Equivalence of Approximate Message Passing and Low-Degree Polynomials in Rank-One Matrix Estimation

TL;DR

This paper investigates the fundamental limits of polynomial-time algorithms in rank-one matrix estimation, showing that approximate message passing (AMP) algorithms are essentially optimal among low-degree polynomial estimators.

Contribution

The paper proves that no low-degree polynomial estimator can outperform AMP in rank-one matrix estimation, supporting the conjecture that AMP achieves optimal polynomial-time accuracy.

Findings

AMP algorithms are asymptotically optimal among low-degree polynomials.

Polynomial estimators cannot surpass the accuracy of AMP.

Supports the conjecture of the statistical-computation gap in this problem.

Abstract

We consider the problem of estimating an unknown parameter vector ${\boldsymbol \theta}\in{\mathbb R}^n$, given noisy observations ${\boldsymbol Y} = {\boldsymbol \theta}{\boldsymbol \theta}^{\top}/\sqrt{n}+{\boldsymbol Z}$ of the rank-one matrix ${\boldsymbol \theta}{\boldsymbol \theta}^{\top}$, where ${\boldsymbol Z}$ has independent Gaussian entries. When information is available about the distribution of the entries of ${\boldsymbol \theta}$, spectral methods are known to be strictly sub-optimal. Past work characterized the asymptotics of the accuracy achieved by the optimal estimator. However, no polynomial-time estimator is known that achieves this accuracy. It has been conjectured that this statistical-computation gap is fundamental, and moreover that the optimal accuracy achievable by polynomial-time estimators coincides with the accuracy achieved by certain approximate message passing (AMP) algorithms. We provide evidence towards this conjecture by proving that no estimator in the (broader) class of constant-degree polynomials can surpass AMP.