replica symmetry breaking analysis

This work investigates the local geometric structure of typical solutions to random regular NAE-SAT in the clustered (1RSB) phase. Addressing the challenge that long-range correlations above the clustering threshold complicate local behavior, we establish, for the first time, a non-asymptotic local weak limit for this sparse CSP in the clustered phase: its limiting distribution is characterized by a tree-based propagation channel defined by the 1RSB belief propagation fixed point, exhibiting pronounced non-Markovian dependencies. We prove that when the solution space is dominated by a few large clusters, the local solution distribution converges tightly at rate (O(n^{-1/2})), surpassing the validity limits of conventional rooted-tree models and Markov approximations. This yields the first rigorous, non-asymptotic local characterization of message-passing algorithms—such as belief propagation—for sparse constraint satisfaction problems in the 1RSB regime.

In Bayesian inference, posterior distributions over high-dimensional correlated variables often exhibit replica symmetry breaking (RSB), challenging conventional assumptions that the Nishimori condition—exact matching between prior and likelihood—guarantees replica symmetry. Method: We integrate the replica-symmetric cavity method, first-order RSB instability analysis, and a geometric formulation of epidemic dynamics on graphs, applied to the canonical inverse problem of “patient-zero identification” in highly contagious S–I epidemic models. Contribution/Results: We construct the first explicit counterexample where RSB occurs despite strict satisfaction of the Nishimori condition. Rigorously proving RSB existence, we quantify the cavity solution’s instability threshold and identify inter-variable correlated disorder as the fundamental origin of RSB. This overturns the long-held belief that Nishimori compliance ensures replica symmetry, establishing a novel reliability criterion for approximate Bayesian inference methods in high-correlation inverse problems.

This work addresses parameter inference and learning in parametric models when the data-generating distribution is unknown or intractable. We propose a novel replica-theoretic framework based on variational Gaussian approximation. Within the grand canonical ensemble, we defer data averaging and replace the conventional quenched average with an empirical average; crucially, the trial Hamiltonian parameters are determined adaptively via a variational principle, eliminating reliance on idealized distributional assumptions. Our key contribution is the explicit incorporation of fluctuation effects into the analysis—establishing, for the first time within the replica method, a rigorous correspondence with information criteria such as AIC and BIC, thereby quantifying how statistical fluctuations govern generalization performance. Theoretical analysis yields exact learning curves for linear regression, and empirical validation confirms the framework’s efficacy in scenarios where standard replica methods break down.

This paper investigates the intrinsic structural properties of individual feedforward neural networks from a statistical mechanics perspective, aiming to uncover nontrivial organizational principles—beyond conventional metrics such as loss and accuracy—that govern data fitting, model capacity, generalization, and robustness. Method: We formulate a Hopfield-like spin-glass model for neural networks and, for the first time, apply replica symmetry breaking (RSB) theory to analyze single network instances. Structural descriptors are defined via inter-replica overlap, yielding computationally tractable, label-free, and training-set-agnostic quantifications of implicit structural complexity solely from network weights. Contribution/Results: The proposed descriptor effectively discriminates between distinct training states and architectures. Empirically, it demonstrates practical utility in model diagnostics, safety verification, and detection of latent vulnerabilities—providing a theoretically grounded, physics-inspired lens for characterizing neural network structure beyond empirical performance metrics.

This work investigates the high-dimensional geometric structure of the solution space achieving zero training error in neural networks—modeled by binary-weight perceptrons—and its evolution with training set size. Using statistical physics methods—including the replica method, Gardner capacity analysis, and geometric characterization of solution spaces—we uncover a phase transition from dense, clustered solutions to sparse, isolated ones. We introduce “linear modal connectivity” as a quantitative measure of the average shape of solution manifolds. Crucially, we identify that algorithmic hardness arises from the disappearance of distant solution clusters precisely at the critical data threshold. Our analysis quantitatively characterizes the SAT/UNSAT phase transition, scaling laws of solution cluster sizes, and local landscape ruggedness. Collectively, these results establish a unified geometric–statistical physical framework for understanding generalization and optimization difficulty in deep learning.

This study challenges the conventional assumption of independent and identically distributed signals by investigating how structured priors with correlations affect Bayesian inference performance. By constructing a planted spin glass model on random regular graphs, where signals are sampled from an Ising model with coupling strength κ, the work systematically analyzes how structured priors—spanning paramagnetic, ferromagnetic, and replica symmetry breaking (RSB) phases—influence signal reconstruction. The analysis reveals, for the first time, that non-separable correlated priors can induce a static RSB phase in the posterior distribution along the Nishimori line, thereby limiting the efficacy of algorithms such as belief propagation. Furthermore, it demonstrates that correlations in the paramagnetic regime lower the reconstruction threshold, while in the ferromagnetic regime, a critical point emerges where observations provide additional informative content.

本文针对线性尺度下的最大平均子矩阵问题，通过开发随机对偶理论框架，提供了该问题的封闭形式上界，并展示了改进方法。

Although high-dimensional non-convex empirical risk functions possess numerous local minima, gradient-based algorithms often converge to solutions near the global optimum; however, the precise characterization of the polynomial-time reachable region remains elusive. This work addresses this gap in the context of multi-index supervised learning models by integrating replica symmetry breaking theory with the Incremental Approximate Message Passing (IAMP) algorithm. Through high-dimensional asymptotic analysis within the empirical risk minimization framework, the study precisely characterizes the training error achievable by IAMP and establishes its quantitative relationship with the test error. The results delineate the performance boundary between computational feasibility and statistical optimality, demonstrating that IAMP achieves optimal performance among all polynomial-time algorithms.

本文提出了一种基于快速混合马尔可夫链和亚临界渗流过程的框架，用于解决高温下stoquastic自旋系统中的快速采样与计数问题。

This study addresses the lack of efficient sampling and partition function approximation algorithms for ferromagnetic two-state systems within certain parameter regimes. By constructing weighted subgraph and random-cluster-like models, the authors establish novel equivalences between these combinatorial structures and the target physical system. Leveraging these connections, they propose the first efficient sampling algorithm tailored to this regime and design a partition function approximation algorithm running in nearly quadratic time on bounded-degree graphs and in polynomial time on general graphs. This represents a significant improvement over the method of Guo et al. (2020). The work integrates techniques from graph theory, statistical physics, and randomized approximation to deliver an enhanced computational framework for analyzing such systems.