Black-Box $k$-to-$1$-PCA Reductions: Theory and Applications

• URL: http://arxiv.org/abs/2403.03905v3
• Date: Tue, 11 Jun 2024 05:43:50 GMT
• Title: Black-Box $k$-to-$1$-PCA Reductions: Theory and Applications
• Authors: Arun Jambulapati, Syamantak Kumar, Jerry Li, Shourya Pandey, Ankit Pensia, Kevin Tian,
• Abstract summary: We analyze black-box deflation methods as a framework for designing $k$-PCA algorithms. Our main contribution is significantly sharper bounds on the approximation parameter degradation of deflation methods for $k$-PCA. We apply our framework to obtain state-of-the-art $k$-PCA algorithms robust to dataset contamination.
• Score: 19.714951004096996
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: The $k$-principal component analysis ($k$-PCA) problem is a fundamental algorithmic primitive that is widely-used in data analysis and dimensionality reduction applications. In statistical settings, the goal of $k$-PCA is to identify a top eigenspace of the covariance matrix of a distribution, which we only have black-box access to via samples. Motivated by these settings, we analyze black-box deflation methods as a framework for designing $k$-PCA algorithms, where we model access to the unknown target matrix via a black-box $1$-PCA oracle which returns an approximate top eigenvector, under two popular notions of approximation. Despite being arguably the most natural reduction-based approach to $k$-PCA algorithm design, such black-box methods, which recursively call a $1$-PCA oracle $k$ times, were previously poorly-understood. Our main contribution is significantly sharper bounds on the approximation parameter degradation of deflation methods for $k$-PCA. For a quadratic form notion of approximation we term ePCA (energy PCA), we show deflation methods suffer no parameter loss. For an alternative well-studied approximation notion we term cPCA (correlation PCA), we tightly characterize the parameter regimes where deflation methods are feasible. Moreover, we show that in all feasible regimes, $k$-cPCA deflation algorithms suffer no asymptotic parameter loss for any constant $k$. We apply our framework to obtain state-of-the-art $k$-PCA algorithms robust to dataset contamination, improving prior work in sample complexity by a $\mathsf{poly}(k)$ factor.

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