in parts of the @@ -42,10 +121,12 @@ phase diagram for these models. However, they neglect another feature: that they are the solutions of the *quenched* complexity, which has not been correctly calculated until now. We agree with the referee that "the complexity of the mixed p-spin glass models" is not a major -breakthrough in and of itself, but believe that the technique for -computing the quenched complexity is a major breakthrough. Just -because this new technique is demonstrated on the simplest toy models -should not discount its importance and potential. +breakthrough in and of itself, we just +chose to demonstrate the problem in simplest toy model. But believe that the technique for +computing the quenched complexity is a major breakthrough +{\bf because it brings in the features of organization of saddles of all +kinds that are invisible in the annealed scheme}. + Referee A states that a connection between the complexity and the equilibrium and dynamical behavior is not made in our paper. Until @@ -55,38 +136,55 @@ spherical models was exciting enough news to be published in PRX 10, 031045 (2020). It is true that our work doesn't solve the problem that paper opened, but it does deepen it by showing definitively that the use of RSB and the quenched complexity are not sufficient to -reestablish a landscape–dynamics connection. We disagree with the -referee's implicit assertion that only clean resolutions, and not the -compelling deepening of problems, are worthy of a broad audience. +reestablish a landscape–dynamics connection. +{\bf One can hardly expect that the structure of saddles at a given energy may be connected +with dynamics (for example in Sherrington Kirkpatrick) if it is unknown}. +%We disagree with the +%referee's implicit assertion that only clean resolutions, and not the +%compelling deepening of problems, are worthy of a broad audience. Report of Referee B -- LY17256/Kent-Dobias -> The paper presents a computation of the complexity in spherical -> spin-glass models. Neither the techniques nor the results are -> sufficiently new and relevant to justify publication on PRL. This is -> not surprising given that the topic has been studied extensively in -> the last thirty years and more, the only novelty with respect to -> previous work is that the results are obtained at zero temperature but -> this is definitively not enough. Essential open problems in the field -> involves dynamics and activated processes and some results have -> appeared recently, instead the analysis of the static landscape, to -> which the present paper is a variation, failed to deliver answers to -> these questions up to now. -> -> I recommend that the paper is not published in PRL while the companion -> paper is worth publishing on PRE. - -We disagree with the statement of Referee B that "the only novelty +{\it The paper presents a computation of the complexity in spherical + spin-glass models. Neither the techniques nor the results are + sufficiently new and relevant to justify publication on PRL. This is + not surprising given that the topic has been studied extensively in + the last thirty years and more, the only novelty with respect to + previous work is that the results are obtained at zero temperature but + this is definitively not enough. Essential open problems in the field + involves dynamics and activated processes and some results have + appeared recently, instead the analysis of the static landscape, to + which the present paper is a variation, failed to deliver answers to + these questions up to now. + + } + +Concerning the statement of Referee B that "the only novelty with respect to previous work is that the results are obtained at zero -temperature". For a system where the quenched and annealed -complexities differ, there has not been a correct calculation of the -quenched complexity at finite temperature. (and, besides our work, -only once or twice at zero temperature, e.g., PRX 9, 011003 (2019).) -Rejecting a paper based on a severe misconception of its contents or -of the state of the field is not appropriate. - -We agree with Referee B's assessment of "[e]ssential open problems in +temperature", we do not know what to make of the referee's statement. +The novelty of the paper is most definitely +not the fact of treating a zero temperature case. +We have added the following phrase, that should clarify the situation: + +{\color{blue} + For simplicity we have concentrated here on the energy, rather +than {\em free-energy} landscape, although the latter is sometimes +more appropriate. From the technical point of view, this makes no fundamental difference, it suffices +to apply the same computation to the Thouless-Andreson-Palmer \cite{Crisanti_1995_Thouless-Anderson-Palmer} (TAP) free energy, instead of the energy. We do not expect new features or technical +complications arise. + +} + + +%For a system where the quenched and annealed +%complexities differ, there has not been a correct calculation of the +%quenched complexity at finite temperature. (and, besides our work, +%only once or twice at zero temperature, e.g., PRX 9, 011003 (2019).) +%Rejecting a paper based on a severe misconception of its contents or +%of the state of the field is not appropriate. + +We agree with Referee B's assessment of "essential open problems in the field," and agree that our work does not deliver answers. However, -delivering answers for essential open problems is not the acceptance +delivering answers for all essential open problems is not the acceptance criterion of PRL. These are - Open a new research area, or a new avenue within an established area. @@ -100,3 +198,11 @@ dynamics. We believe that its essential step is through the introduction of a new technique, calculation of the quenched complexity, which we believe will have significant impact as it is applied to more complicated models. + +\closing{Sincerely,} + +\vspace{1em} + +\end{letter} + +\end{document} -- cgit v1.2.3-70-g09d2 From a8850fdb22cec36eb0f4981c4ff66163b34e990d Mon Sep 17 00:00:00 2001 From: "kurchan.jorge" Date: Mon, 23 Jan 2023 10:36:35 +0000 Subject: Update on Overleaf. --- frsb_kac-rice_letter.tex | 14 ++++++++------ response.txt | 9 +++++---- 2 files changed, 13 insertions(+), 10 deletions(-) diff --git a/frsb_kac-rice_letter.tex b/frsb_kac-rice_letter.tex index 27025e7..f41e8ca 100644 --- a/frsb_kac-rice_letter.tex +++ b/frsb_kac-rice_letter.tex @@ -26,12 +26,14 @@ \affiliation{Laboratoire de Physique de l'Ecole Normale Supérieure, Paris, France} \begin{abstract} - Complexity is a measure of the number of stationary points in complex - landscapes. We derive a general solution for the complexity of mean-field - complex landscapes. It incorporates Parisi's solution for the ground state, - as it should. Using this solution, we count the stationary points of two representative - models. Including - replica symmetry breaking uncovers s the full organization of saddles in terms of their energies and stabilities encountered in generic models. + {\color{red} Complex landscapes are defined as those having a proliferation of saddle points. + The question of their number and organization has been the object of long-standing attention, in particular centered around Gaussian mean-field potentials, + which includes glass and spin glass models. +The annealed approximation is by now well understood, but is exact for a restricted subset of these problems. Here we derive the exact quenched +solution for the general case, which incorporates Parisi's solution for the ground state, + as it should. More importantly, including + replica symmetry breaking uncovers the full organization of saddles in terms of their energies and stabilities, a structure that is lost in the annealed approximation. This structure should be a guide for the identification + of relevant activated processes in relaxational or driven dynamics.} %These examples demonstrate the consistency of the %solution and reveal that the signature of replica symmetry breaking at high %energy densities is found in high-index saddles, not minima. diff --git a/response.txt b/response.txt index 058a6f8..424aec8 100644 --- a/response.txt +++ b/response.txt @@ -40,10 +40,11 @@ \opening{Dear Dr.