From 5834e535839406e6200f6ebdf464a3b0584f081d Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Mon, 10 Mar 2025 17:00:23 -0300 Subject: 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. --- referee_response.md | 8 +++++--- 1 file changed, 5 insertions(+), 3 deletions(-) (limited to 'referee_response.md') 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 -- cgit v1.2.3-70-g09d2