arXiv subjects

Zhangsong Li

Publications and source records attributed to Zhangsong Li.

At least 19 recordsLinked to original sources

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 Gaussian planted submatrix model. The observation is $$ \boldsymbol{Y}= \frac{\lambda}{\sqrt{n}} \boldsymbol{\theta} \boldsymbol{\theta}^{\top}+\boldsymbol{W}, $$ where the coordinates of $\boldsymbol{\theta}$ are independent $\mathsf{Ber}(\rho)$ variables and $\boldsymbol{W}$ is symmetric with independent standard Gaussian upper-triangular entries. For every fixed $\lambda>0$ and $\rho\in(0,1)$, 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 $\rho-q_{\mathsf{amp}}/\lambda$, the limiting error of Bayes approximate message passing (AMP). This extends the constant-degree result of Montanari and Wein~\cite{montanari2025equivalence} for the Bernoulli prior. 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 therefore resolves the Bernoulli rank-one case of the growing-degree AMP-equivalence question discussed in~\cite{wein2025computational, maleki2026high}. The proof constructs a low-degree certificate using \emph{conditional} joint cumulants of the signal coordinates and their products. Specifically, we condition on 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. Most of the arguments in this paper were generated using GPT-6 Astra.

math.ST↗

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 $\alpha$-th power of that candidate's vote count in the preceding round. For $\alpha=1$, 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 $\alpha=1$. It was known that when $\alpha=1$, $\mathcal{T}$ is typically of order $n$. We prove that for $\alpha>1$, it drops to order $\textit{loglog n.}$ In contrast, for $\alpha<1$, $\mathcal{T}$ is typically at least $\exp(\Omega(n))$. We also prove a sharp phase transition in the identity of the winner when $\alpha>1$. For every positive integer $k$, if $2^{1/k}<\alpha<2^{1/(k-1)}$ (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

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

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↗

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 $$ X=\tfrac{\lambda}{\sqrt{n}} xu^{\top}+W, \quad Y=\tfrac{\mu}{\sqrt{n}} yv^{\top}+Z $$ where the spikes $x,y$ have correlation $\rho$. Specifically, we consider two fundamental models: (1) Correlated spiked Wigner model with signal-to-noise ratio $\lambda,\mu$; (2) Correlated spiked $n*N$ Wishart (covariance) model with signal-to-noise ratio $\sqrt\lambda,\sqrt\mu$. 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 $$ F(\lambda,\mu,\rho,\gamma)=\max\Big\{ \frac{ \lambda^2 }{ \gamma }, \frac{ \mu^2 }{ \gamma }, \frac{ \lambda^2 \rho^2 }{ \gamma-\lambda^2+\lambda^2 \rho^2 } + \frac{ \mu^2 \rho^2 }{ \gamma-\mu^2+\mu^2 \rho^2 } \Big\} \,. $$ We prove our algorithm succeeds for the correlated spiked Wigner model whenever $F(\lambda,\mu,\rho,1)>1$, and succeeds for the correlated spiked Wishart model whenever $F(\lambda,\mu,\rho,\tfrac{n}{N})>1$. 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 $F(\lambda,\mu,\rho,1)<1$ for the correlated spiked Wigner model and when $F(\lambda,\mu,\rho,\tfrac{n}{N})<1$ 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

A Smooth Computational Transition in Tensor PCA

We propose an efficient algorithm for tensor PCA based on counting a specific family of weighted hypergraphs. For the order-$p$ tensor PCA problem where $p \geq 3$ is a fixed integer, we show that when the signal-to-noise ratio is $\lambda n^{-\frac{p}{4}}$ where $\lambda=\Omega(1)$, our algorithm succeeds and runs in time $n^{C+o(1)}$ where $C=C(\lambda)$ is a constant depending on $\lambda$. This algorithm improves a poly-logarithmic factor compared to previous algorithms based on the Sum-of-Squares hierarchy \cite{HSS15} or based on the Kikuchi hierarchy in statistical physics \cite{WEM19}. Furthermore, our result shows a smooth tradeoff between the signal-to-noise ratio and the computational cost in this problem, thereby confirming a conjecture posed in \cite{KWB22}.