~Curatolo,} - -Because neither referee addressed (or criticized) the scientific content of our paper, -nor substantively addressed its presentation, we have only modified -the manuscripts in order to highlight the importance of having a full solution. +Enclosed please find a revised manuscript. +Neither referee addressed (or criticized) the scientific content of our paper, +nor substantively addressed its presentation. We have followed their comments +in the direction of highlighting the importance of having a full solution. In particular +we have emphasized that going to the full replica treatment We have thus added the paragraph: -- cgit v1.2.3-70-g09d2 From 3df1436bf703bc9ab05f72f16cbd54a7a0dfb693 Mon Sep 17 00:00:00 2001 From: "kurchan.jorge" Date: Mon, 23 Jan 2023 10:37:09 +0000 Subject: Update on Overleaf. --- response.txt | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/response.txt b/response.txt index 424aec8..1a830e8 100644 --- a/response.txt +++ b/response.txt @@ -44,7 +44,7 @@ Enclosed please find a revised manuscript. Neither referee addressed (or criticized) the scientific content of our paper, nor substantively addressed its presentation. We have followed their comments in the direction of highlighting the importance of having a full solution. In particular -we have emphasized that going to the full replica treatment +we have emphasized that going to the full replica treatment uncovers a phase-space structure that needs to be taken into account, and that is absent in the annealed treatment. We have thus added the paragraph: -- cgit v1.2.3-70-g09d2 From 148b89a29e563e104f6681716d23a2111bebf4a5 Mon Sep 17 00:00:00 2001 From: "kurchan.jorge" Date: Mon, 23 Jan 2023 10:40:15 +0000 Subject: Update on Overleaf. --- frsb_kac-rice_letter.tex | 2 +- response.txt | 2 +- 2 files changed, 2 insertions(+), 2 deletions(-) diff --git a/frsb_kac-rice_letter.tex b/frsb_kac-rice_letter.tex index f41e8ca..b6e7b8b 100644 --- a/frsb_kac-rice_letter.tex +++ b/frsb_kac-rice_letter.tex @@ -32,7 +32,7 @@ The annealed approximation is by now well understood, but is exact for a restricted subset of these problems. Here we derive the exact quenched solution for the general case, which incorporates Parisi's solution for the ground state, as it should. More importantly, including - replica symmetry breaking uncovers the full organization of saddles in terms of their energies and stabilities, a structure that is lost in the annealed approximation. This structure should be a guide for the identification + replica symmetry breaking uncovers the full distribution of saddles at given energy in terms of their stabilities, a structure that is lost in the annealed approximation. This structure should be a guide for the identification of relevant activated processes in relaxational or driven dynamics.} %These examples demonstrate the consistency of the %solution and reveal that the signature of replica symmetry breaking at high diff --git a/response.txt b/response.txt index 1a830e8..1709e15 100644 --- a/response.txt +++ b/response.txt @@ -41,7 +41,7 @@ \opening{Dear Dr.~Curatolo,} Enclosed please find a revised manuscript. -Neither referee addressed (or criticized) the scientific content of our paper, +Neither referee criticized the scientific content of our paper, nor substantively addressed its presentation. We have followed their comments in the direction of highlighting the importance of having a full solution. In particular we have emphasized that going to the full replica treatment uncovers a phase-space structure that needs to be taken into account, and that is absent in the annealed treatment. -- cgit v1.2.3-70-g09d2 From c4182fa022ceebd61cfbfe258fbc0f3294ea72cb Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Mon, 23 Jan 2023 11:56:07 +0100 Subject: Changed file format of response letter. --- response.tex | 209 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ response.txt | 209 ----------------------------------------------------------- 2 files changed, 209 insertions(+), 209 deletions(-) create mode 100644 response.tex delete mode 100644 response.txt diff --git a/response.tex b/response.tex new file mode 100644 index 0000000..1709e15 --- /dev/null +++ b/response.tex @@ -0,0 +1,209 @@ +\documentclass[a4paper]{letter} + +\usepackage[utf8]{inputenc} % why not type "Bézout" with unicode? +\usepackage[T1]{fontenc} % vector fonts plz +\usepackage{newtxtext,newtxmath} % Times for PR +\usepackage[ + colorlinks=true, + urlcolor=purple, + linkcolor=black, + citecolor=black, + filecolor=black, +]{hyperref} % ref and cite links with pretty colors +\usepackage{xcolor} +\usepackage[style=phys]{biblatex} + +\renewcommand{\thefootnote}{\fnsymbol{footnote}} + +\addbibresource{frsb_kac-rice.bib} + +\signature{ + \vspace{-6\medskipamount} + \smallskip + Jaron Kent-Dobias \& Jorge Kurchan +} + +\address{ + Laboratoire de Physique\\ + Ecole Normale Sup\'erieure\\ + 24 rue Lhomond\\ + 75005 Paris +} + +\begin{document} +\begin{letter}{ + Agnese I.~Curatolo, Ph.D.\\ + Physical Review Letters\\ + 1 Research Road\\ + Ridge, NY 11961 +} + +\opening{Dear Dr.~Curatolo,} + +Enclosed please find a revised manuscript. +Neither referee criticized the scientific content of our paper, +nor substantively addressed its presentation. We have followed their comments +in the direction of highlighting the importance of having a full solution. In particular +we have emphasized that going to the full replica treatment uncovers a phase-space structure that needs to be taken into account, and that is absent in the annealed treatment. + + +We have thus added the paragraph: + +{\color{red} +Having a full, exact (`quenched') solution of the generic problem is not +primarily a matter of {\em accuracy}. +Very basic structural questions are omitted in the approximate `annealed' solution. What is lost is the nature, at any given +energy (or free energy) level, of the stationary points in a generic energy function: at low energies are they basically all minima, with an exponentially small number of saddles, or +-- as we show here -- do they consist of a mixture of saddles whose index -- the number of unstable directions -- is a smoothly distributed number? +These questions need to be answered for the understanding of the relevance of more complex objects such as +barrier crossing (which barriers?) \cite{Ros_2019_Complexity, Ros_2021_Dynamical}, or the fate of long-time dynamics +(which end in what kind of target states?). + +} + +Both referees find that our paper is clearly written but technical, and +that its topic of "the different RSB schemes" are not suitable for a +broad audience. This is surprising to the authors, since a quick +search on Google Scholar reveals several recent PRLs with heavy use of +RSB schemes. + +We would also like to submit to the referees that it is somewhat +incongruous that the solution to a problem that had remained open for 42 years -- during which it was always present in articles in PRL and PRX -- is rejected +because it demands of the readers a slightly longer attention span. + + + + +\begin{enumerate} + \item PRL has been publishing articles on precisely this problem in the + last 30 years.\footfullcite{Fyodorov_2004_Complexity, Bray_2007_Statistics, Fyodorov_2012_Critical, Wainrib_2013_Topological} + \item These works were often limited by the fact that general landscapes (for + which an annealed solution is not exact) were inaccessible. It is perhaps + true that the final solution of an open problem may often be more technical + than the previous ones. +\end{enumerate} + + + +Below, we respond to the referees' comments. + +{\it Report of Referee A -- LY17256/Kent-Dobias + The authors consider spin glass models with mixed p-spin interactions + on the N-Sphere and calculate the number of stationary points, the + logarithm of which yields the complexity. The disorder average of this + logarithm is computed with the replica trick, and for different model + variants different replica symmetry breaking (RSB) solutions are + obtained. A new feature of the solutions, in contrast to previous + replica symmetric calculations, is that RSB must occur in parts of the + energy-stability phase diagram. + + The paper is clearly written although the content is rather technical + and probably only accessible to experts in mean field spin glass + models and the different RSB schemes developed in this field. In + connection with the well-studied p=3 spin glass model it is briefly + mentioned that the complexity and its transitions as a function of + energy and/or stability is relevant for the equilibrium and the + dynamical behavior of this model – but such a connection has not been + made here. + + Therefore, I feel that the results presented here are only interesting + for group of experts and I do not assess the finding that the + complexity of mixed p-spin glass models shows RSB as a major + breakthrough in the field. Therefore, the manuscript is not suitable + for publication in Phys. Rev. Lett., and the publication of the + accompanying longer paper, submitted to PRE, is sufficient to + disseminate the results summarized in this manuscript.