Computational and statistical lower bounds for low-rank estimation under general inhomogeneous noise
This work addresses fundamental limits in statistical inference for researchers in machine learning and theoretical computer science, but it is incremental as it builds on prior results on inhomogeneous noise models.
The paper studies computational and statistical limits for low-rank estimation under general inhomogeneous noise, showing that when a spectral algorithm fails to detect a signal, no low-degree polynomial algorithm can succeed, providing evidence for a computational hardness conjecture. It also proves sharp information-theoretic lower bounds for new signal distributions and suggests the spectral algorithm might be optimal for general noise profiles beyond block structures.
Recent work has generalized several results concerning the well-understood spiked Wigner matrix model of a low-rank signal matrix corrupted by additive i.i.d. Gaussian noise to the inhomogeneous case, where the noise has a variance profile. In particular, for the special case where the variance profile has a block structure, a series of results identified an effective spectral algorithm for detecting and estimating the signal, identified the threshold signal strength required for that algorithm to succeed, and proved information-theoretic lower bounds that, for some special signal distributions, match the above threshold. We complement these results by studying the computational optimality of this spectral algorithm. Namely, we show that, for a much broader range of signal distributions, whenever the spectral algorithm cannot detect a low-rank signal, then neither can any low-degree polynomial algorithm. This gives the first evidence for a computational hardness conjecture of Guionnet, Ko, Krzakala, and Zdeborová (2023). With similar techniques, we also prove sharp information-theoretic lower bounds for a class of signal distributions not treated by prior work. Unlike all of the above results on inhomogeneous models, our results do not assume that the variance profile has a block structure, and suggest that the same spectral algorithm might remain optimal for quite general profiles. We include a numerical study of this claim for an example of a smoothly-varying rather than piecewise-constant profile. Our proofs involve analyzing the graph sums of a matrix, which also appear in free and traffic probability, but we require new bounds on these quantities that are tighter than existing ones for non-negative matrices, which may be of independent interest.