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We consider the problem of estimating an unknown parameter vector${\boldsymbol \theta}\in{\mathbb R}^n$, given noisy observations ${\boldsymbolY} = {\boldsymbol \theta}{\boldsymbol \theta}^{\top}/\sqrt{n}+{\boldsymbol Z}$of the rank-one matrix ${\boldsymbol \theta}{\boldsymbol \theta}^{\top}$, where${\boldsymbol Z}$ has independent Gaussian entries. When information isavailable about the distribution of the entries of ${\boldsymbol theta}$,spectral methods are known to be strictly sub-optimal. Past work characterizedthe asymptotics of the accuracy achieved by the optimal estimator. However, nopolynomial-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 estimatorscoincides with the accuracy achieved by certain approximate message passing(AMP) algorithms. We provide evidence towards this conjecture by proving thatno estimator in the (broader) class of constant-degree polynomials can surpassAMP.

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