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.

In this paper, assuming a natural strengthening of 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, solving a remaining problem in \cite{DDL23+}; (2) the detection problem between the correlated sparse stochastic block model $\mathcal S(n,\tfrac{\lambda}{n};k,\epsilon;s)$ 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, solving 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.

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 $\epsilon n * \epsilon n$ principle minor of $A,B$, respectively. We propose a vector-approximate message passing (vector-AMP) algorithm that succeeds in polynomial time as long as the correlation $\rho$ between $(A,B)$ is a non-vanishing constant and $\epsilon = o\big( \tfrac{1}{(\log n)^{20}} \big)$. The main methodological inputs for our result are the iterative random graph matching algorithm proposed in \cite{DL22+, DL23+} and the spectral cleaning 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.

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.

We propose an efficient algorithm for matching two correlated Erd\H{o}s--R\'enyi graphs with $n$ vertices whose edges are correlated through a latent vertex correspondence. When the edge density $q= n^{- \alpha+o(1)}$ for a constant $\alpha \in [0,1)$, 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 $\approx 0.338$).

Motivated by the problem of matching vertices in two correlated Erd\H{o}s-R\'enyi graphs, we study the problem of matching two correlated Gaussian Wigner matrices. We propose an iterative matching algorithm, which succeeds in polynomial time as long as the correlation between the two Gaussian matrices does not vanish. Our result is the first polynomial time algorithm that solves a graph matching type of problem when the correlation is an arbitrarily small constant.