math.ST↗

arXiv2025

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 $n^{\delta+o(1)}$, 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 $p=n^{-1+\alpha+o(1)}$ and each edge not in the projection is added with probability $q=n^{-1+\beta+o(1)}$. For all constant $d$, we establish sharp thresholds for both detection (distinguishing the noisy projection from an Erd\H{o}s-R\'enyi 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↗

arXiv2025

Asymptotic diameter of preferential attachment model

We study the asymptotic diameter of the preferential attachment model $\operatorname{PA}\!_n^{(m,\delta)}$ with parameters $m \ge 2$ and $\delta > 0$. Building on the recent work \cite{VZ25}, we prove that the diameter of $G_n \sim \operatorname{PA}\!_n^{(m,\delta)}$ is $(1+o(1))\log_\nu n$ with high probability, where $\nu$ is the exponential growth rate of the local weak limit of $G_n$. Our result confirms the conjecture in \cite{VZ25} and closes the remaining gap in understanding the asymptotic diameter of preferential attachment graphs with general parameters $m \ge 1$ and $\delta >-m$. Our proof follows a general recipe that relates the diameter of a random graph to its typical distance, which we expect to have applicability in a broader range of models.

math.PR↗

arXiv2025

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 $Y=\tfrac{1}{\sqrt{1+\sigma^2}}(\Pi_* X Q_* + \sigma Z)$, where $X$ is an $n*d$ standard Gaussian design matrix, $Z$ is an $n*m$ Gaussian noise matrix, $\Pi_*$ 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 $\sigma=0$, provided with $D^4=o\big( \tfrac{d}{m} \big)$. (2) When $m=d$ and $\sigma=\omega(1)$, we show that all degree-$D$ polynomials fail to distinguish these two models provided with $D=o(\sigma)$. (3) When $m=d$ and $\sigma=o(1)$, 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 $\sigma$, and the computational complexity of the testing task.

stat.ML↗

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 $\mathcal S(n,\tfrac{\lambda}{n},\epsilon;s)$ subsampled from a common parent stochastic block model with two symmetric communities, average degree $\lambda=O(1)$, divergence parameter $\epsilon\in (0,1)$ and subsampling probability $s$. For all $\epsilon\in(0,1)$ and $\Delta>0$, we construct a statistic based on the combination of two low-degree polynomials and show that there exists a sufficiently small constant $\delta=\delta(\epsilon,\lambda,\Delta)>0$ such that if $\epsilon^2 \lambda s>1+\Delta$ and $s>\sqrt{\alpha}-\delta$ where $\alpha\approx 0.338$ is Otter's constant, this statistic can distinguish this model and a pair of independent stochastic block models $\mathcal S(n,\tfrac{\lambda s}{n},\epsilon)$ 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↗

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 Erd\H{o}s-R\'enyi graphs $\mathcal G(n,q;\rho)$ when the edge-density $q=n^{-1+o(1)}$ and the correlation $\rho<\sqrt{\alpha}$ 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 $\mathcal S(n,\tfrac{\lambda}{n};k,\epsilon;s)$ and a pair of independent stochastic block models $\mathcal S(n,\tfrac{\lambda s}{n};k,\epsilon)$ when $\epsilon^2 \lambda s<1$ lies below the Kesten-Stigum (KS) threshold and $s<\sqrt{\alpha}$ 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 $\mathbb{P}(\mathcal A(\mathsf Y)=1)=\Omega(1)$. 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↗

arXiv2024

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 $\epsilon n * \epsilon n$ 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 $\rho$ between $(A,B)$ is a non-vanishing constant and $\epsilon = o\big( \tfrac{1}{(\log n)^{20}} \big)$. 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↗

arXiv2024

A computational transition for detecting correlated stochastic block models by low-degree polynomials

