Global Convergence of Adaptive Sensing for Principal Eigenvector Estimation

TL;DR

This paper proves that an adaptive sensing variant of Oja's algorithm can efficiently and reliably estimate the principal eigenvector in high-dimensional noisy data streams, with convergence guarantees and reduced measurement costs.

Contribution

It provides the first theoretical convergence guarantees for adaptive sensing in noisy subspace tracking, with a simplified proof technique and practical implications.

Findings

Achieves global convergence with noisy data

Requires fewer measurements per iteration

Converges at a rate comparable to minimax bounds

Abstract

This paper addresses the challenge of efficient principal component analysis (PCA) in high-dimensional spaces by analyzing a compressively sampled variant of Oja's algorithm with adaptive sensing. Traditional PCA methods incur substantial computational costs that scale poorly with data dimensionality, whereas subspace tracking algorithms like Oja's offer more efficient alternatives but typically require full-dimensional observations. We analyze a variant where, at each iteration, only two compressed measurements are taken: one in the direction of the current estimate and one in a random orthogonal direction. We prove that this adaptive sensing approach achieves global convergence in the presence of noise when tracking the leading eigenvector of a datastream with eigengap $\Delta=\lambda_1-\lambda_2$. Our theoretical analysis demonstrates that the algorithm experiences two phases: (1) a warmup phase requiring $O(\lambda_1\lambda_2d^2/\Delta^2)$ iterations to achieve a constant-level alignment with the true eigenvector, followed by (2) a local convergence phase where the sine alignment error decays at a rate of $O(\lambda_1\lambda_2d^2/\Delta^2 t)$ for iterations $t$. The guarantee aligns with existing minimax lower bounds with an added factor of $d$ due to the compressive sampling. This work provides the first convergence guarantees in adaptive sensing for subspace tracking with noise. Our proof technique is also considerably simpler than those in prior works. The results have important implications for applications where acquiring full-dimensional samples is challenging or costly.