} + + + +Referee A correctly points out that one new feature of the solutions +outlined in our manuscript is that RSB must occur in parts of the +phase diagram for these models. However, they neglect another feature: +that they are the solutions of the *quenched* complexity, which has +not been correctly calculated until now. We agree with the referee +that "the complexity of the mixed p-spin glass models" is not a major +breakthrough in and of itself, we just +chose to demonstrate the problem in simplest toy model. But believe that the technique for +computing the quenched complexity is a major breakthrough +{\bf because it brings in the features of organization of saddles of all +kinds that are invisible in the annealed scheme}. + + +Referee A states that a connection between the complexity and the +equilibrium and dynamical behavior is not made in our paper. Until +recently, this connection was taken for granted, and the demonstration +that the standard correspondence does not hold in the mixed p-spin +spherical models was exciting enough news to be published in PRX 10, +031045 (2020). It is true that our work doesn't solve the problem that +paper opened, but it does deepen it by showing definitively that the +use of RSB and the quenched complexity are not sufficient to +reestablish a landscape–dynamics connection. +{\bf One can hardly expect that the structure of saddles at a given energy may be connected +with dynamics (for example in Sherrington Kirkpatrick) if it is unknown}. +%We disagree with the +%referee's implicit assertion that only clean resolutions, and not the +%compelling deepening of problems, are worthy of a broad audience. + +Report of Referee B -- LY17256/Kent-Dobias +{\it The paper presents a computation of the complexity in spherical + spin-glass models. Neither the techniques nor the results are + sufficiently new and relevant to justify publication on PRL. This is + not surprising given that the topic has been studied extensively in + the last thirty years and more, the only novelty with respect to + previous work is that the results are obtained at zero temperature but + this is definitively not enough. Essential open problems in the field + involves dynamics and activated processes and some results have + appeared recently, instead the analysis of the static landscape, to + which the present paper is a variation, failed to deliver answers to + these questions up to now. + + } + +Concerning the statement of Referee B that "the only novelty +with respect to previous work is that the results are obtained at zero +temperature", we do not know what to make of the referee's statement. +The novelty of the paper is most definitely +not the fact of treating a zero temperature case. +We have added the following phrase, that should clarify the situation: + +{\color{blue} + For simplicity we have concentrated here on the energy, rather +than {\em free-energy} landscape, although the latter is sometimes +more appropriate. From the technical point of view, this makes no fundamental difference, it suffices +to apply the same computation to the Thouless-Andreson-Palmer \cite{Crisanti_1995_Thouless-Anderson-Palmer} (TAP) free energy, instead of the energy. We do not expect new features or technical +complications arise. + +} + + +%For a system where the quenched and annealed +%complexities differ, there has not been a correct calculation of the +%quenched complexity at finite temperature. (and, besides our work, +%only once or twice at zero temperature, e.g., PRX 9, 011003 (2019).) +%Rejecting a paper based on a severe misconception of its contents or +%of the state of the field is not appropriate. + +We agree with Referee B's assessment of "essential open problems in +the field," and agree that our work does not deliver answers. However, +delivering answers for all essential open problems is not the acceptance +criterion of PRL. These are + + - Open a new research area, or a new avenue within an established area. + - Solve, or make essential steps towards solving, a critical problem. + - Introduce techniques or methods with significant impact. + - Be of unusual intrinsic interest to PRL's broad audience. + +We believe our manuscript makes essential steps toward solving the +critical problem of connecting analysis of the static landscape to +dynamics. We believe that its essential step is through the +introduction of a new technique, calculation of the quenched +complexity, which we believe will have significant impact as it is +applied to more complicated models. + +\closing{Sincerely,} + +\vspace{1em} + +\end{letter} + +\end{document} diff --git a/response.txt b/response.txt deleted file mode 100644 index 1709e15..0000000 --- a/response.txt +++ /dev/null @@ -1,209 +0,0 @@ -\documentclass[a4paper]{letter} - -\usepackage[utf8]{inputenc} % why not type "Bézout" with unicode? -\usepackage[T1]{fontenc} % vector fonts plz -\usepackage{newtxtext,newtxmath} % Times for PR -\usepackage[ - colorlinks=true, - urlcolor=purple, - linkcolor=black, - citecolor=black, - filecolor=black, -]{hyperref} % ref and cite links with pretty colors -\usepackage{xcolor} -\usepackage[style=phys]{biblatex} - -\renewcommand{\thefootnote}{\fnsymbol{footnote}} - -\addbibresource{frsb_kac-rice.bib} - -\signature{ - \vspace{-6\medskipamount} - \smallskip - Jaron Kent-Dobias \& Jorge Kurchan -} - -\address{ - Laboratoire de Physique\\ - Ecole Normale Sup\'erieure\\ - 24 rue Lhomond\\ - 75005 Paris -} - -\begin{document} -\begin{letter}{ - Agnese I.~Curatolo, Ph.D.\\ - Physical Review Letters\\ - 1 Research Road\\ - Ridge, NY 11961 -} - -\opening{Dear Dr.~Curatolo,} - -Enclosed please find a revised manuscript. -Neither referee criticized the scientific content of our paper, -nor substantively addressed its presentation. We have followed their comments -in the direction of highlighting the importance of having a full solution. In particular -we have emphasized that going to the full replica treatment uncovers a phase-space structure that needs to be taken into account, and that is absent in the annealed treatment. - - -We have thus added the paragraph: - -{\color{red} -Having a full, exact (`quenched') solution of the generic problem is not -primarily a matter of {\em accuracy}. -Very basic structural questions are omitted in the approximate `annealed' solution. What is lost is the nature, at any given -energy (or free energy) level, of the stationary points in a generic energy function: at low energies are they basically all minima, with an exponentially small number of saddles, or --- as we show here -- do they consist of a mixture of saddles whose index -- the number of unstable directions -- is a smoothly distributed number? -These questions need to be answered for the understanding of the relevance of more complex objects such as -barrier crossing (which barriers?) \cite{Ros_2019_Complexity, Ros_2021_Dynamical}, or the fate of long-time dynamics -(which end in what kind of target states?). - -} - -Both referees find that our paper is clearly written but technical, and -that its topic of "the different RSB schemes" are not suitable for a -broad audience. This is surprising to the authors, since a quick -search on Google Scholar reveals several recent PRLs with heavy use of -RSB schemes. - -We would also like to submit to the referees that it is somewhat -incongruous that the solution to a problem that had remained open for 42 years -- during which it was always present in articles in PRL and PRX -- is rejected -because it demands of the readers a slightly longer attention span. - - - - -\begin{enumerate} - \item PRL has been publishing articles on precisely this problem in the - last 30 years.\footfullcite{Fyodorov_2004_Complexity, Bray_2007_Statistics, Fyodorov_2012_Critical, Wainrib_2013_Topological} - \item These works were often limited by the fact that general landscapes (for - which an annealed solution is not exact) were inaccessible. It is perhaps - true that the final solution of an open problem may often be more technical - than the previous ones. -\end{enumerate} - - - -Below, we respond to the referees' comments. - -{\it Report of Referee A -- LY17256/Kent-Dobias - The authors consider spin glass models with mixed p-spin interactions - on the N-Sphere and calculate the number of stationary points, the - logarithm of which yields the complexity. The disorder average of this - logarithm is computed with the replica trick, and for different model - variants different replica symmetry breaking (RSB) solutions are - obtained. A new feature of the solutions, in contrast to previous - replica symmetric calculations, is that RSB must occur in parts of the - energy-stability phase diagram. - - The paper is clearly written although the content is rather technical - and probably only accessible to experts in mean field spin glass - models and the different RSB schemes developed in this field. In - connection with the well-studied p=3 spin glass model it is briefly - mentioned that the complexity and its transitions as a function of - energy and/or stability is relevant for the equilibrium and the - dynamical behavior of this model – but such a connection has not been - made here. - - Therefore, I feel that the results presented here are only interesting - for group of experts and I do not assess the finding that the - complexity of mixed p-spin glass models shows RSB as a major - breakthrough in the field. Therefore, the manuscript is not suitable - for publication in Phys. Rev. Lett., and the publication of the - accompanying longer paper, submitted to PRE, is sufficient to - disseminate the results summarized in this manuscript.