Search arXiv⌕ Search

arXiv subjects

Zhangsong Li

Publications and source records attributed to Zhangsong Li.

At least 19 recordsLinked to original sources
arXiv2026

Near linear mixing of random-cluster Glauber dynamics on random regular graphs

We study the random-cluster model on locally treelike graphs at low temperatures. This is a model of dependent percolation parametrized by an edge probability $p \in (0,1)$ and a clustering weight $q \in [1,\infty)$, generalizing independent Bernoulli percolation ($q=1$) and closely related to the classical ferromagnetic Ising and Potts spin systems at integer $q$. For $q>2$, at parameter $p$ below the tree uniqueness threshold $p_{\mathsf u}(q,d)$, it is expected that sampling is easy and local Markov chains mix rapidly on all graphs with degree bounded by $d$. On typical graphs, the same is predicted at $p>p_{\mathsf s}(q,d)$, where $p_{\mathsf s}(q,d)$ is a second uniqueness transition point on the $d$-regular wired tree. Our result confirms this prediction for random regular graphs, showing that the random-cluster Glauber dynamics mix in time $O(n^{1+o(1)})$ on a random $d$-regular graph for all fixed $q>2$ and $p>p_{\mathsf s}$. For comparison, the best prior sampling results for these models that held for all $p>p_{\mathsf s}$ required $q\gtrsim \log d$ \cite{BGPZ26}. The key difficulty is that simply utilizing the contraction of cavity messages on $d$-regular trees does not suffice to cover the entire regime $q>2$. To address this issue, motivated by \cite{LS26}, we develop a careful bootstrap argument to show the shrinkage of disagreement under grand coupling. The core ideas of the proof were generated using GPT-6 and the Pharos system; the proof was then refined and simplified by the author.

math.PR↗
arXiv2026

Free energy fluctuations of symmetric perceptron models up to and including criticality

We study the free energy fluctuations of spherical and Ising perceptron models with general even activations. These include the symmetric binary perceptron, the $U$-perceptron, and the bipartite Sherrington--Kirkpatrick model. Our main result is a Gaussian CLT for the free energy that holds both in the ``high-temperature'' interior of a second moment RS regime and on its ``critical'' boundary. At criticality, the CLT has scale $(\frac{\log N}{6})^{1/2}$, which matches the free energy fluctuations of the critical Sherrington--Kirkpatrick model and suggests this scaling is universal. The proof shows that the perceptron model's free energy fluctuations are explained, up to an additive $o(1)$ error with high probability, by those of a corresponding spherical Hopfield model, and that this relation persists even at criticality. This proof strategy builds on a critical reweighted moment method developed by the first and third authors \cite{du2026fluctuations} to identify the critical fluctuations of the SK free energy, which links these fluctuations to the spectral edge spacing of the disorder.

math.PR↗
arXiv2026

Improved rapid phase ordering for Ising dynamics on random regular graphs

We study Glauber dynamics for the Ising model on random $d$-regular graphs with $n$ vertices at low temperature, in the regime $β\geq C/d$ where $C,d$ are sufficiently large constants. In this regime, the full mixing time is exponential in $n$, because of the bottleneck between the two dominant phases: configurations with a majority of plus spins and configurations with a majority of minus spins. We show that, for sufficiently large $d$, if the dynamics is initialized with an excess of at least $ε_{C,d} n$ plus spins over minus spins, then it quasi-equilibrates in time $O(\log n)$ to the Gibbs measure conditioned on positive magnetization. In addition, the required bias $ε_{C,d}$ can be chosen so that $ε_{C,d}\to 0$ as $d,C\to\infty$. Previous bound of \cite{GSS25+} in the Ising and large-$d$ setting required $β\gtrsim (\log d)/d$. A key difficulty is that, even from the all-plus initialization, one must control the propagation of negative information in spacetime in a low-temperature model with strong local correlations. We address this by developing a new bootstrap approach to proving subcriticality of disagreement clusters under the grand coupling.

math.PR↗
arXiv2026

Almost Sharp Equivalence between Approximate Message Passing and Low-Degree Polynomials

