Almost-Optimal Local-Search Methods for Sparse Tensor PCA

• URL: http://arxiv.org/abs/2506.09959v1
• Date: Wed, 11 Jun 2025 17:33:25 GMT
• Title: Almost-Optimal Local-Search Methods for Sparse Tensor PCA
• Authors: Max Lovig, Conor Sheehan, Konstantinos Tsirkas, Ilias Zadik,
• Abstract summary: Local-search methods are widely employed in statistical applications.<n>Recent studies have revealed a significant "local-computational" gap.<n>We propose a series of local-search methods that provably "close" this gap.
• Score: 10.249620437941
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Local-search methods are widely employed in statistical applications, yet interestingly, their theoretical foundations remain rather underexplored, compared to other classes of estimators such as low-degree polynomials and spectral methods. Of note, among the few existing results recent studies have revealed a significant "local-computational" gap in the context of a well-studied sparse tensor principal component analysis (PCA), where a broad class of local Markov chain methods exhibits a notable underperformance relative to other polynomial-time algorithms. In this work, we propose a series of local-search methods that provably "close" this gap to the best known polynomial-time procedures in multiple regimes of the model, including and going beyond the previously studied regimes in which the broad family of local Markov chain methods underperforms. Our framework includes: (1) standard greedy and randomized greedy algorithms applied to the (regularized) posterior of the model; and (2) novel random-threshold variants, in which the randomized greedy algorithm accepts a proposed transition if and only if the corresponding change in the Hamiltonian exceeds a random Gaussian threshold-rather that if and only if it is positive, as is customary. The introduction of the random thresholds enables a tight mathematical analysis of the randomized greedy algorithm's trajectory by crucially breaking the dependencies between the iterations, and could be of independent interest to the community.

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