} - - - -Referee A correctly points out that one new feature of the solutions -outlined in our manuscript is that RSB must occur in parts of the -phase diagram for these models. However, they neglect another feature: -that they are the solutions of the *quenched* complexity, which has -not been correctly calculated until now. We agree with the referee -that "the complexity of the mixed p-spin glass models" is not a major -breakthrough in and of itself, we just -chose to demonstrate the problem in simplest toy model. But believe that the technique for -computing the quenched complexity is a major breakthrough -{\bf because it brings in the features of organization of saddles of all -kinds that are invisible in the annealed scheme}. - - -Referee A states that a connection between the complexity and the -equilibrium and dynamical behavior is not made in our paper. Until -recently, this connection was taken for granted, and the demonstration -that the standard correspondence does not hold in the mixed p-spin -spherical models was exciting enough news to be published in PRX 10, -031045 (2020). It is true that our work doesn't solve the problem that -paper opened, but it does deepen it by showing definitively that the -use of RSB and the quenched complexity are not sufficient to -reestablish a landscape–dynamics connection. -{\bf One can hardly expect that the structure of saddles at a given energy may be connected -with dynamics (for example in Sherrington Kirkpatrick) if it is unknown}. -%We disagree with the -%referee's implicit assertion that only clean resolutions, and not the -%compelling deepening of problems, are worthy of a broad audience. - -Report of Referee B -- LY17256/Kent-Dobias -{\it The paper presents a computation of the complexity in spherical - spin-glass models. Neither the techniques nor the results are - sufficiently new and relevant to justify publication on PRL. This is - not surprising given that the topic has been studied extensively in - the last thirty years and more, the only novelty with respect to - previous work is that the results are obtained at zero temperature but - this is definitively not enough. Essential open problems in the field - involves dynamics and activated processes and some results have - appeared recently, instead the analysis of the static landscape, to - which the present paper is a variation, failed to deliver answers to - these questions up to now. - - } - -Concerning the statement of Referee B that "the only novelty -with respect to previous work is that the results are obtained at zero -temperature", we do not know what to make of the referee's statement. -The novelty of the paper is most definitely -not the fact of treating a zero temperature case. -We have added the following phrase, that should clarify the situation: - -{\color{blue} - For simplicity we have concentrated here on the energy, rather -than {\em free-energy} landscape, although the latter is sometimes -more appropriate. From the technical point of view, this makes no fundamental difference, it suffices -to apply the same computation to the Thouless-Andreson-Palmer \cite{Crisanti_1995_Thouless-Anderson-Palmer} (TAP) free energy, instead of the energy. We do not expect new features or technical -complications arise. - -} - - -%For a system where the quenched and annealed -%complexities differ, there has not been a correct calculation of the -%quenched complexity at finite temperature. (and, besides our work, -%only once or twice at zero temperature, e.g., PRX 9, 011003 (2019).) -%Rejecting a paper based on a severe misconception of its contents or -%of the state of the field is not appropriate. - -We agree with Referee B's assessment of "essential open problems in -the field," and agree that our work does not deliver answers. However, -delivering answers for all essential open problems is not the acceptance -criterion of PRL. These are - - - Open a new research area, or a new avenue within an established area. - - Solve, or make essential steps towards solving, a critical problem. - - Introduce techniques or methods with significant impact. - - Be of unusual intrinsic interest to PRL's broad audience. - -We believe our manuscript makes essential steps toward solving the -critical problem of connecting analysis of the static landscape to -dynamics. We believe that its essential step is through the -introduction of a new technique, calculation of the quenched -complexity, which we believe will have significant impact as it is -applied to more complicated models. - -\closing{Sincerely,} - -\vspace{1em} - -\end{letter} - -\end{document} -- cgit v1.2.3-70-g09d2 From 4045e729291d6a589fd48c79a154be9f81a1f2ad Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Mon, 23 Jan 2023 12:00:35 +0100 Subject: Fancier formatting. --- response.tex | 93 ++++++++++++++++++++++++++++++++---------------------------- 1 file changed, 49 insertions(+), 44 deletions(-) diff --git a/response.tex b/response.tex index 1709e15..4917d67 100644 --- a/response.tex +++ b/response.tex @@ -87,33 +87,35 @@ because it demands of the readers a slightly longer attention span. Below, we respond to the referees' comments. -{\it Report of Referee A -- LY17256/Kent-Dobias - The authors consider spin glass models with mixed p-spin interactions - on the N-Sphere and calculate the number of stationary points, the - logarithm of which yields the complexity. The disorder average of this - logarithm is computed with the replica trick, and for different model - variants different replica symmetry breaking (RSB) solutions are - obtained. A new feature of the solutions, in contrast to previous - replica symmetric calculations, is that RSB must occur in parts of the - energy-stability phase diagram. - - The paper is clearly written although the content is rather technical - and probably only accessible to experts in mean field spin glass - models and the different RSB schemes developed in this field. In - connection with the well-studied p=3 spin glass model it is briefly - mentioned that the complexity and its transitions as a function of - energy and/or stability is relevant for the equilibrium and the - dynamical behavior of this model – but such a connection has not been - made here. - - Therefore, I feel that the results presented here are only interesting - for group of experts and I do not assess the finding that the - complexity of mixed p-spin glass models shows RSB as a major - breakthrough in the field. Therefore, the manuscript is not suitable - for publication in Phys. Rev. Lett., and the publication of the - accompanying longer paper, submitted to PRE, is sufficient to - disseminate the results summarized in this manuscript.