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diff --git a/response.md b/response.md new file mode 100644 index 0000000..351a221 --- /dev/null +++ b/response.md @@ -0,0 +1,184 @@ + +For reviewer 2 + +> On page 9, the author points out that there are solutions with complexity 0 +> that do not show an extensive barrier "in any situation". First, this "in any +> situation" is quite unclear. Does the author mean above and below the +> threshold energy? Does this solution exist even at high energy? +> Can the author comment on what this solution could imply? + +The sense of "any situation" means "for a reference point of any energy and +stability." This includes energies above and below the threshold energy, at +stabilities that imply saddles, minima, or marginal minima, and even for +combinations of energy and stability where the complexity of stationary points +is negative. + +The two-point complexity is computed under the condition that the reference +point exists. Given that the reference point exists, there is at least one +point that can be found at zero overlap with the reference: itself. This +reasoning alone rationalizes why we should find a solution with Σ₁₂ = 0, q = 0, +and E₀ = E₁, μ₀ = μ₁ for any E₀ and μ₀. + +The interpretation is more subtle. Two different stationary points cannot lie +at the same point, but the complexity calculation only resolves numbers of +points that are exponential in N and differences in overlap that are linear in +N. Therefore, the complexity calculation is compatible with many stationary +points being contained in the subextensive region of dimension Δq = O(1) around +any reference point. We can reason as to where these extremely near neighbors +are likely or unlikely to exist in specific conditions, but the complexity +calculation cannot rule them out. + +This is point is not crucial to anything in the paper, except to make more +precise the statement that non-threshold marginal minima are separated by a gap +in their overlap. Because marginal minima have very flat directions, they are +good candidates for possessing these extremely near neighbors, and this might +lead one to say they are not isolated. However, if such extremely near +neighbors exist, they are irrelevant to dynamics: the entire group is isolated, +since the complexity of similar stationary points at a small but extensive +overlap further is negative. + +> At the technical level, I am confused by one of the constraints imposed in +> eq.16, when \sigma_1 couples with all replicated s_a. I was expecting a sum +> of \sigma_b.s_a over a and b. This may represent a rotation applied to all +> replicas along a reference direction, which is probably what the author did, +> but there is no comment about that. In general, it would have helped to +> specify the constraints enforced instead of writing simply "Lagrange +> multipliers" before eq.16. + +The fact noticed by the referee that only σ₁ appears in the scalar product with +sₐ in equation (16) of the original manuscript was not introduced in that +equation, but instead was introduced in equation (10) of the original +manuscript. Right after that equation, the special status of σ₁ was clarified. +This arises because of the structure of equation (9): in that equation, the +logarithmic expression being averaged depends only on σ, which corresponds with +σ₁ in the following equation. σ₂ through σₘ correspond to σ', which is +replicated (m - 1) times to bring the partition function into the numerator. +Therefore, there is a clear reason behind the asymmetry among the replicas +associated with the reference spin, and it was not due to an ad-hoc +transformation as suggested by the referee. + +> The author analyses the problem using the Franz-Parisi potential, however, +> this analysis does not seem to matter in the paper. We can read a comment at +> the end of Sec.3.1 but without actual implications. It should be either +> removed or expanded, at the moment it seems just without purpose. + +The analysis of the Franz-Parisi potential has been moved to an appendix, with +a more explanatory discussion of our interest in it included in the manuscript. + +> "We see arrangements of barriers relative to each other, perhaps...". Why +> "perhaps"? Second, where is this analysis carried out? In the results +> section, the author analyses stable minima and marginal states, I don't know +> where to look. Adding a reference would have helped. + +> After eq.3, the author comments on the replica ansatz, but this is out of +> place. We are still introducing the model. It would be better to have it at +> the end of the section (where indeed the author comes back to the same +> concept) or remove it entirely. + +> fig.1, add a caption under each figure saying what they are (oriented +> saddles, oriented minima, etc), it is much easier to read. + +The suggestion of the referee was good and was implemented in the new manuscript. + +> fig.2, elaborate a bit more in the main text. This is introduced at the of +> the