We prove a sharp lower bound for growing-degree polynomial estimation in the symmetric rank-one matrix estimation problem. The observation is \[ \boldsymbol{Y}=\frac{1}{\sqrt{n}} \boldsymbolθ \boldsymbolθ^{\top}+\boldsymbol{W}, \] where the coordinates of $\boldsymbolθ$ are independent with a common, fixed sub-Gaussian distribution $π_Θ$ satisfying $\mathbb E[Θ]\neq0$, and $\boldsymbol{W}$ is symmetric with independent standard Gaussian upper-triangular entries. For every such prior, we give an explicit finite-dimensional bound implying that every sequence of polynomial estimators of degree $D(n)=o(n^{1/60})$ has normalized mean-square error with limit inferior at least $\mathbb E[Θ^2]-q_{\mathsf{amp}}$, the limiting error of Bayes approximate message passing (AMP). This extends the constant-degree equivalence of Montanari and Wein~\cite{montanari2025equivalence} to growing degree under the same prior assumptions. Combined with their polynomial approximation of fixed-iteration AMP, the bound identifies the exact limiting low-degree MMSE whenever $D(n)\to\infty$ within this range. It thereby answers the growing-degree AMP-equivalence question discussed in~\cite{wein2025computational, maleki2026high} for nonzero-mean sub-Gaussian priors in the stated degree range. We also obtain a moment-dependent finite-dimensional lower bound for any nonzero-mean prior with all moments finite. The proof bounds low-degree MMSE using \emph{conditional} joint cumulants and a refined application of Jensen's inequality. The conditional cumulants are defined using an auxiliary Gaussian channel $\boldsymbol{R}$ calibrated to the AMP fixed point. This retains signal dependence that is lost in unconditional cumulant bounds and produces the cancellations needed for quantitative control as the degree grows.

math.ST↗
arXiv2026

Graph Matching in Random Geometric Graphs I: Efficient Algorithms

We study the recovery of the hidden vertex correspondence between two correlated dense random geometric graphs. Each graph is generated from $n$ latent unit vectors in $\mathbb R^d$; the vectors $u_i^A$ and $u_i^B$ of corresponding vertices satisfy $\langle u_i^A,u_i^B\rangle=ρ$, and the vertex labels of the second graph are permuted by an unknown permutation $π$. For every fixed correlation $ρ\in(0,1]$ and every edge density bounded away from zero and one, we give a randomized polynomial time algorithm that recovers $π$ exactly with probability tending to one whenever $d\ge(\log n)^L$, where $L$ depends only on the correlation and the density bounds. The algorithm assigns to each vertex a signature formed by counting decorated paths with a constant number of seeds. The number of such seeded templates grows exponentially with the path length, with an arbitrarily large constant base, so the resulting statistics amplify an arbitrarily small constant correlation. We control the dependencies created by the latent geometry in two ways: when $d$ is larger than $n/(\log n)^{L_0}$ for a suitable constant $L_0$, by comparing local moments with those of correlated Erdős--Rényi graphs; for smaller $d$, by spectral estimation of the latent geometry combined with a harmonic correction. We also provide low-degree evidence for a tradeoff between the correlation and the computational cost.

math.PR↗
arXiv2026

Efficient Posterior Sampling for $\mathbb Z_2$ Synchronization

Consider the $\mathbb Z_2$ synchronization problem \[ \boldsymbol{Y} = \fracλ{\sqrt n} θθ^{\top} + \boldsymbol{Z}, \] where $θ$ is uniform on $\{-1,1\}^n$ and $\boldsymbol{Z}$ is an independent Gaussian Wigner matrix with off-diagonal variance one. We give a polynomial-time posterior sampling algorithm for every fixed $λ>1$, for which the conditional output law converges to the posterior in total variation, in expectation over the observation. The construction combines sequential TAP proposals with an independence Metropolis correction. The key is to control signed overlaps after logarithmic pinning and conditional TAP approximations along a random revealing path, which give an efficiently evaluable proposal with a polynomial density-ratio bound outside a set of vanishing posterior mass. To the best of our knowledge, this is the first polynomial-time posterior sampler for $\mathbb Z_2$ synchronization with a total-variation guarantee throughout the supercritical regime. For comparison, the diffusion-based sampler of \cite{montanari2023posterior} gives normalized Wasserstein guarantees at sufficiently large fixed signal-to-noise ratio.

