Paper
Low-Degree Method Fails to Predict Robust Subspace Recovery
Authors
He Jia, Aravindan Vijayaraghavan
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.
Metadata
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Raw Data (Debug)
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"raw_xml": "<entry>\n <id>http://arxiv.org/abs/2603.02594v1</id>\n <title>Low-Degree Method Fails to Predict Robust Subspace Recovery</title>\n <updated>2026-03-03T04:42:13Z</updated>\n <link href='https://arxiv.org/abs/2603.02594v1' rel='alternate' type='text/html'/>\n <link href='https://arxiv.org/pdf/2603.02594v1' rel='related' title='pdf' type='application/pdf'/>\n <summary>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.</summary>\n <category scheme='http://arxiv.org/schemas/atom' term='stat.ML'/>\n <category scheme='http://arxiv.org/schemas/atom' term='cs.CC'/>\n <category scheme='http://arxiv.org/schemas/atom' term='cs.DS'/>\n <category scheme='http://arxiv.org/schemas/atom' term='cs.LG'/>\n <published>2026-03-03T04:42:13Z</published>\n <arxiv:comment>27 pages, 1 figure</arxiv:comment>\n <arxiv:primary_category term='stat.ML'/>\n <author>\n <name>He Jia</name>\n </author>\n <author>\n <name>Aravindan Vijayaraghavan</name>\n </author>\n </entry>"
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