Generative Principal Component Analysis

TL;DR

This paper introduces a generative model-based approach to principal component analysis, proposing a quadratic estimator and a modified power method that achieve near-optimal statistical rates and fast convergence, with demonstrated improvements on image datasets.

Contribution

The paper develops a generative PCA framework with a quadratic estimator and a novel power method variant, providing theoretical guarantees and empirical validation.

Findings

Quadratic estimator achieves a statistical rate of √(k log L / m).

Modified power method converges exponentially under certain conditions.

Method outperforms classic power methods on image datasets.

Abstract

In this paper, we study the problem of principal component analysis with generative modeling assumptions, adopting a general model for the observed matrix that encompasses notable special cases, including spiked matrix recovery and phase retrieval. The key assumption is that the underlying signal lies near the range of an $L$-Lipschitz continuous generative model with bounded $k$-dimensional inputs. We propose a quadratic estimator, and show that it enjoys a statistical rate of order $\sqrt{\frac{k\log L}{m}}$, where $m$ is the number of samples. We also provide a near-matching algorithm-independent lower bound. Moreover, we provide a variant of the classic power method, which projects the calculated data onto the range of the generative model during each iteration. We show that under suitable conditions, this method converges exponentially fast to a point achieving the above-mentioned statistical rate. We perform experiments on various image datasets for spiked matrix and phase retrieval models, and illustrate performance gains of our method to the classic power method and the truncated power method devised for sparse principal component analysis.