The planted XY model: thermodynamics and inference

TL;DR

This work analyzes a fully connected planted XY model, linking spin-glass physics with angular synchronization in inference. It derives a replica-symmetric phase diagram via AMP and SE, identifying paramagnetic, ferromagnetic, mixed, and spin-glass regimes and showing AMP convergence coincides with RS stability, while deviations occur off the Nishimori line. To address RS shortcomings, the authors develop an approximate survey propagation (ASP) framework and its state evolution to quantify metastable states through the complexity $\Sigma(f)$ and to determine the 1RSB free entropy, including the Parisi parameter via $s$. Numerically, ASP reveals metastable structure beyond RS, extends inference performance beyond AMP, and clarifies when 1RSB accurately captures the Gibbs measure, with implications for mean-field inference problems and phase transitions in disordered systems.

Abstract

In this paper we study a fully connected planted spin glass named the planted XY model. Motivation for studying this system comes both from the spin glass field and the one of statistical inference where it models the angular synchronization problem. We derive the replica symmetric (RS) phase diagram in the temperature, ferromagnetic bias plane using the approximate message passing (AMP) algorithm and its state evolution (SE). While the RS predictions are exact on the Nishimori line (i.e. when the temperature is matched to the ferromagnetic bias), they become inaccurate when the parameters are mismatched, giving rise to a spin glass phase where AMP is not able to converge. To overcome the defects of the RS approximation we carry out a one-step replica symmetry breaking (1RSB) analysis based on the approximate survey propagation (ASP) algorithm. Exploiting the state evolution of ASP, we count the number of metastable states in the measure, derive the 1RSB free entropy and find the behavior of the Parisi parameter throughout the spin glass phase.

Figures (8)

• Figure 1: Numerical results for AMP and SE on the Nishimori line. All plots show the values of $m,q$ at convergence. (a): comparing $m$ and $q$ from AMP to verify that $m=q$; also verifying that they both agree with SE (continuous line). The initial condition for AMP is uninformative ($m^{t=0}=0$). (b): the two curves represent the value of $m$ to which SE converges with respective initial condition $m^{t=0}=0, q^{t=0}=1$ and $m^{t=0}=1, q^{t=0}=1$; the fact that they're equal shows that there is a unique fixed point of SE. (c): Bethe free entropy as a function of $m$; the red cross on each curve marks the unique stationary point to which SE converges; the uniqueness of the SE fixed point is a direct consequence of the free entropy having exactly one stationary point.
• Figure 2: Comparison between AMP ($N=1000$) and SE, for (a) on and (b) outside the Nishimori line. The red line represents the behavior of SE while each of the 40 gray lines is an independent AMP run. State evolution tracks accurately both $m$ and $q$. All the runs are initialized with a configuration where each spin's value is picked uniform over the unit circle.
• Figure 3: Phase diagram obtained from SE. The dashed line in both panels is the Nishimori line. (a): the light blue region is paramagnetic phase, where $m=q=0$. It is delimited by the curve $\lambda=\min(1,\hat{\lambda}^{-1})$. The pink bottom right region is the spin glass phase (evaluated by the RS solution): here $m=0$ while $q>0$. The upper curve delimiting the spin glass phase (evaluated on the RS level) converges to $\lambda=4/\pi$ for $\hat{\lambda}\to\infty$. In the top right we find the ferromagnetic region where both $m$, $q$ are strictly positive and the RS solution is stable. The remaining slice between the ferromagnetic and spin glass phase is called the mixed phase. (b): The yellow area, which encompasses the spin glass phase and the mixed phase, is the region where the RS stability parameter $c$ is negative and thus AMP does not converge, or equivalently the RS solution is unstable.
• Figure 4: Heatmap representing $m$ and $q$ for (a) AMP ($N=500$) and (b) SE as a function of $\lambda$, $\hat{\lambda}$.
• Figure 5: Convergence properties of AMP. To measure the rate of change of the estimator, we introduce the quantity $\Delta \hat{x}^t=\frac{1}{N}\sqrt{\sum_i|\hat{x}^{(t)}_i-\hat{x}_i^{(t-1)}|^2}$. We consider AMP's iterations to have converged when $\Delta \hat{x}^t< 10^{-5}$. (a): convergence time of AMP over the $\lambda,\hat{\lambda}$ plane. The convergence time is capped at $300$, each pixel represents the average convergence time over 25 independent runs. (b): heatmap of $c_{\text{AMP}}$ computed at the final time, each pixel represents the average convergence time over 25 independent runs, (c): heatmap of $c_{\text{SE}}$ computed when SE has converged. (b-e): we pick two points in the $\lambda,\hat{\lambda}$ plane (marked by white crosses in the center panel) and plot $\Delta \hat{x}^t$ for 40 independent AMP runs. When $\Delta \hat{x}^t$ does not decay to zero, AMP does not converge.