math.PR↗
arXiv2026

The Conclave Process

We introduce a stochastic model for the papal conclave in which $n$ cardinals vote repeatedly among themselves until one cardinal receives all the votes. In each round, the probability that a cardinal votes for a given candidate is proportional to the $α$-th power of that candidate's vote count in the preceding round. For $α=1$, the model reduces to the Wright-Fisher model and is dual to Kingman's n-coalescent. We reveal a sharp transition in the absorption time $\mathcal{T}$ at $α=1$. It was known that when $α=1$, $\mathcal{T}$ is typically of order $n$. We prove that for $α>1$, it drops to order $\textit{loglog n.}$ In contrast, for $α<1$, $\mathcal{T}$ is typically at least $\exp(Ω(n))$. We also prove a sharp phase transition in the identity of the winner when $α>1$. For every positive integer $k$, if $2^{1/k}<α<2^{1/(k-1)}$ (where we write $2^{1/0} = +\infty$), with probability tending to 1 as $n\to\infty$, the eventual winner is the unique leader after round $k$. These results show that reinforced voting processes reach consensus remarkably quickly even for large electorates.

math.PR↗
arXiv2026

On Injectivity of Phase Retrieval

In this short note, we prove that if $A \in \mathbb C^{N \times M}$ with $N=4M-5$ has i.i.d.\ standard complex Gaussian entries, then the probability that the phase retrieval map generated by $A$ is not injective is positive. This proves Part (1) of a conjecture of Cynthia Vinzant, which was later restated by Afonso S. Bandeira in \cite{BDL+26}. The main result of this paper was obtained using generative AI, in particular the Rethlas system.

math.FA↗
arXiv2026

Robust Random Graph Matching in Dense Graphs via an Approximate Message Passing Type Algorithm

In this paper, we focus on the matching recovery problem between a pair of correlated Gaussian Wigner matrices with a latent vertex correspondence. We are particularly interested in a robust version of this problem such that our observation is a perturbed input $(A+E,B+F)$ where $(A,B)$ is a pair of correlated Gaussian Wigner matrices and $E,F$ are adversarially chosen matrices supported on an unknown $εn * εn$ principal minor of $A,B$, respectively. We propose an approximate message passing (AMP) type iterative algorithm that succeeds in polynomial time as long as the correlation $ρ$ between $(A,B)$ is a non-vanishing constant and $ε= o\big( \tfrac{1}{(\log n)^{20}} \big)$. A key distinction from standard AMP is the introduction of a time-dependent matrix multiplication step within the iteration, which simultaneously enlarges the feature dimension and cancels the correlation during the iteration. The main methodological inputs for our result are the iterative random graph matching algorithm proposed in \cite{DL22+, DL23+} and the spectral preprocessing procedure proposed in \cite{IS24+}. To the best of our knowledge, our algorithm is the first efficient random graph matching type algorithm that is robust under any adversarial perturbations of $n^{1-o(1)}$ size.

stat.ML↗
arXiv2026

Algorithmic Contiguity from Low-Degree Heuristic II: Predicting Detection-Recovery Gaps