} - +Report of Referee A -- LY17256/Kent-Dobias +\begin{quote} + \it + The authors consider spin glass models with mixed p-spin interactions + on the N-Sphere and calculate the number of stationary points, the + logarithm of which yields the complexity. The disorder average of this + logarithm is computed with the replica trick, and for different model + variants different replica symmetry breaking (RSB) solutions are + obtained. A new feature of the solutions, in contrast to previous + replica symmetric calculations, is that RSB must occur in parts of the + energy-stability phase diagram. + + \hspace{2em}The paper is clearly written although the content is rather technical + and probably only accessible to experts in mean field spin glass + models and the different RSB schemes developed in this field. In + connection with the well-studied p=3 spin glass model it is briefly + mentioned that the complexity and its transitions as a function of + energy and/or stability is relevant for the equilibrium and the + dynamical behavior of this model – but such a connection has not been + made here. + + \hspace{2em}Therefore, I feel that the results presented here are only interesting + for group of experts and I do not assess the finding that the + complexity of mixed p-spin glass models shows RSB as a major + breakthrough in the field. Therefore, the manuscript is not suitable + for publication in Phys. Rev. Lett., and the publication of the + accompanying longer paper, submitted to PRE, is sufficient to + disseminate the results summarized in this manuscript. +\end{quote} Referee A correctly points out that one new feature of the solutions @@ -145,19 +147,20 @@ with dynamics (for example in Sherrington Kirkpatrick) if it is unknown}. %compelling deepening of problems, are worthy of a broad audience. Report of Referee B -- LY17256/Kent-Dobias -{\it The paper presents a computation of the complexity in spherical - spin-glass models. Neither the techniques nor the results are - sufficiently new and relevant to justify publication on PRL. This is - not surprising given that the topic has been studied extensively in - the last thirty years and more, the only novelty with respect to - previous work is that the results are obtained at zero temperature but - this is definitively not enough. Essential open problems in the field - involves dynamics and activated processes and some results have - appeared recently, instead the analysis of the static landscape, to - which the present paper is a variation, failed to deliver answers to - these questions up to now. - - } +\begin{quote} + \textit{The paper presents a computation of the complexity in spherical + spin-glass models. Neither the techniques nor the results are + sufficiently new and relevant to justify publication on PRL. This is + not surprising given that the topic has been studied extensively in + the last thirty years and more, the only novelty with respect to + previous work is that the results are obtained at zero temperature but + this is definitively not enough. Essential open problems in the field + involves dynamics and activated processes and some results have + appeared recently, instead the analysis of the static landscape, to + which the present paper is a variation, failed to deliver answers to + these questions up to now. + } +\end{quote} Concerning the statement of Referee B that "the only novelty with respect to previous work is that the results are obtained at zero @@ -188,10 +191,12 @@ the field," and agree that our work does not deliver answers. However, delivering answers for all essential open problems is not the acceptance criterion of PRL. These are - - Open a new research area, or a new avenue within an established area. - - Solve, or make essential steps towards solving, a critical problem. - - Introduce techniques or methods with significant impact. - - Be of unusual intrinsic interest to PRL's broad audience. +\begin{itemize} + \item Open a new research area, or a new avenue within an established area. + \item Solve, or make essential steps towards solving, a critical problem. + \item Introduce techniques or methods with significant impact. + \item Be of unusual intrinsic interest to PRL's broad audience. +\end{itemize} We believe our manuscript makes essential steps toward solving the critical problem of connecting analysis of the static landscape to -- cgit v1.2.3-70-g09d2 From df2fbf0ee2627c67b0212ac6d549a042038abbe7 Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Mon, 23 Jan 2023 12:09:24 +0100 Subject: Some formatting changes and new references. --- response.tex | 9 ++++----- 1 file changed, 4 insertions(+), 5 deletions(-) diff --git a/response.tex b/response.tex index 4917d67..3454eaf 100644 --- a/response.tex +++ b/response.tex @@ -49,17 +49,16 @@ we have emphasized that going to the full replica treatment uncovers a phase-spa We have thus added the paragraph: -{\color{red} +\begin{quote} Having a full, exact (`quenched') solution of the generic problem is not primarily a matter of {\em accuracy}. Very basic structural questions are omitted in the approximate `annealed' solution. What is lost is the nature, at any given energy (or free energy) level, of the stationary points in a generic energy function: at low energies are they basically all minima, with an exponentially small number of saddles, or -- as we show here -- do they consist of a mixture of saddles whose index -- the number of unstable directions -- is a smoothly distributed number? These questions need to be answered for the understanding of the relevance of more complex objects such as -barrier crossing (which barriers?) \cite{Ros_2019_Complexity, Ros_2021_Dynamical}, or the fate of long-time dynamics +barrier crossing (which barriers?) \footfullcite{Ros_2019_Complexity, Ros_2021_Dynamical}, or the fate of long-time dynamics (which end in what kind of target states?). - -} +\end{quote} Both referees find that our paper is clearly written but technical, and that its topic of "the different RSB schemes" are not suitable for a @@ -76,7 +75,7 @@ because it demands of the readers a slightly longer attention span. \begin{enumerate} \item PRL has been publishing articles on precisely this problem in the - last 30 years.\footfullcite{Fyodorov_2004_Complexity, Bray_2007_Statistics, Fyodorov_2012_Critical, Wainrib_2013_Topological} + last 30 years.\footfullcite{Fyodorov_2004_Complexity, Bray_2007_Statistics, Fyodorov_2012_Critical, Wainrib_2013_Topological, Dennis_2020_Jamming} \item These works were often limited by the fact that general landscapes (for which an annealed solution is not exact) were inaccessible. It is perhaps true that the final solution of an open problem may often be more technical -- cgit v1.2.3-70-g09d2 From 255cd496d8ba4a115ba670a64c8445a8b9c2f485 Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Mon, 23 Jan 2023 12:34:04 +0100 Subject: More small changes. --- frsb_kac-rice.bib | 2 +- response.tex | 7 ++----- 2 files changed, 3 insertions(+), 6 deletions(-) diff --git a/frsb_kac-rice.bib b/frsb_kac-rice.bib index 37ffb2d..6a2b5e8 100644 --- a/frsb_kac-rice.bib +++ b/frsb_kac-rice.bib @@ -294,7 +294,7 @@ @article{Crisanti_1995_Thouless-Anderson-Palmer, author = {Crisanti, A. and Sommers, H.-J.}, - title = {Thouless-Anderson-Palmer Approach to the Spherical $p$-Spin Spin Glass Model}, + title = {{Thouless}-{Anderson}-{Palmer} Approach to the Spherical $p$-Spin Spin Glass Model}, journal = {Journal de Physique I}, publisher = {EDP Sciences}, year = {1995}, diff --git a/response.tex b/response.tex index 3454eaf..6abde6c 100644 --- a/response.tex +++ b/response.tex @@ -126,7 +126,7 @@ that "the complexity of the mixed p-spin glass models" is not a major breakthrough in and of itself, we just chose to demonstrate the problem in simplest toy model. But believe that the technique for computing the quenched complexity is a major breakthrough -{\bf because it brings in the features of organization of saddles of all +\textit{because it brings in the features of organization of saddles of all kinds that are invisible in the annealed scheme}. @@ -168,15 +168,12 @@ The novelty of the paper is most definitely not the fact of treating a zero temperature case. We have added the following phrase, that should clarify the situation: -{\color{blue} For simplicity we have concentrated here on the energy, rather than {\em free-energy} landscape, although the latter is sometimes more appropriate. From the technical point of view, this makes no fundamental difference, it suffices -to apply the same computation to the Thouless-Andreson-Palmer \cite{Crisanti_1995_Thouless-Anderson-Palmer} (TAP) free energy, instead of the energy. We do not expect new features or technical +to apply the same computation to the Thouless-Andreson-Palmer (TAP) free energy, \footfullcite{Crisanti_1995_Thouless-Anderson-Palmer} instead of the energy. We do not expect new features or technical complications arise. -} - %For a system where the quenched and annealed %complexities differ, there has not been a correct calculation of the -- cgit v1.2.3-70-g09d2 From 2f2cda1c805ef73408b5a9e3bc3dcab1b8fffe7d Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Mon, 23 Jan 2023 14:22:31 +0100 Subject: Removed some comments and changed some formatting. --- response.tex | 19 ++++++------------- 1 file changed, 6 insertions(+), 13 deletions(-) diff --git a/response.tex b/response.tex index 6abde6c..13c220f 100644 --- a/response.tex +++ b/response.tex @@ -86,8 +86,10 @@ because it demands of the readers a slightly longer attention span. Below, we respond to the referees' comments. -Report of Referee A -- LY17256/Kent-Dobias \begin{quote} + \begin{center} + Report