Black-Field $okay$-to-$1$-PCA Reductions: Concept and Functions


arXiv:2403.03905v1 Announce Kind: cross
Summary: The $okay$-principal element evaluation ($okay$-PCA) downside is a elementary algorithmic primitive that’s widely-used in knowledge evaluation and dimensionality discount purposes. In statistical settings, the purpose of $okay$-PCA is to determine a high eigenspace of the covariance matrix of a distribution, which we solely have implicit entry to through samples. Motivated by these implicit settings, we analyze black-box deflation strategies as a framework for designing $okay$-PCA algorithms, the place we mannequin entry to the unknown goal matrix through a black-box $1$-PCA oracle which returns an approximate high eigenvector, beneath two fashionable notions of approximation. Regardless of being arguably essentially the most pure reduction-based strategy to $okay$-PCA algorithm design, such black-box strategies, which recursively name a $1$-PCA oracle $okay$ occasions, had been beforehand poorly-understood.
Our predominant contribution is considerably sharper bounds on the approximation parameter degradation of deflation strategies for $okay$-PCA. For a quadratic kind notion of approximation we time period ePCA (power PCA), we present deflation strategies undergo no parameter loss. For an alternate well-studied approximation notion we time period cPCA (correlation PCA), we tightly characterize the parameter regimes the place deflation strategies are possible. Furthermore, we present that in all possible regimes, $okay$-cPCA deflation algorithms undergo no asymptotic parameter loss for any fixed $okay$. We apply our framework to acquire state-of-the-art $okay$-PCA algorithms sturdy to dataset contamination, enhancing prior work each in pattern complexity and approximation high quality.

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