The low-degree polynomial framework has emerged as a powerful tool for providing evidence of statistical-computational gaps in high-dimensional inference. For detection problems, the standard approach bounds the low-degree advantage through an explicit orthonormal basis. However, this method does not extend naturally to estimation tasks, and thus fails to capture the \emph{detection-recovery gap phenomenon} that arises in many high-dimensional problems. Although several important advances have been made to overcome this limitation \cite{SW22, SW25, CGGV25+}, the existing approaches often rely on delicate, model-specific combinatorial arguments. In this work, we develop a general approach for obtaining \emph{conditional computational lower bounds} for recovery problems from mild bounds on low-degree testing advantage. Our method combines the notion of algorithmic contiguity in \cite{Li25} with a cross-validation reduction in \cite{DHSS25} that converts successful recovery into a hypothesis test with lopsided success probabilities. In contrast to prior unconditional lower bounds, our argument is conceptually simple, flexible, and largely model-independent. We apply this framework to several canonical inference problems, including planted submatrix, planted dense subgraph, stochastic block model, multi-frequency angular synchronization, orthogonal group synchronization, and multi-layer stochastic block model. In the first three settings, our method recovers existing low-degree lower bounds for recovery in \cite{SW22, SW25} via a substantially simpler argument. In the latter three, it gives new evidence for conjectured computational thresholds including the persistence of detection-recovery gaps. Together, these results suggest that mild control of low-degree advantage is often sufficient to explain computational barriers for recovery in high-dimensional statistical models.

math.ST↗
arXiv2026

The Umeyama algorithm for matching correlated Gaussian geometric models in the low-dimensional regime

Motivated by the problem of matching two correlated random geometric graphs, we study the problem of matching two Gaussian geometric models correlated through a latent node permutation. Specifically, given an unknown permutation $π^*$ on $\{1,\ldots,n\}$ and given $n$ i.i.d. pairs of correlated Gaussian vectors $\{X_{π^*(i)},Y_i\}$ in $\mathbb{R}^d$ with noise parameter $σ$, we consider two types of (correlated) weighted complete graphs with edge weights given by $A_{i,j}=\langle X_i,X_j \rangle$, $B_{i,j}=\langle Y_i,Y_j \rangle$. The goal is to recover the hidden vertex correspondence $π^*$ based on the observed matrices $A$ and $B$. For the low-dimensional regime where $d=O(\log n)$, Wang, Wu, Xu, and Yolou [WWXY22+] established the information thresholds for exact and almost exact recovery in matching correlated Gaussian geometric models. They also conducted numerical experiments for the classical Umeyama algorithm. In our work, we prove that this algorithm achieves exact recovery of $π^*$ when the noise parameter $σ=o(d^{-3}n^{-2/d})$, and almost exact recovery when $σ=o(d^{-3}n^{-1/d})$. Our results approach the information thresholds up to a $\operatorname{poly}(d)$ factor in the low-dimensional regime.

math.ST↗
arXiv2026

Detection and Reconstruction of a Random Hypergraph from Noisy Graph Projection

For a $d$-uniform random hypergraph on $n$ vertices in which hyperedges are included i.i.d.\ so that the average degree in the hypergraph is $n^{δ+o(1)}$, the projection of such a hypergraph is a graph on the same $n$ vertices where an edge connects two vertices if and only if they belong to a same hyperedge. In this work, we study the inference problem where the observation is a \emph{noisy} version of the graph projection where each edge in the projection is kept with probability $p=n^{-1+α+o(1)}$ and each edge not in the projection is added with probability $q=n^{-1+β+o(1)}$. For all constant $d$, we establish sharp thresholds for both detection (distinguishing the noisy projection from an Erdős-Rényi random graph with edge density $q$) and reconstruction (estimating the original hypergraph). Notably, our results reveal a \emph{detection-reconstruction gap} phenomenon in this problem. Our work also answers a problem raised in \cite{BGPY25+}.

math.ST↗
arXiv2026

Fundamental Limits of Community Detection in Contextual Multi-Layer Stochastic Block Models

We consider the problem of community detection from the joint observation of a high-dimensional covariate matrix and $L$ sparse networks, all encoding noisy, partial information about the latent community labels of $n$ subjects. In the asymptotic regime where the networks have constant average degree and the number of features $p$ grows proportionally with $n$, we derive a sharp threshold under which detecting and estimating the subject labels is possible. Our results extend the work of \cite{MN23} to the constant-degree regime with noisy measurements, and also resolve a conjecture in \cite{YLS24+} when the number of networks is a constant. Our information-theoretic lower bound is obtained via a novel comparison inequality between Bernoulli and Gaussian moments, as well as a statistical variant of the ``recovery to chi-square divergence reduction'' argument inspired by \cite{DHSS25}. On the algorithmic side, we design efficient algorithms based on counting decorated cycles and decorated paths and prove that they achieve the sharp threshold for both detection and weak recovery. In particular, our results show that there is no statistical-computational gap in this setting.

