The complexity of spherical p-spin models - a second moment approach

By Eliran Subag. 2015

134 citations recorded by the source corpus.

Recently, Auffinger, Ben Arous, and \v{C}ern\'y initiated the study of critical points of the Hamiltonian in the spherical pure $p$-spin spin glass model, and established connections between those and several notions from the physics literature. Denoting the number of critical values less than $Nu$ by $\mbox{Crt}_{N}(u)$, they computed the asymptotics of $\frac{1}{N}\log(\mathbb{E}\mbox{Crt}_{N}(u))$, as $N$, the dimension of the sphere, goes to $\infty$. We compute the asymptotics of the corresponding second moment and show that, for $p\geq3$ and sufficiently negative $u$, it matches the first moment: \[ \mathbb{E}\left\{ \left(\mbox{Crt}_{N}\left(u\right)\right)^{2}\right\} /\left(\vphantom{\left(\mbox{Crt}_{N}\left(u\right)\right)^{2}}\mathbb{E}\left\{ \mbox{Crt}_{N}\left(u\right)\right\} \right)^{2}\to1. \] As an immediate consequence we obtain that $\mbox{Crt}_{N}(u)/\mathbb{E}\{ \mbox{Crt}_{N}(u)\} \to 1$, in $L^2$ and thus in probability. For any $u$ for which $\mathbb{E}\mbox{Crt}_{N}(u)$ does not tend to $0$ we prove that the moments match on an exponential scale.

Subfields: Condensed Matter Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/caa3b0c8303ec880abc8b6b8184baa493a8d2ee1

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In Condensed Matter Physics

In Statistics and Probability

Citations in the corpus

The Landscape of the Spiked Tensor Model (2017, methodology): “ces. However, for the pure noise case = 0, it was expected that the number of critical points concentrates around its expectation. This was recently conrmed by Subag via a second moment calculation [Sub15]. (See also [Sub17] and [SZ17] for additional information about the landscape geometry.) Finall”
On the energy landscape of deep networks (2015, background): “A-4 hinges upon the observation in Subag (2015) that the second negative term (apart from the first GOE term) in the Hessian at a critical point (A-11) does not change the eigenvalue distribution of the GOE much.”
The Effect of Gradient Noise on the Energy Landscape of Deep Networks (2015, background): “The proof of Lemma S-6 hinges upon the observation in Subag (2015) that the additional negative term in the Hessian at a critical point (S-15) does not change the eigenvalue distribution of the GOE much.”