of Referee A -- LY17256/Kent-Dobias + \end{center} \it The authors consider spin glass models with mixed p-spin interactions on the N-Sphere and calculate the number of stationary points, the @@ -141,12 +143,11 @@ use of RSB and the quenched complexity are not sufficient to reestablish a landscape–dynamics connection. {\bf One can hardly expect that the structure of saddles at a given energy may be connected with dynamics (for example in Sherrington Kirkpatrick) if it is unknown}. -%We disagree with the -%referee's implicit assertion that only clean resolutions, and not the -%compelling deepening of problems, are worthy of a broad audience. -Report of Referee B -- LY17256/Kent-Dobias \begin{quote} + \begin{center} + Report of Referee B -- LY17256/Kent-Dobias + \end{center} \textit{The paper presents a computation of the complexity in spherical spin-glass models. Neither the techniques nor the results are sufficiently new and relevant to justify publication on PRL. This is @@ -174,14 +175,6 @@ more appropriate. From the technical point of view, this makes no fundamental di to apply the same computation to the Thouless-Andreson-Palmer (TAP) free energy, \footfullcite{Crisanti_1995_Thouless-Anderson-Palmer} instead of the energy. We do not expect new features or technical complications arise. - -%For a system where the quenched and annealed -%complexities differ, there has not been a correct calculation of the -%quenched complexity at finite temperature. (and, besides our work, -%only once or twice at zero temperature, e.g., PRX 9, 011003 (2019).) -%Rejecting a paper based on a severe misconception of its contents or -%of the state of the field is not appropriate. - We agree with Referee B's assessment of "essential open problems in the field," and agree that our work does not deliver answers. However, delivering answers for all essential open problems is not the acceptance -- cgit v1.2.3-70-g09d2 From 463f9092809df7e2f76479ee2217078bdaee2692 Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Mon, 23 Jan 2023 14:31:45 +0100 Subject: Some condesing of the language. --- response.tex | 42 ++++++++++++++++++------------------------ 1 file changed, 18 insertions(+), 24 deletions(-) diff --git a/response.tex b/response.tex index 13c220f..4addddf 100644 --- a/response.tex +++ b/response.tex @@ -61,27 +61,21 @@ barrier crossing (which barriers?) \footfullcite{Ros_2019_Complexity, Ros_2021_D \end{quote} Both referees find that our paper is clearly written but technical, and -that its topic of "the different RSB schemes" are not suitable for a +that its topic of ``the different RSB schemes'' are not suitable for a broad audience. This is surprising to the authors, since a quick search on Google Scholar reveals several recent PRLs with heavy use of RSB schemes. -We would also like to submit to the referees that it is somewhat -incongruous that the solution to a problem that had remained open for 42 years -- during which it was always present in articles in PRL and PRX -- is rejected -because it demands of the readers a slightly longer attention span. - - - - -\begin{enumerate} - \item PRL has been publishing articles on precisely this problem in the - last 30 years.\footfullcite{Fyodorov_2004_Complexity, Bray_2007_Statistics, Fyodorov_2012_Critical, Wainrib_2013_Topological, Dennis_2020_Jamming} - \item These works were often limited by the fact that general landscapes (for - which an annealed solution is not exact) were inaccessible. It is perhaps - true that the final solution of an open problem may often be more technical - than the previous ones. -\end{enumerate} - +We would also like to submit to the referees that it is somewhat incongruous +that the solution to a problem that had remained open for 42 years -- during +which it was always present in articles in PRL +\footfullcite{Fyodorov_2004_Complexity, Bray_2007_Statistics, +Fyodorov_2012_Critical, Wainrib_2013_Topological, Dennis_2020_Jamming}-- is +rejected because it demands of the readers a slightly longer attention span. +These previous works were often limited by the fact that general landscapes +(for which an annealed solution is not exact) were inaccessible. It is perhaps +true that the final solution of an open problem may often be more technical +than the previous ones. Below, we respond to the referees' comments. @@ -113,7 +107,7 @@ Below, we respond to the referees' comments. for group of experts and I do not assess the finding that the complexity of mixed p-spin glass models shows RSB as a major breakthrough in the field. Therefore, the manuscript is not suitable - for publication in Phys. Rev. Lett., and the publication of the + for publication in Phys.\ Rev.\ Lett., and the publication of the accompanying longer paper, submitted to PRE, is sufficient to disseminate the results summarized in this manuscript. \end{quote} @@ -122,9 +116,9 @@ Below, we respond to the referees' comments. Referee A correctly points out that one new feature of the solutions outlined in our manuscript is that RSB must occur in parts of the phase diagram for these models. However, they neglect another feature: -that they are the solutions of the *quenched* complexity, which has +that they are the solutions of the \textit{quenched} complexity, which has not been correctly calculated until now. We agree with the referee -that "the complexity of the mixed p-spin glass models" is not a major +that ``the complexity of the mixed p-spin glass models'' is not a major breakthrough in and of itself, we just chose to demonstrate the problem in simplest toy model. But believe that the technique for computing the quenched complexity is a major breakthrough @@ -162,9 +156,9 @@ with dynamics (for example in Sherrington Kirkpatrick) if it is unknown}. } \end{quote} -Concerning the statement of Referee B that "the only novelty +Concerning the statement of Referee B that ``the only novelty with respect to previous work is that the results are obtained at zero -temperature", we do not know what to make of the referee's statement. +temperature,'' we do not know what to make of the referee's statement. The novelty of the paper is most definitely not the fact of treating a zero temperature case. We have added the following phrase, that should clarify the situation: @@ -175,8 +169,8 @@ more appropriate. From the technical point of view, this makes no fundamental di to apply the same computation to the Thouless-Andreson-Palmer (TAP) free energy, \footfullcite{Crisanti_1995_Thouless-Anderson-Palmer} instead of the energy. We do not expect new features or technical complications arise. -We agree with Referee B's assessment of "essential open problems in -the field," and agree that our work does not deliver answers. However, +We agree with Referee B's assessment of ``essential open problems in +the field,'' and agree that our work does not deliver answers. However, delivering answers for all essential open problems is not the acceptance criterion of PRL. These are -- cgit v1.2.3-70-g09d2 From 87cc37c5c6cd8d1d3a29d1cda94ad9e1f9ec790f Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Mon, 23 Jan 2023 16:38:05 +0100 Subject: Some rewriting. --- frsb_kac-rice_letter.tex | 141 ++++++++++++++++++++--------------------------- 1 file changed, 59 insertions(+), 82 deletions(-) diff --git a/frsb_kac-rice_letter.tex b/frsb_kac-rice_letter.tex index b6e7b8b..5c29523 100644 --- a/frsb_kac-rice_letter.tex +++ b/frsb_kac-rice_letter.tex @@ -26,17 +26,16 @@ \affiliation{Laboratoire de Physique de l'Ecole Normale Supérieure, Paris, France} \begin{abstract} - {\color{red} Complex landscapes are defined as those having a proliferation of saddle points. - The question of their number and organization has been the object of long-standing attention, in particular centered around Gaussian mean-field potentials, - which includes glass and spin glass models. -The annealed approximation is by now well understood, but is exact for a restricted subset of these problems. Here we derive the exact quenched -solution for the general case, which incorporates Parisi's solution for the ground state, - as it should. More importantly, including - replica symmetry breaking uncovers the full distribution of saddles at given energy in terms of their stabilities, a structure that is lost in the annealed approximation. This structure should be a guide for the identification - of relevant activated processes in relaxational or driven dynamics.