A Friendly Tutorial on Mean-Field Spin Glass Techniques for Non-Physicists

Abstract

This tutorial is based on lecture notes written for a class taught in the Statistics Department at Stanford in the Winter Quarter of 2017. The objective was to provide a working knowledge of some of the techniques developed over the last 40 years by theoretical physicists and mathematicians to study mean field spin glasses and their applications to high-dimenensional statistics and statistical learning.

Figures (9)

• Figure 1: The phase diagram of the spiked matrix model (case $k=2$ of the spiked tensor).
• Figure 2: Spiked tensor model with $k=3$, on the Bayes line. Left: Mutual information. Right: Order parameter.
• Figure 3: Same as in Figure \ref{['fig:PspinBayes']}: zoom around the critical point.
• Figure 4: On the left, we plot $f_1$ as a function of $q_1$, $k=3$, $m=0.5$. In the high-temperature regime (shown in BROWN) \ref{['eq:q_fixedpt']} does not have any solution. New solutions appear at $T= T_d(m)$ (shown in RED). For $T<T_d(m)$ (shown in BLUE), there are two roots---the larger root is physically relevant. On the right, we plot the dependence of $q_{1*}(\beta,m)$ on $T$. The distinct curves represent different values of $m$.
• Figure 5: We plot $g(m)$ vs. $m$ for $k=3$. In subplot (a), we consider $T > T_d(1)$; the only physically relevant solution is $q_1=0$ and we are back to the paramagnetic solution. In (b), we consider $T_s < T < T_d(1)$; $m(T)$ is represented in BLUE. The function $g$ restricted to the interval $[m(T),1]$ is non-negative, and thus the replica symmetric free energy is correct in this regime. Finally (c) represents the setting $T<T_s$. $g(m)$ on the interval $[m(T),1]$ (represented by the BLUE and RED points attains negative values). Thus the 1RSB free energy strictly improves on the replica symmetric approximation.

Theorems & Definitions (65)

• Lemma 1.1.1
• proof
• Lemma 1.1.2
• Conjecture 1.2.1
• Remark 1.2.1
• Remark 1.2.2
• Remark 1.2.3
• Theorem 1: lesieur2017statistical
• Definition 1.4.1: Replicas
• Lemma 1.5.1: Kac-Rice formula adler2007random