Detection of correlation in a pair of random graphs is a fundamental statistical and computational problem that has been extensively studied in recent years. In this work, we consider a pair of correlated (sparse) stochastic block models $\mathcal{S}(n,\tfrac{\lambda}{n};k,\epsilon;s)$ that are subsampled from a common parent stochastic block model $\mathcal S(n,\tfrac{\lambda}{n};k,\epsilon)$ with $k=O(1)$ symmetric communities, average degree $\lambda=O(1)$, divergence parameter $\epsilon$, and subsampling probability $s$. For the detection problem of distinguishing this model from a pair of independent Erd\H{o}s-R\'enyi graphs with the same edge density $\mathcal{G}(n,\tfrac{\lambda s}{n})$, we focus on tests based on \emph{low-degree polynomials} of the entries of the adjacency matrices, and we determine the threshold that separates the easy and hard regimes. More precisely, we show that this class of tests can distinguish these two models if and only if $s> \min \{ \sqrt{\alpha}, \frac{1}{\lambda \epsilon^2} \}$, where $\alpha\approx 0.338$ is the Otter's constant and $\frac{1}{\lambda \epsilon^2}$ is the Kesten-Stigum threshold. Combining a reduction argument in \cite{Li25+}, our hardness result also implies low-degree hardness for partial recovery and detection (to independent block models) when $s< \min \{ \sqrt{\alpha}, \frac{1}{\lambda \epsilon^2} \}$. Finally, our proof of low-degree hardness is based on a conditional variant of the low-degree likelihood calculation.

math.PR↗

arXiv2024

A polynomial-time iterative algorithm for random graph matching with non-vanishing correlation

We propose an efficient algorithm for matching two correlated Erdős--Rényi graphs with $n$ vertices whose edges are correlated through a latent vertex correspondence. When the edge density $q= n^{- α+o(1)}$ for a constant $α\in [0,1)$, we show that our algorithm has polynomial running time and succeeds to recover the latent matching as long as the edge correlation is non-vanishing. This is closely related to our previous work on a polynomial-time algorithm that matches two Gaussian Wigner matrices with non-vanishing correlation, and provides the first polynomial-time random graph matching algorithm (regardless of the regime of $q$) when the edge correlation is below the square root of the Otter's constant (which is $\approx 0.338$).

cs.DS↗

arXiv2024

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 $\pi^*$ on $\{1,\ldots,n\}$ and given $n$ i.i.d. pairs of correlated Gaussian vectors $\{X_{\pi^*(i)},Y_i\}$ in $\mathbb{R}^d$ with noise parameter $\sigma$, 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 $\pi^*$ 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 $\pi^*$ when the noise parameter $\sigma=o(d^{-3}n^{-2/d})$, and almost exact recovery when $\sigma=o(d^{-3}n^{-1/d})$. Our results approach the information thresholds up to a $\operatorname{poly}(d)$ factor in the low-dimensional regime.

math.ST↗

arXiv2023

Low-Degree Hardness of Detection for Correlated Erd\H{o}s-R\'enyi Graphs

Given two Erd\H{o}s-R\'enyi graphs with $n$ vertices whose edges are correlated through a latent vertex correspondence, we study complexity lower bounds for the associated correlation detection problem for the class of low-degree polynomial algorithms. We provide evidence that any degree-$O(\rho^{-1})$ polynomial algorithm fails for detection, where $\rho$ is the edge correlation. Furthermore, in the sparse regime where the edge density $q=n^{-1+o(1)}$, we provide evidence that any degree-$d$ polynomial algorithm fails for detection, as long as $\log d=o\big( \frac{\log n}{\log nq} \wedge \sqrt{\log n} \big)$ and the correlation $\rho<\sqrt{\alpha}$ where $\alpha\approx 0.338$ is the Otter's constant. Our result suggests that several state-of-the-art algorithms on correlation detection and exact matching recovery may be essentially the best possible.

cs.DS↗