} - %These examples demonstrate the consistency of the - %solution and reveal that the signature of replica symmetry breaking at high - %energy densities is found in high-index saddles, not minima. + Complex landscapes are defined by their many saddle points. Determining their + number and organization is a long-standing problem, in particular for + tractable Gaussian mean-field potentials, which include glass and spin glass + models. The annealed approximation is well understood, but is generally not exact. Here we derive the exact + quenched solution for the general case, which incorporates Parisi's solution + for the ground state, as it should. More importantly, the quenched solution + correctly uncovers the full distribution of saddles at a given energy, a + structure that is lost in the annealed approximation. This structure should + be a guide for the accurate identification of the relevant activated + processes in relaxational or driven dynamics. \end{abstract} \maketitle @@ -48,49 +47,47 @@ size of the system \cite{Maillard_2020_Landscape, Ros_2019_Complex, Altieri_2021_Properties}. Though they are often called `rough landscapes' to evoke the intuitive image of many minima in something like a mountain range, the metaphor to topographical landscapes is strained by the reality that these -complex landscapes exist in very high dimensions. -Many interesting versions of the problem have been treated, and the subject has -evolved into an active field of probability -theory \cite{Auffinger_2012_Random, Auffinger_2013_Complexity, -BenArous_2019_Geometry} and has been applied to -energy functions inspired by molecular biology, evolution, and machine learning -\cite{Maillard_2020_Landscape, Ros_2019_Complex, Altieri_2021_Properties}. - - - - - -The computation of the number of metastable states in this setting -was pioneered forty years ago by Bray and Moore -\cite{Bray_1980_Metastable}, who proposed the first calculation for the -Sherrington--Kirkpatrick model, in one of the -first applications of any replica symmetry breaking (RSB) scheme. As was clear -from the later results by Parisi \cite{Parisi_1979_Infinite}, their result was not exact, and the problem has -been open ever since. To date the program of -computing the statistics of stationary points---minima, saddle points, and -maxima---of mean-field complex landscapes has been only carried out in an exact form for a relatively small subset of -models, including most notably the (pure) $p$-spin model ($p>2$) -\cite{Rieger_1992_The, Crisanti_1995_Thouless-Anderson-Palmer, Cavagna_1997_An, Cavagna_1998_Stationary}. +complex landscapes exist in very high dimensions. Many interesting versions +of the problem have been treated, and the subject has evolved into an active +field of probability theory \cite{Auffinger_2012_Random, + Auffinger_2013_Complexity, BenArous_2019_Geometry} and has been applied to + energy functions inspired by molecular biology, evolution, and machine + learning \cite{Maillard_2020_Landscape, Ros_2019_Complex, + Altieri_2021_Properties}. + +The computation of the number of metastable states in such a landscape was +pioneered forty years ago by Bray and Moore \cite{Bray_1980_Metastable} on the +Sherrington--Kirkpatrick (SK) model in one of the first applications of any +replica symmetry breaking (RSB) scheme. As was clear from the later results by +Parisi \cite{Parisi_1979_Infinite}, their result was not exact, and the +problem has been open ever since. To date the program of computing the +statistics of stationary points---minima, saddle points, and maxima---of +mean-field complex landscapes has been only carried out in an exact form for a +relatively small subset of models, including most notably the (pure) $p$-spin spherical +model ($p>2$) \cite{Rieger_1992_The, Crisanti_1995_Thouless-Anderson-Palmer, +Cavagna_1997_An, Cavagna_1998_Stationary}. {\color{red} -Having a full, exact (`quenched') solution of the generic problem is not -primarily a matter of {\em accuracy}. -Very basic structural questions are omitted in the approximate `annealed' solution. What is lost is the nature, at any given -energy (or free energy) level, of the stationary points in a generic energy function: at low energies are they basically all minima, with an exponentially small number of saddles, or --- as we show here -- do they consist of a mixture of saddles whose index -- the number of unstable directions -- is a smoothly distributed number? -These questions need to be answered for the understanding of the relevance of more complex objects such as -barrier crossing (which barriers?) \cite{Ros_2019_Complexity, Ros_2021_Dynamical}, or the fate of long-time dynamics -(which end in what kind of target states?). - - - - - +Having a full, exact (`quenched') solution of the generic problem is not +primarily a matter of {\em accuracy}. Very basic structural questions are +omitted in the approximate `annealed' solution. What is lost is the nature, +at any given energy (or free energy) level, of the stationary points in a +generic energy function: at low energies are they basically all minima, with an +exponentially small number of saddles, or -- as we show here -- do they consist +of a mixture of saddles whose index -- the number of unstable directions -- is +a smoothly distributed number? These questions need to be answered for the +understanding of the relevance of more complex objects such as barrier crossing +(which barriers?) \cite{Ros_2019_Complexity, Ros_2021_Dynamical}, or the fate +of long-time dynamics (which end in what kind of target states?). In fact, we show that the +state of dynamics in generic cases is limited to energies \emph{at which saddles +are exponentially more numerous than minima}. In this paper we present what we argue is the general replica ansatz for the number of stationary points of generic mean-field models, which we expect to -include the Sherrington--Kirkpatrick model. This allows us -to clarify the rich structure of all the saddles, and in particular the lowest ones. The interpretation of a Parisi ansatz itself, in this context must be recast in a way that makes sense for the order parameters involved. +include the SK model. This allows us to clarify the rich structure of all the +saddles, and in particular the lowest ones. The interpretation of a Parisi +ansatz itself, in this context must be recast in a way that makes sense for the +order parameters involved. } @@ -108,25 +105,20 @@ complications arise. In this paper and its longer companion, we share the first results for the complexity with nontrivial hierarchy \cite{Kent-Dobias_2022_How}. Using a general form for the solution detailed in a companion article, we describe the -structure of landscapes with a 1RSB complexity and a full RSB complexity -%\footnote{The Thouless--Anderson--Palmer (TAP) complexity is the complexity of - % a kind of mean-field free energy. Because of some deep thermodynamic - % relationships between the TAP complexity and the equilibrium free energy, the -%TAP complexity can be computed with extensions of the equilibrium method. As a -%result, the TAP complexity has been previously computed for nontrivial -%hierarchical structure.}. - -For definiteness, we consider the standard example of the mixed $p$-spin spherical models, with Hamiltonian -\begin{equation} \label{eq:hamiltonian} - H(\mathbf s)=-\sum_p\frac1{p!}\sum_{i_1\cdots i_p}^NJ^{(p)}_{i_1\cdots i_p}s_{i_1}\cdots s_{i_p} +structure of landscapes with a 1RSB complexity and a full RSB complexity. + +For definiteness, we consider the standard example of the mixed $p$-spin +spherical models, which exhibit a zoo of orders and phases. These models can be +defined by taking a random Gaussian Hamiltonian $H$ defined on the $N-1$ sphere +and with a covariance that depends on only the dot product (or overlap) between +two configurations. For $s_1,s_2\in S^{N-1}$, +\begin{equation} \label{eq:covariance} + \overline{H(\mathbf s_1)H(\mathbf s_2)}=Nf\left(\frac{\mathbf s_1\cdot\mathbf s_2}N\right), \end{equation} - $\mathbf s\in\mathbb R^N$ confined to the $N-1$ sphere -$\{|\mathbf s\|^2=N\}$. The coupling coefficients $J$ are taken at random, with -zero mean and variance $\overline{(J^{(p)})^2}=a_pp!/2N^{p-1}$ chosen so that -the energy is typically extensive. The overbar will always denote an average -over the coefficients $J$. The factors $a_p$ in the variances are freely chosen -constants that define the particular model. For instance, the so-called `pure' -models have $a_p=1$ for some $p$ and all others zero. +where $f$ is a function with positive coefficients. This uniquely defines the +distribution over Hamiltonians $H$. The overbar will always denote an average +over the functions $H$. The choice of function $f$ fixes the model. For +instance, the `pure' $p$-spin models have $f(q)=\frac12q^p$. The complexity of the $p$-spin models has been extensively studied by physicists and mathematicians. Among physicists, the bulk of work has been on @@ -139,21 +131,6 @@ complexity have been used to study many geometric properties of the pure models, from the relative position of stationary points to one another to shape and prevalence of instantons \cite{Ros_2019_Complexity, Ros_2021_Dynamical}. -{\color{green} {\bf eliminate?