The Algorithmic Phase Transition in Symmetric Correlated Spiked Wigner Model

• URL: http://arxiv.org/abs/2511.06040v2
• Date: Fri, 14 Nov 2025 01:35:57 GMT
• Title: The Algorithmic Phase Transition in Symmetric Correlated Spiked Wigner Model
• Authors: Zhangsong Li,
• Abstract summary: We study the computational task of detecting and estimating correlated signals in a pair of spiked Wigner matrices.<n>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.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We study the computational task of detecting and estimating correlated signals in a pair of spiked Wigner matrices. Our model consists of observations $$ X = \tfracλ{\sqrt{n}} xx^{\top} + W \,, \quad Y = \tfracμ{\sqrt{n}} yy^{\top} + Z \,. $$ where $x,y \in \mathbb R^n$ are signal vectors with norm $\|x\|,\|y\| \approx\sqrt{n}$ and correlation $\langle x,y \rangle \approx ρ\|x\|\|y\|$, while $W,Z$ are independent Gaussian Wigner matrices. We propose an efficient algorithm that succeeds whenever $F(λ,μ,ρ)>1$, where $$ F(λ,μ,ρ)=\max\Big\{ λ,μ, \frac{ λ^2 ρ^2 }{ 1-λ^2+λ^2 ρ^2 } + \frac{ μ^2 ρ^2 }{ 1-μ^2+μ^2 ρ^2 } \Big\} \,. $$ 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 result with evidence for a matching computational lower bound. In particular, we prove that when $F(λ,μ,ρ)<1$, all algorithms based on {\em low-degree polynomials} fails to distinguish $(X,Y)$ with two independent Wigner matrices. This low-degree analysis strongly suggests that $F(λ,μ,ρ)=1$ is the precise computation threshold for this problem.

Related papers

• Information-Computation Tradeoffs for Noiseless Linear Regression with Oblivious Contamination [65.37519531362157]
  We show that any efficient Statistical Query algorithm for this task requires VSTAT complexity at least $tildeOmega(d1/2/alpha2)$.
  arXiv  Detail & Related papers  (2025-10-12T15:42:44Z)
• Algorithmic contiguity from low-degree conjecture and applications in correlated random graphs [0.0]
  We provide evidence of computational hardness for two problems.<n>One of the main ingredient in our proof is to derive certain forms of emphalgorithm contiguity between two probability measures.<n>This framework provides a useful tool for performing reductions between different tasks.
  arXiv  Detail & Related papers  (2025-02-14T00:24:51Z)
• Robust random graph matching in Gaussian models via vector approximate message passing [0.0]
  We propose an efficient random graph matching type algorithm that is robust under any adversarial perturbations of $n1-o(1)$ size.<n>Our algorithm is the first efficient random graph matching type algorithm that is robust under any adversarial perturbations of $n1-o(1)$ size.
  arXiv  Detail & Related papers  (2024-12-21T03:15:38Z)
• The Communication Complexity of Approximating Matrix Rank [50.6867896228563]
  We show that this problem has randomized communication complexity $Omega(frac1kcdot n2log|mathbbF|)$. As an application, we obtain an $Omega(frac1kcdot n2log|mathbbF|)$ space lower bound for any streaming algorithm with $k$ passes.
  arXiv  Detail & Related papers  (2024-10-26T06:21:42Z)
• A computational transition for detecting correlated stochastic block models by low-degree polynomials [13.396246336911842]
  We consider a pair of correlated block models $mathcalS(n,tfraclambdan;k,epsilon;s)$ that are subsampled from a common parent block model $mathcal S(n,tfraclambdan;k,epsilon)$.<n>For the detection problem of distinguishing this model from a pair of independent ErdHos'enyi graphs, we focus on tests based on emphlow-degrees of the entries of the adjacency
  arXiv  Detail & Related papers  (2024-09-02T06:14:05Z)
• Provably learning a multi-head attention layer [55.2904547651831]
  Multi-head attention layer is one of the key components of the transformer architecture that sets it apart from traditional feed-forward models. In this work, we initiate the study of provably learning a multi-head attention layer from random examples. We prove computational lower bounds showing that in the worst case, exponential dependence on $m$ is unavoidable.
  arXiv  Detail & Related papers  (2024-02-06T15:39:09Z)
• Solving Dense Linear Systems Faster Than via Preconditioning [1.8854491183340518]
  We show that our algorithm has an $tilde O(n2)$ when $k=O(n0.729)$. In particular, our algorithm has an $tilde O(n2)$ when $k=O(n0.729)$. Our main algorithm can be viewed as a randomized block coordinate descent method.
  arXiv  Detail & Related papers  (2023-12-14T12:53:34Z)
• Randomized and Deterministic Attention Sparsification Algorithms for Over-parameterized Feature Dimension [18.57735939471469]
  We consider the sparsification of the attention problem. For any super large feature dimension, we can reduce it down to the size nearly linear in length of sentence.
  arXiv  Detail & Related papers  (2023-04-10T05:52:38Z)
• Learning a Single Neuron with Adversarial Label Noise via Gradient Descent [50.659479930171585]
  We study a function of the form $mathbfxmapstosigma(mathbfwcdotmathbfx)$ for monotone activations. The goal of the learner is to output a hypothesis vector $mathbfw$ that $F(mathbbw)=C, epsilon$ with high probability.
  arXiv  Detail & Related papers  (2022-06-17T17:55:43Z)
• Spectral properties of sample covariance matrices arising from random matrices with independent non identically distributed columns [50.053491972003656]
  It was previously shown that the functionals $texttr(AR(z))$, for $R(z) = (frac1nXXT- zI_p)-1$ and $Ain mathcal M_p$ deterministic, have a standard deviation of order $O(|A|_* / sqrt n)$. Here, we show that $|mathbb E[R(z)] - tilde R(z)|_F
  arXiv  Detail & Related papers  (2021-09-06T14:21:43Z)
• The Complexity of Sparse Tensor PCA [1.90365714903665]
  For any $1 leq leq k$, our algorithms recover the sparse vector for signal-to-noise ratio $lambda geq tildemathcalO (sqrtt cdot (k/t)p/2)$ in time. Even in the restricted case of PCA, known algorithms only recover the sparse vectors for $lambda geq tildemathcalO(k cdot r) while our algorithms require $lambda ge
  arXiv  Detail & Related papers  (2021-06-11T10:57:00Z)
• Optimal Spectral Recovery of a Planted Vector in a Subspace [80.02218763267992]
  We study efficient estimation and detection of a planted vector $v$ whose $ell_4$ norm differs from that of a Gaussian vector with the same $ell$ norm. We show that in the regime $n rho gg sqrtN$, any spectral method from a large class (and more generally, any low-degree of the input) fails to detect the planted vector.
  arXiv  Detail & Related papers  (2021-05-31T16:10:49Z)
• The Average-Case Time Complexity of Certifying the Restricted Isometry Property [66.65353643599899]
  In compressed sensing, the restricted isometry property (RIP) on $M times N$ sensing matrices guarantees efficient reconstruction of sparse vectors. We investigate the exact average-case time complexity of certifying the RIP property for $Mtimes N$ matrices with i.i.d. $mathcalN(0,1/M)$ entries.
  arXiv  Detail & Related papers  (2020-05-22T16:55:01Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.