math.ST↗
arXiv2026

Improved Computational Lower Bound of Estimation for Multi-Frequency Group Synchronization

We study the computational phase transition in a multi-frequency group synchronization problem, where pairwise relative measurements of group elements are observed across multiple frequency channels and corrupted by Gaussian noise. Using the framework of \emph{low-degree polynomial algorithms}, we analyze the task of estimating the structured signal in such observations. We show that, assuming the low-degree heuristic, in synchronization models over the circle group $\mathsf{SO}(2)$, a simple spectral method is computationally optimal among all polynomial-time estimators when the number of frequencies satisfies $L=n^{o(1)}$. This significantly extends prior work \cite{KBK24+}, which only applied to a fixed constant number of frequencies. Together with known upper bounds on the statistical threshold \cite{PWBM18a}, our results establish the existence of a \emph{statistical-to-computational gap} in this model when the number of frequencies is sufficiently large.

math.ST↗
arXiv2026

A Computational Transition for Detecting Multivariate Shuffled Linear Regression by Low-Degree Polynomials

In this paper, we study the problem of multivariate shuffled linear regression, where the correspondence between predictors and responses in a linear model is obfuscated by a latent permutation. Specifically, we investigate the model $Y=\tfrac{1}{\sqrt{1+σ^2}}(Π_* X Q_* + σZ)$, where $X$ is an $n*d$ standard Gaussian design matrix, $Z$ is an $n*m$ Gaussian noise matrix, $Π_*$ is an unknown $n*n$ permutation matrix, and $Q_*$ is an unknown $d*m$ on the Grassmanian manifold satisfying $Q_*^{\top} Q_* = \mathbb I_m$. Consider the hypothesis testing problem of distinguishing this model from the case where $X$ and $Y$ are independent Gaussian random matrices of sizes $n*d$ and $n*m$, respectively. Our results reveal a phase transition phenomenon in the performance of low-degree polynomial algorithms for this task. (1) When $m=o(d)$, we show that all degree-$D$ polynomials fail to distinguish these two models even when $σ=0$, provided with $D^4=o\big( \tfrac{d}{m} \big)$. (2) When $m=d$ and $σ=ω(1)$, we show that all degree-$D$ polynomials fail to distinguish these two models provided with $D=o(σ)$. (3) When $m=d$ and $σ=o(1)$, we show that there exists a constant-degree polynomial that strongly distinguish these two models. These results establish a smooth transition in the effectiveness of low-degree polynomial algorithms for this problem, highlighting the interplay between the dimensions $m$ and $d$, the noise level $σ$, and the computational complexity of the testing task.

stat.ML↗
arXiv2025

Computational Lower Bounds for Correlated Random Graphs via Algorithmic Contiguity

In this paper, assuming the low-degree conjecture, we provide evidence of computational hardness for two problems: (1) the (partial) matching recovery problem in the sparse correlated Erdős-Rényi graphs $\mathcal G(n,q;ρ)$ when the edge-density $q=n^{-1+o(1)}$ and the correlation $ρ<\sqrtα$ lies below the Otter's threshold, this resolves a remaining problem in \cite{DDL23+}; (2) the detection problem between a pair of correlated sparse stochastic block models $\mathcal S(n,\tfracλ{n};k,ε;s)$ and a pair of independent stochastic block models $\mathcal S(n,\tfrac{λs}{n};k,ε)$ when $ε^2 λs<1$ lies below the Kesten-Stigum (KS) threshold and $s<\sqrtα$ lies below the Otter's threshold, this resolves a remaining problem in \cite{CDGL24+}. One of the main ingredient in our proof is to derive certain forms of \emph{algorithmic contiguity} between two probability measures based on bounds on their low-degree advantage. To be more precise, consider the high-dimensional hypothesis testing problem between two probability measures $\mathbb{P}$ and $\mathbb{Q}$ based on the sample $\mathsf Y$. We show that if the low-degree advantage $\mathsf{Adv}_{\leq D} \big( \frac{\mathrm{d}\mathbb{P}}{\mathrm{d}\mathbb{Q}} \big)=O(1)$, then (assuming the low-degree conjecture) there is no efficient algorithm $\mathcal A$ such that $\mathbb{Q}(\mathcal A(\mathsf Y)=0)=1-o(1)$ and $\mathbb{P}(\mathcal A(\mathsf Y)=1)=Ω(1)$. This framework provides a useful tool for performing reductions between different inference tasks, without requiring a strengthened version of the low-degree conjecture as in \cite{MW23+, DHSS25+}.

