Low-Degree Method Fails to Predict Robust Subspace Recovery

TL;DR

This paper demonstrates that the low-degree polynomial framework fails to predict the computational complexity of a specific robust subspace recovery problem, highlighting limitations in its universality for certain high-dimensional tasks.

Contribution

It introduces a natural hypothesis testing problem where low-degree methods fail, yet a simple polynomial-time algorithm exists, challenging the low-degree conjecture's predictive power.

Findings

Low-degree moments match up to degree O(√(log n)/log log n).

The problem is polynomial-time solvable but not predicted by low-degree methods.

The results challenge the universality of low-degree methods in predicting computational hardness.

Abstract

The low-degree polynomial framework has been highly successful in predicting computational versus statistical gaps for high-dimensional problems in average-case analysis and machine learning. This success has led to the low-degree conjecture, which posits that this method captures the power and limitations of efficient algorithms for a wide class of high-dimensional statistical problems. We identify a natural and basic hypothesis testing problem in $\mathbb{R}^n$ which is polynomial time solvable, but for which the low-degree polynomial method fails to predict its computational tractability even up to degree $k=n^{\Omega(1)}$. Moreover, the low-degree moments match exactly up to degree $k=O(\sqrt{\log n/\log\log n})$. Our problem is a special case of the well-studied robust subspace recovery problem. The lower bounds suggest that there is no polynomial time algorithm for this problem. In contrast, we give a simple and robust polynomial time algorithm that solves the problem (and noisy variants of it), leveraging anti-concentration properties of the distribution. Our results suggest that the low-degree method and low-degree moments fail to capture algorithms based on anti-concentration, challenging their universality as a predictor of computational barriers.