Changes addressing report #3, weakness #1.

Added a paragraph to the text discussing the content of previous work on
this model.

2 files changed, 16 insertions, 9 deletions

diff --git a/referee_response.md b/referee_response.md
index d1f1f44..f0ca4a3 100644
--- a/referee_response.md
+++ b/referee_response.md
@@ -32,17 +32,10 @@ Recommendation
  1. Ok.
  2. Ok.
  3. At the moment, when the manuscript is typeset Fig. 2 is on the same page as the description of the topological phases containing the requested information. Therefore, adding them to the caption feel redundant. However, if the referee feels strongly that the information should appear in both places the modification can be made.
+ 4. Ok.
 
 # Report #3
 
-Strengths
-
-1- The study addresses a gap in understanding the loss landscape of CS problems, a field that has been challenging due to technical complexity.
-2- The application of the Kac-Rice formula to study Euler characteristics in high-dimensional settings is, to my knowledge, novel and simplifies the calculations considerably. This approach could potentially be extended to other problems where progress has been hindered by the sign of the determinant.
-3- Technically, this remains a difficult problem, highlighting the significance of the result.
-4- The results provide valuable insights, identifying a complex picture of the landscape and several regimes that may help elucidate the dynamics.
-5- The final question addressed in the paper is significant, as it has remained elusive in previous studies. While I am not fully convinced by the author's argument (detailed later), it introduces valuable new elements to the discussion.
-Weaknesses
 
 1- The paper provides insufficient discussion of previous work. The author condenses key background and literature into two brief sentences (at the start of the second paragraph of Section 1 and at the end of page 2). Even for readers familiar with these references, this is hard to parse. I had to go into the bibliography and see which paper the author was referring to in order to follow. I strongly recommend expanding the introduction and discussing prior work in greater detail to provide adequate context for the reader.
 2- The interpretation of magnetization m is unclear. While briefly mentioned at the beginning of Section 2.2, the explanation is insufficient. Since there is no planting in this problem, the physical meaning of an arbitrary random direction is still unclear to me.
@@ -74,7 +67,7 @@ Ask for minor revision
     formatting: perfect
     grammar: perfect
 
- 1. Ok
+ 1. A discussion of the previous literature on this model has been added in the introduction.
  2. Ok
  3. Ok
    * The referee is wrong to say that the Euler characteristic of a hypersphere is 2 independent of dimension. The Euler characteristic of all odd-dimensional manifolds is zero. Consider the cell complex on *S*₁ [pictured here](https://kent-dobias.com/files/S_1.png). The Euler characteristic calculated using the alternating sum over the number of cells of increasing dimension is χ(*S*₁) = 1 – 1 = 0.
diff --git a/topology.tex b/topology.tex
index 56af0c6..32f1a6a 100644
--- a/topology.tex
+++ b/topology.tex
@@ -171,6 +171,20 @@ studied properties of the cost function
   \mathscr C(\mathbf x)=\frac12\sum_{k=1}^M\big[V_k(\mathbf x)-V_0\big]^2
 \end{equation}
 which achieves zero only for configurations that satisfy all the constraints.
+Introduced in Ref.~\cite{Fyodorov_2019_A}, the existence of solutions and the
+geometric structure of the cost function were studied for the linear problem in
+a series of papers \cite{Fyodorov_2019_A, Fyodorov_2020_Counting,
+Fyodorov_2022_Optimization} and later reviewed \cite{Vivo_2024_Random}. Some
+work on the equilibrium measure of the cost function in the nonlinear problem
+was made in Ref.~\cite{Tublin_2022_A}, and the problem was solved in
+Ref.~\cite{Urbani_2023_A}. Subsequent work has studied varied dynamics applied
+to the cost function, including gradient descent, Hessian descent, Langevin,
+stochastic gradient descent, and approximate message passing
+\cite{Kamali_2023_Dynamical, Kamali_2023_Stochastic, Montanari_2023_Solving,
+Montanari_2024_On}. Finally, some progress has been made on aspects of the
+geometric structure of the cost function in the nonlinear problem
+\cite{Kent-Dobias_2024_Conditioning, Kent-Dobias_2024_Algorithm-independent}.
+
 From the perspective of the cost function, the set of solutions looks like a network of flat canyons at the bottom of the cost landscape.
 Here we dispense with the cost function and study the set of solutions
 directly. This set can be written as