Low-Degree Method Fails to Predict Robust Subspace Recovery

• URL: http://arxiv.org/abs/2603.02594v1
• Date: Tue, 03 Mar 2026 04:42:13 GMT
• Title: Low-Degree Method Fails to Predict Robust Subspace Recovery
• Authors: He Jia, Aravindan Vijayaraghavan,
• Abstract summary: We show that the low-degree method and low-degree moments fail to capture algorithms based on anti-concentration.<n>Our results challenge the universality as a predictor of computational barriers.
• Score: 11.09360259927697
• License: http://creativecommons.org/licenses/by/4.0/
• 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 $k=n^{Ω(1)}$. 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.

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