section without any comment. + +> fig.3, "the dot-dashed lines on both plots depict the trajectory of the solid +> line on the other plot", which one? + +> fig.3, "In this case, the points lying nearest to the reference minimum are +> saddle with mu\<mu, but with energies smaller than the threshold energy", so? +> What is the implication? This misses a conclusion. + +> Sec.3.1, the author comments on the similarity with the pure model, without +> explaining what is similar. What should we expect on the p-spin? At least the +> relevant aspects. It would also be useful to plot a version of Fig.3 for the +> p-spin. It would make the discussion easier to follow. + +> "the nearest neighbour points are always oriented saddles", where do I see this? + +> the sentence "like in the pure models, the emergence [...]" is extremely hard +> to parse and the paragraph ends without a conclusion. What are the +> consequences? + +> at page 9, the author talk about \Sigma_12 that however has not been defined yet. + +> this section starts without explaining what is the strategy to solve the +> problem. Explaining how the following subsection will contribute to the +> solution without entering into the details of the computation would be of +> great help. + +> "This replica symmetry will be important later" how? Either we have an +> explanation following or it should be removed. + +> at the end of a step it would be good to wrap everything up. For instance, +> sec.4.2 ends with "we do not include these details, which are standard" at +> least give a reference. Second, add the final result. + +> "there is a desert where none are found" -> solutions are exponentially rare +> (or something else) + +> I would suggest a rewriting, especially the last sessions (4-6). I understand +> the intention of removing simple details, but they should be replaced by +> comments. The impression (which can be wrong but gives the idea) is of some +> working notes where simple steps have been removed, resulting in +> hard-to-follow computations. Finally, I would also recommend moving these +> sections to an appendix (after acknowledgement and funding). + +For reviewer 1 + +> i) On page 7, when referring to the set of marginal states that attract +> dynamics "as evidenced by power-law relaxations", it would be convenient to +> provide references for this statement. + +The evidence of power-law relaxation to marginal minima is contained in G. +Folena and F. Zamponi, On weak ergodicity breaking in mean-field spin glasses, +SciPost Physics 15(3), 109 (2023). In the original manuscript this work was +cited at the end of the sentence, but the sentence has now be rephrased and the +specific point about power-law relaxation has been removed to improve clarity. + +> ii) On the same page, the author refers to a quadratic pseudo-gap in the +> complexity function associated with marginal states. It would be helpful to +> have some more indication of how this was derived or, again, to provide +> appropriate references. + +The form of the pseudo-gap in overlap for marginal states above the threshold +energy is demonstrated in the subsection on the expansion of the complexity in +the near neighborhood (equation 40 in the original manuscript). + +> iii) Section 4 “Calculation of the two-point complexity”. The author states +> that conditioning the Hessian matrix of the stationary points to have a given +> energy and given stability properties influences the statistics of points +> only at the sub-leading order. It would be valuable to clarify the conditions +> under which this occurs. I was thus wondering whether the author can +> straightforwardly generalize such a computation and give some insights in the +> case of a sparse (no longer fully connected) model. + +CITE VALENTINA?? + +> iv) Eq. (34) is quite complicated and difficult to grasp by eye. I thus +> wonder whether the numerical protocol is robust enough to be sure that by +> initializing differently, not exactly at q=0, the same solution is always +> found. How sensitive is the protocol to the choice of initial conditions? + +DO A LITTLE EXPERIMENT + +> v) In Section 5, the analysis of an isolated eigenvalue, which can be +> attributed to a low-rank perturbation in the Hessian matrix, is discussed. +> The technique results from a generalization of a paper recently published by +> H. Ikeda, restricted to a quadratic model though. It would be worthwhile to +> discuss how many of these predictions can be extended to models defined by a +> double-well potential or to optimization problems relying on non-quadratic +> functions (such as ReLu, sigmoid). + +ALWAYS QUADRATIC! + +Requested changes + +> I found the paper interesting but quite technical in some points. Moving the +> saddle-point computations and part of the analysis (see for instance on pages +> 11-13 and 18-20) to the supplement would make it easier to capture the main +> results, especially for general readers without extensive expertise in the +> replica trick and these models. + +OK |