stat.ML↗
arXiv2025

The Algorithmic Phase Transition in Correlated Spiked Models

We study the computational task of detecting and estimating correlated signals in a pair of spiked matrices $$ X=\tfracλ{\sqrt{n}} xu^{\top}+W, \quad Y=\tfracμ{\sqrt{n}} yv^{\top}+Z $$ where the spikes $x,y$ have correlation $ρ$. Specifically, we consider two fundamental models: (1) Correlated spiked Wigner model with signal-to-noise ratio $λ,μ$; (2) Correlated spiked $n*N$ Wishart (covariance) model with signal-to-noise ratio $\sqrtλ,\sqrtμ$. We propose an efficient detection and estimation algorithm based on counting a specific family of edge-decorated cycles. The algorithm's performance is governed by the function $$ F(λ,μ,ρ,γ)=\max\Big\{ \frac{ λ^2 }{ γ}, \frac{ μ^2 }{ γ}, \frac{ λ^2 ρ^2 }{ γ-λ^2+λ^2 ρ^2 } + \frac{ μ^2 ρ^2 }{ γ-μ^2+μ^2 ρ^2 } \Big\} \,. $$ We prove our algorithm succeeds for the correlated spiked Wigner model whenever $F(λ,μ,ρ,1)>1$, and succeeds for the correlated spiked Wishart model whenever $F(λ,μ,ρ,\tfrac{n}{N})>1$. Our result shows that an algorithm can leverage the correlation between the spikes to detect and estimate the signals even in regimes where efficiently recovering either $x$ from ${X}$ alone or $y$ from ${Y}$ alone is believed to be computationally infeasible. We complement our algorithmic results with evidence for a matching computational lower bound. In particular, we prove that when $F(λ,μ,ρ,1)<1$ for the correlated spiked Wigner model and when $F(λ,μ,ρ,\tfrac{n}{N})<1$ for the spiked Wishart model, all algorithms based on low-degree polynomials fails to distinguish $({X},{Y})$ with two independent noise matrices. This strongly suggests that $F=1$ is the precise computation threshold for our models.

math.ST↗
arXiv2025

Detecting Correlation Efficiently in Stochastic Block Models: Breaking Otter's Threshold in the Entire Supercritical Regime

Consider a pair of sparse correlated stochastic block models $\mathcal S(n,\tfracλ{n},ε;s)$ subsampled from a common parent stochastic block model with two symmetric communities, average degree $λ=O(1)$, divergence parameter $ε\in (0,1)$ and subsampling probability $s$. For all $ε\in(0,1)$ and $Δ>0$, we construct a statistic based on the combination of two low-degree polynomials and show that there exists a sufficiently small constant $δ=δ(ε,λ,Δ)>0$ such that if $ε^2 λs>1+Δ$ and $s>\sqrtα-δ$ where $α\approx 0.338$ is Otter's constant, this statistic can distinguish this model and a pair of independent stochastic block models $\mathcal S(n,\tfrac{λs}{n},ε)$ with probability $1-o(1)$. We also provide an efficient algorithm that approximates this statistic in polynomial time. The crux of our statistic's construction lies in a carefully curated family of multigraphs called \emph{decorated trees}, which enables effective aggregation of the community signal and graph correlation by leveraging the counts of the same decorated tree while suppressing the undesirable correlations among counts of different decorated trees. We believe such construction may be of independent interest.

cs.DS↗