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Analyzing dynamics and average case complexity in the spherical Sherrington-Kirkpatrick model: a focus on extreme eigenvectors

Tingzhou Yu

TL;DR

This work analyzes the dynamics and average-case complexity of the spherical Sherrington–Kirkpatrick model in the large-$N$ limit. It derives CHSCK-type integro-differential equations for the empirical correlation $K(t,s)$ and energy $H(t)$, revealing a dynamical phase transition at $\beta_c=c/4$ and providing explicit limiting forms for $H(t)$ in different temperature regimes. The paper also establishes hitting-time bounds for both gradient-descent dynamics and the power-iteration method when identifying extreme eigenvectors of Wigner matrices, showing $T_\varepsilon$ scales as $\Theta(N^{2/3})$ up to logarithmic factors under standard moment conditions. By tying aging phenomena in spin-glass dynamics to concrete eigenvector computation questions, it connects stochastic dynamics and random-matrix theory with practical algorithmic complexity insights, supported by semicircle-law asymptotics and Bessel-function analyses.

Abstract

We explore Langevin dynamics in the spherical Sherrington-Kirkpatrick model, delving into the asymptotic energy limit. Our approach involves integro-differential equations, incorporating the Crisanti-Horner-Sommers-Cugliandolo-Kurchan equation from spin glass literature, to analyze the system's size and its temperature-dependent phase transition. Additionally, we conduct an average case complexity analysis, establishing hitting time bounds for the bottom eigenvector of a Wigner matrix. Our investigation also includes the power iteration algorithm, examining its average case complexity in identifying the top eigenvector overlap, with comprehensive complexity bounds.

Analyzing dynamics and average case complexity in the spherical Sherrington-Kirkpatrick model: a focus on extreme eigenvectors

TL;DR

This work analyzes the dynamics and average-case complexity of the spherical Sherrington–Kirkpatrick model in the large- limit. It derives CHSCK-type integro-differential equations for the empirical correlation and energy , revealing a dynamical phase transition at and providing explicit limiting forms for in different temperature regimes. The paper also establishes hitting-time bounds for both gradient-descent dynamics and the power-iteration method when identifying extreme eigenvectors of Wigner matrices, showing scales as up to logarithmic factors under standard moment conditions. By tying aging phenomena in spin-glass dynamics to concrete eigenvector computation questions, it connects stochastic dynamics and random-matrix theory with practical algorithmic complexity insights, supported by semicircle-law asymptotics and Bessel-function analyses.

Abstract

We explore Langevin dynamics in the spherical Sherrington-Kirkpatrick model, delving into the asymptotic energy limit. Our approach involves integro-differential equations, incorporating the Crisanti-Horner-Sommers-Cugliandolo-Kurchan equation from spin glass literature, to analyze the system's size and its temperature-dependent phase transition. Additionally, we conduct an average case complexity analysis, establishing hitting time bounds for the bottom eigenvector of a Wigner matrix. Our investigation also includes the power iteration algorithm, examining its average case complexity in identifying the top eigenvector overlap, with comprehensive complexity bounds.
Paper Structure (10 sections, 20 theorems, 107 equations, 2 figures)

This paper contains 10 sections, 20 theorems, 107 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Notation:
  3. Asymptotic energy dynamics in the spherical SK model
  4. Main results
  5. Proof of Theorem \ref{['thm: limit_energy']}
  6. Proof of Theorem \ref{['thm:ntoin_2']}
  7. Proof of \ref{['coro_hk']}
  8. Gradient descent in spherical SK model: hitting time analysis
  9. Power iteration method: a hitting time perspective
  10. Acknowledgement:

Key Result

Theorem 2.1

arous2001aging Assume that the initial data $\{X_0^i\}_{1\le i\le N}$ are i.i.d with law $\mu_0$ so that $\mathbb{E}_{X\sim \mu_0} [e^{\alpha X}]<\infty$ for some $\alpha>0$. Fix $T\ge 0$. As $N\to \infty$, $K_N$ converges almost surely to deterministic limits $K$. Recall that $\mu_D$ is the semicir where $\pi^{\infty}=\mu_D\otimes \mu_0$ and here we write $K(s):=K(s.s)$.

Figures (2)

  • Figure 1: In this figure, we set $c=1$ and plot the limiting behavior of the function $H(t)$ as $t$ approaches infinity. $H(t)$ exhibits a jump discontinuity in the phase transition at the critical inverse temperature $\beta_c=0.25$.
  • Figure 2: In this figure, we set $c=1$ and plot the limiting behavior of the function $H(t)/K(t)$ as $t$ approaches infinity. There is a jump discontinuity in the phase transition at the critical inverse temperature $\beta_c=0.25$.

Theorems & Definitions (38)

  • Definition 1.1
  • Remark 1.2
  • Theorem 2.1
  • Remark 2.2
  • Remark 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Corollary 2.6
  • Definition 2.7
  • Lemma 2.8
  • ...and 28 more