Changes addressing report #3, comment #3.2

Added a footnote discussing the intuition for interpreting a large Euler
characteristic as a union manifold rather than a product manifold.

2 files changed, 33 insertions, 5 deletions

diff --git a/referee_response.md b/referee_response.md
index f0ca4a3..e39df9b 100644
--- a/referee_response.md
+++ b/referee_response.md
@@ -67,11 +67,13 @@ Ask for minor revision
     formatting: perfect
     grammar: perfect
 
+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.
+
  1. A discussion of the previous literature on this model has been added in the introduction.
- 2. Ok
- 3. Ok
+ 2. A discussion of how to interpret the order parameter *m* has been added to the end of section 2.1.
+ 3. See the comments below.
    * 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.
-   * Ok
+   * In this manuscript we present what we consider to be the simplest interpretation of the calculation, but the referee is correct to point out that a large Euler characteristic could indicate a complicated product manifold as well as one with many connected components, or other exotic manifolds besides. Our intuition for this is that applying one constraint amounts to taking a smooth, non-self-intersecting slice of a sphere, which should typically produce spheres of one fewer dimension. Repeating this reasoning recursively leads to the conclusion that the result is mostly unions of spheres all the way down. This schematic argument has been added to the manuscript as a footnote in section 2.3. As to what dynamics might look like in a problem where the manifold of solutions were actually a nontrivial product manifold, we have no idea.
  4. The referee points out that previous work on gradient descent in the spherical spin glasses studied gradient descent from both uniformly random initial conditions ("infinite" temperature) and initial conditions drawn from a Boltzmann distribution at some finite temperature, and found that the final state of the dynamics reached marginal minima in a range of energies depending on the initial condition. The conjecture in this manuscript seeks only to explain the upper energy of this range, that associated with gradient descent from a uniformly random initial condition. Presumably there are a variety of behaviors observable by choosing initial conditions using a variety of initial distributions, Boltzmann or otherwise, and one day we may hope to address such questions using similar approaches to this paper. However, this is not addressed here. A small discussion of this point has been added to the manuscript.
    * A paragraph addressing what might occur in planted models has been added to the manuscript.
  5. Make a supplementary materials file
diff --git a/topology.tex b/topology.tex
index 32f1a6a..f89a6ca 100644
--- a/topology.tex
+++ b/topology.tex
@@ -322,7 +322,7 @@ and the number of variables, and $R_m$ is a function of $m$ given by
   \Bigg]
   \end{aligned}
 \end{equation}
-This function is plotted in Fig.~\ref{fig:action} for a selection of
+The effective action \eqref{eq:S.m} is plotted in Fig.~\ref{fig:action} for a selection of
 parameters. To finish evaluating the integral by the saddle-point
 approximation, the action should be maximized with respect to $m$. If $m_*$ is
 such a maximum, then the resulting average Euler characteristic is
@@ -354,6 +354,30 @@ parameters are varied.
   } \label{fig:action}
 \end{figure}
 
+The order parameter $m$ may appear similar to the magnetization that appears in
+problems that have a signal or spike, where it gives the overlap of a
+configuration with the hidden signal. Here $\textbf x_0$ is no signal, but a
+direction chosen uniformly at random and with no significance to the set of solutions. If in this problem a feature of the
+action is present at some value $m$, it should be interpreted as indicating
+that, with overwhelming probability, the feature has a proximity to a typical
+point in configuration space given by the overlap $m$. For instance, for $m$
+sufficiently close to one $\mathcal S_\chi(m)$ is always negative, which is a
+result of the absence of any stationary points contributing to the Euler
+characteristic at those overlaps. Given a random height axis $\mathbf x_0$, the
+nearest point to $\mathbf x_0$ on the solution manifold will be the absolute
+maximum of the height function, and therefore contribute to the Euler
+characteristic. Therefore, we can interpret the region of negative action in
+the vicinity of $m=1$ to mean that there is a typical distance between the
+solution manifold and a randomly drawn point in configuration space, and that
+it is vanishingly unlikely to draw a point in configuration space uniformly at
+random and find it any closer to the solution manifold than this.\footnote{
+  Other properties of the set of solutions could be studied by drawing $\mathbf
+  x_0$ from an alternative distribution, like the Boltzmann distribution of the
+  cost function, from the set of its stationary points, or from the solution
+  manifold itself. While the value of the Euler characteristic would not
+  change, the dependence of the effective action on $m$ would change.
+}
+
 \subsection{Features of the effective action}
 
 The order parameter $m$ is the overlap of the
@@ -546,7 +570,9 @@ than the shattering value $V_\text{sh}$ and $\mathcal S(0)>0$. Above the
 shattering transition the effective action is real everywhere, and its value at
 the equator is the dominant contribution. Large connected components of the
 manifold may or may not exist, but small disconnected components outnumber
-holes.
+holes.\footnote{
+  We interpret the large Euler characteristic to indicate a manifold with many (topologically) spherical disconnected components because the manifold is formed by the process of repeatedly taking non-self-intersecting slices of the previous manifold, starting with a sphere. Therefore, an outcome consisting mostly of (topological) spheres seems most plausible. However, a large Euler characteristic is also consistent with a variety of connected product manifolds, among other exotic possibilities. Definitely ruling out such scenarios is not within the scope of this paper.
+}
 
 \paragraph{Regime V: \boldmath{$\overline{\chi(\Omega)}$} very small.}