1 files changed, 184 insertions, 0 deletions
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