} The variance of the couplings implies that the covariance of the energy with -itself depends on only the dot product (or overlap) between two configurations. -In particular, one finds -\begin{equation} \label{eq:covariance} - \overline{H(\mathbf s_1)H(\mathbf s_2)}=Nf\left(\frac{\mathbf s_1\cdot\mathbf s_2}N\right), -\end{equation} -where $f$ is defined by the series -\begin{equation} - f(q)=\frac12\sum_pa_pq^p. -\end{equation} -One needn't start with a Hamiltonian like -\eqref{eq:hamiltonian}, defined as a series: instead, the covariance rule -\eqref{eq:covariance} can be specified for arbitrary, non-polynomial $f$, as in -the `toy model' of M\'ezard and Parisi \cite{Mezard_1992_Manifolds}. In fact, defined this way the mixed spherical model encompasses all isotropic Gaussian fields on the sphere.} - The family of spherical models thus defined is quite rich, and by varying the covariance $f$ nearly any hierarchical structure can be found in equilibrium. Because of a correspondence between the ground state complexity -- cgit v1.2.3-70-g09d2 From 5c203019791f877f16c7f4819693c8f28b472310 Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Thu, 26 Jan 2023 15:58:54 +0100 Subject: Some writing edits. --- frsb_kac-rice_letter.tex | 89 ++++++++++++++++++++++-------------------------- 1 file changed, 40 insertions(+), 49 deletions(-) diff --git a/frsb_kac-rice_letter.tex b/frsb_kac-rice_letter.tex index 5c29523..c7ef913 100644 --- a/frsb_kac-rice_letter.tex +++ b/frsb_kac-rice_letter.tex @@ -29,13 +29,14 @@ Complex landscapes are defined by their many saddle points. Determining their number and organization is a long-standing problem, in particular for tractable Gaussian mean-field potentials, which include glass and spin glass - models. The annealed approximation is well understood, but is generally not exact. Here we derive the exact - quenched solution for the general case, which incorporates Parisi's solution - for the ground state, as it should. More importantly, the quenched solution - correctly uncovers the full distribution of saddles at a given energy, a - structure that is lost in the annealed approximation. This structure should - be a guide for the accurate identification of the relevant activated - processes in relaxational or driven dynamics. + models. The annealed approximation is well understood, but is generally not + exact. Here we describe the exact quenched solution for the general case, + which incorporates Parisi's solution for the ground state, as it should. More + importantly, the quenched solution correctly uncovers the full distribution + of saddles at a given energy, a structure that is lost in the annealed + approximation. This structure should be a guide for the accurate + identification of the relevant activated processes in relaxational or driven + dynamics. \end{abstract} \maketitle @@ -67,58 +68,48 @@ relatively small subset of models, including most notably the (pure) $p$-spin sp model ($p>2$) \cite{Rieger_1992_The, Crisanti_1995_Thouless-Anderson-Palmer, Cavagna_1997_An, Cavagna_1998_Stationary}. -{\color{red} Having a full, exact (`quenched') solution of the generic problem is not -primarily a matter of {\em accuracy}. Very basic structural questions are -omitted in the approximate `annealed' solution. What is lost is the nature, -at any given energy (or free energy) level, of the stationary points in a -generic energy function: at low energies are they basically all minima, with an -exponentially small number of saddles, or -- as we show here -- do they consist -of a mixture of saddles whose index -- the number of unstable directions -- is -a smoothly distributed number? These questions need to be answered for the -understanding of the relevance of more complex objects such as barrier crossing -(which barriers?) \cite{Ros_2019_Complexity, Ros_2021_Dynamical}, or the fate -of long-time dynamics (which end in what kind of target states?). In fact, we show that the -state of dynamics in generic cases is limited to energies \emph{at which saddles -are exponentially more numerous than minima}. +primarily a matter of {\em accuracy}. Basic structural questions are +omitted in the approximate `annealed' solution. What is lost is the nature of +the stationary points at a given energy level: at low energies are they +basically all minima, with an exponentially small number of saddles, or (as +we show here) do they consist of a mixture of saddles whose index (the +number of unstable directions) is a smoothly distributed number? These +questions need to be answered if one hopes to correctly describe more complex +objects such as barrier crossing (which barriers?) \cite{Ros_2019_Complexity, +Ros_2021_Dynamical} or the fate of long-time dynamics (that end in which kind +of states?). In this paper we present what we argue is the general replica ansatz for the number of stationary points of generic mean-field models, which we expect to include the SK model. This allows us to clarify the rich structure of all the -saddles, and in particular the lowest ones. The interpretation of a Parisi -ansatz itself, in this context must be recast in a way that makes sense for the -order parameters involved. - -} - -{\color{blue} - For simplicity we have concentrated here on the energy, rather -than {\em free-energy} landscape, although the latter is sometimes -more appropriate. From the technical point of view, this makes no fundamental difference, it suffices -to apply the same computation to the Thouless-Andreson-Palmer \cite{Crisanti_1995_Thouless-Anderson-Palmer} (TAP) free energy, instead of the energy. We do not expect new features or technical -complications arise. - -} - - - -In this paper and its longer companion, we share the first results for the -complexity with nontrivial hierarchy \cite{Kent-Dobias_2022_How}. Using a -general form for the solution detailed in a companion article, we describe the -structure of landscapes with a 1RSB complexity and a full RSB complexity. +saddles, and in particular the lowest ones. Using a +general form for the solution detailed in a companion article \cite{Kent-Dobias_2022_How}, we describe the +structure of landscapes with a 1RSB complexity and a full RSB complexity. The interpretation of a Parisi +ansatz itself must be recast to make sense of the new order parameters +involved. + +For simplicity we concentrate on the energy, rather than {\em +free-energy}, landscape, although the latter is sometimes more appropriate. From +the technical point of view, this makes no fundamental difference and it suffices +to apply the same computation to the Thouless--Anderson--Palmer +\cite{Crisanti_1995_Thouless-Anderson-Palmer} (TAP) free energy, instead of the +energy. We do not expect new features or technical complications to arise. For definiteness, we consider the standard example of the mixed $p$-spin -spherical models, which exhibit a zoo of orders and phases. These models can be -defined by taking a random Gaussian Hamiltonian $H$ defined on the $N-1$ sphere -and with a covariance that depends on only the dot product (or overlap) between -two configurations. For $s_1,s_2\in S^{N-1}$, +spherical models, which exhibit a zoo of disordered phases. These models can be +defined by drawing a random Hamiltonian $H$ from a distribution of isotropic +Gaussian fields defined on the $N-1$ sphere. Isotropy implies that the +covariance in energies between two configurations depends on only their dot +product (or overlap), so for $\mathbf s_1,\mathbf s_2\in +S^{N-1}$, \begin{equation} \label{eq:covariance} \overline{H(\mathbf s_1)H(\mathbf s_2)}=Nf\left(\frac{\mathbf s_1\cdot\mathbf s_2}N\right), \end{equation} -where $f$ is a function with positive coefficients. This uniquely defines the -distribution over Hamiltonians $H$. The overbar will always denote an average -over the functions $H$. The choice of function $f$ fixes the model. For -instance, the `pure' $p$-spin models have $f(q)=\frac12q^p$. +where $f$ is a function with positive coefficients. The overbar will always +denote an average over the functions $H$. The choice of function $f$ uniquely +fixes the model. For instance, the `pure' $p$-spin models have +$f(q)=\frac12q^p$. The complexity of the $p$-spin models has been extensively studied by physicists and mathematicians. Among physicists, the bulk of work has been on -- cgit v1.2.3-70-g09d2