Comment addressing report #2, first question.

Addressed the contribution of complex-m_* saddle points when they exist.

1 files changed, 3 insertions, 13 deletions

diff --git a/referee_response.md b/referee_response.md
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@@ -6,18 +6,6 @@
 
 # Report #2
 
-1- The work deals with a constraint satisfaction problem that was introduced recently, which serves as a toy model for both confluent tissues and non-linear regression in high dimension; it addresses a problem (the characterization of the topology of the solution space) that is of interest for this model, but also beyond.
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-2 - The author points out a limit in which the constraint satisfaction problem maps into well known mean-field models of glasses, allowing them to formulate a conjecture on the dynamical meaning of the shattering transition in this limit.
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-3 - The appendices contain some careful analysis (e.g. of lower order contributions which contribute to the refactor of the average Euler characteristics). The main text remains very readable, the content is well distributed between main and appendices.
-Weaknesses
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-1 - The comparison with previous results in the literature and the discussion on the instability towards RSB phases could be clarified further (see comments below).
-Report
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-The work characterizes the topology of the manifold of solutions of a high-dimensional, random constant satisfaction problem, which was introduced recently in the literature in connection to problems of confluent tissues and of non-linear regression in high dimension. The topology is characterized by computing the average Euler characteristics of the solution manifold, in the limit of large dimensionality N of the configuration space. The analysis allows to identify five distinct regimes as a function of the model's control parameter V0. These regimes differ by the magnitude of the average Euler characteristics (exponential in N vs order one in N), by its sign and by the regions of configuration space that contribute dominantly to it. A scaling limit is also discussed, in which this analysis corresponds to the characterization of the topology of level sets of the energy landscape of spherical p-spin models. Within this context, the author proposes a dynamical interpretation of the "shattering energy”, which corresponds to the energy level where a transition occurs between different topological phases, as being predictive of the dynamic threshold.
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 The manuscript is clear and self-contained. Below are some questions and requests for clarifications:
 
 (i) It is argue in the paper that in the regime V<V_sh, where the action at m=0 is complex, that in the regime where the action at, the solution should not be discarded as it indicates a negative average Euler characteristic. This argument is supported by the calculation of the second moment and the equality (18). Regarding the m* solutions, the action at m* becomes complex above V_on: is there an interpretation for these solutions in the regime V> V_on?
@@ -40,7 +28,9 @@ Recommendation
 
 Publish (meets expectations and criteria for this Journal)
 
- 1. Ok, complex m^* solutions
+(ii) It is mentioned that the naive satisfiability threshold predicted from the vanishing of (12) coincides with the threshold obtained within the replica symmetric analysis of the cost function (3). By reading the manuscript, I have missed if/how the satisfiability threshold arising from the analysis of the average Euler characteristics compares with the threshold obtained from the zero-temperature analysis of the equilibrium problem with energy (3): could the Author comment on this?
+
+ 1. In fact, the action is not complex when evaluated at m_* for V² > V_on² even though m_* itself becomes complex: the action remains real but becomes negative in this regime. This means that the contribution of these complex-m_* solutions in this regime shrinks with increasing N, and rather than representing a subleading but exponentially large (or even order 1) contribution to the Euler characteristic, their contribution is negligible.
  2.
  3.
  4. Maybe??