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| -rw-r--r-- | frsb_kac-rice_letter.tex | 36 | ||||
| -rw-r--r-- | response.tex | 35 |
2 files changed, 34 insertions, 37 deletions
diff --git a/frsb_kac-rice_letter.tex b/frsb_kac-rice_letter.tex index bdcec74..a7f9f11 100644 --- a/frsb_kac-rice_letter.tex +++ b/frsb_kac-rice_letter.tex @@ -26,10 +26,10 @@ \affiliation{Laboratoire de Physique de l'Ecole Normale Supérieure, Paris, France} \begin{abstract} - Complex landscapes are defined by their many saddle points. Determining their + Complex landscapes are characterized 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 + models. The annealed approximation is well understood, but is generically 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 @@ -51,16 +51,16 @@ 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 + Auffinger_2013_Complexity, BenArous_2019_Geometry} that 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 +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 +Parisi \cite{Parisi_1979_Infinite}, their result was only approximate, 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 @@ -68,7 +68,7 @@ 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}. -Having a full, exact (`quenched') solution of the generic problem is not +Having a full `quenched' solution of the generic problem is not 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 @@ -77,7 +77,7 @@ 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 +Ros_2021_Dynamical} or the fate of long-time dynamics (that targets which kind of states?). In this paper we present what we argue is the general replica ansatz for the @@ -117,20 +117,20 @@ physicists and mathematicians. Among physicists, the bulk of work has been on Crisanti_1995_Thouless-Anderson-Palmer, Cavagna_1997_An, Cavagna_1997_Structure, Cavagna_1998_Stationary, Cavagna_2005_Cavity, 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 +Auffinger_2013_Complexity, BenArous_2019_Geometry}. 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 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 -and the entropy at zero temperature, any hierarchical structure in the -equilibrium should be reflected in the complexity. +The family of spherical models thus defined is rich, by varying the +covariance $f$ any hierarchical structure can be found in +equilibrium. Because of the correspondence between the ground state complexity +and the equilibrium entropy, any hierarchical structure in +equilibrium should be reflected in the computation. The complexity is calculated using the Kac--Rice formula, which counts the stationary points using a $\delta$-function weighted by a Jacobian -\cite{Kac_1943_On, Rice_1939_The}. The count is given by +\cite{Kac_1943_On, Rice_1939_The}. It is given by \begin{equation} \begin{aligned} \mathcal N(E, \mu) @@ -156,9 +156,9 @@ is a Wigner semicircle of radius $\mu_\mathrm m=\sqrt{4f''(1)}$ centered at $\mu $\mu>\mu_\mathrm m$, stationary points are minima whose sloppiest eigenvalue is $\mu-\mu_\mathrm m$. When $\mu=\mu_\mathrm m$, the stationary points are marginal minima with flat directions. When $\mu<\mu_\mathrm m$, the stationary points are saddles with -indexed fixed to within order one (fixed macroscopic index). +index fixed to within order one (fixed macroscopic index). -It's worth reviewing the complexity for the best-studied case of the pure model +It is worth reviewing the complexity for the best-studied case of the pure model for $p\geq3$ \cite{Cugliandolo_1993_Analytical}. Here, because the covariance is a homogeneous polynomial, $E$ and $\mu$ cannot be fixed separately, and one implies the other: $\mu=pE$. Therefore at each energy there is only one kind of @@ -346,7 +346,7 @@ complexity of the ground state, predicting that the complexity of minima vanishes at a higher energy than the complexity of saddles, with both at a lower energy than the equilibrium ground state. The 1RSB complexity resolves these problems, shown in Fig.~\ref{fig:2rsb.contour}. It predicts the same ground state as equilibrium and with a -ground state stability $\mu_0=6.480\,764\ldots>\mu_\mathrm m$. It predicts that +ground state stability $\mu_0=6.480\,764\ldots>\mu_\mathrm m$. Also, the complexity of marginal minima (and therefore all saddles) vanishes at $E_\mathrm m$, which is very slightly greater than $E_0$. Saddles become dominant over minima at a higher energy $E_\mathrm{th}$. The 1RSB complexity @@ -356,7 +356,7 @@ $E_\mathrm{max}$. The numeric values for all these energies are listed in Table~\ref{tab:energies}. For minima, the complexity does -not inherit a 1RSB description until the energy is with in a close vicinity of +not inherit a 1RSB description until the energy is within a close vicinity of the ground state. On the other hand, for high-index saddles the complexity becomes described by 1RSB at quite high energies. This suggests that when sampling a landscape at high energies, high index saddles may show a sign of diff --git a/response.tex b/response.tex index 4addddf..69a5531 100644 --- a/response.tex +++ b/response.tex @@ -47,7 +47,7 @@ in the direction of highlighting the importance of having a full solution. In p 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: +Among other changes, we have added the paragraph: \begin{quote} Having a full, exact (`quenched') solution of the generic problem is not @@ -60,25 +60,28 @@ barrier crossing (which barriers?) \footfullcite{Ros_2019_Complexity, Ros_2021_D (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 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 +It is perhaps +true that the final solution of an open problem may often be more technical +than the previous ones. +But we would 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. +(for which an annealed solution is not exact) were inaccessible. Below, we respond to the referees' comments. +A comprehensive accounting of the changes to our manuscript can be found appended to this letter. \begin{quote} \begin{center} @@ -115,8 +118,7 @@ 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 \textit{quenched} complexity, which has +phase diagram for these models they are indeed 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 breakthrough in and of itself, we just @@ -163,25 +165,20 @@ 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: +\begin{quote} 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 (TAP) free energy, \footfullcite{Crisanti_1995_Thouless-Anderson-Palmer} instead of the energy. We do not expect new features or technical complications arise. +\end{quote} 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 - -\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} +the field,'' and agree that our work does not deliver all answers. However, +delivering all answers for all essential open problems is not the acceptance +criterion of PRL. -We believe our manuscript makes essential steps toward solving the +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 |