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Global Convergence of Adaptive Sensing for Principal Eigenvector Estimation

Alex Saad-Falcon, Brighton Ancelin, Justin Romberg

TL;DR

The paper tackles efficient principal component analysis in high dimensions by proposing a compressive, adaptive sensing variant of Oja's algorithm that uses two measurements per iteration: one along the current eigenvector estimate and one in a random orthogonal direction. It proves global convergence to the leading eigenvector under noise, characterizing a two-phase dynamic consisting of a warmup period of $t_0=O(λ_1λ_2 d^2/Δ^2)$ and a local phase with sine-alignment decay at rate $O( (λ_1λ_2 d^2)/(Δ^2 t) )$, up to a $d$-factor penalty due to compression. The main contributions are the first convergence guarantees for adaptive sensing in noisy subspace tracking and a simplified proof technique that connects compressive measurements to existing minimax bounds. The results hold practical significance for applications where obtaining full-dimension measurements is costly or impractical, such as radar, MRI, genomics, and large-scale recommender systems, enabling scalable, adaptive PCA with provable performance. Overall, the work advances theory and practice of online, resource-efficient subspace tracking under realistic measurement constraints.

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 $Δ=λ_1-λ_2$. Our theoretical analysis demonstrates that the algorithm experiences two phases: (1) a warmup phase requiring $O(λ_1λ_2d^2/Δ^2)$ 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 $O(λ_1λ_2d^2/Δ^2 t)$ 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.

Global Convergence of Adaptive Sensing for Principal Eigenvector Estimation

TL;DR

The paper tackles efficient principal component analysis in high dimensions by proposing a compressive, adaptive sensing variant of Oja's algorithm that uses two measurements per iteration: one along the current eigenvector estimate and one in a random orthogonal direction. It proves global convergence to the leading eigenvector under noise, characterizing a two-phase dynamic consisting of a warmup period of and a local phase with sine-alignment decay at rate , up to a -factor penalty due to compression. The main contributions are the first convergence guarantees for adaptive sensing in noisy subspace tracking and a simplified proof technique that connects compressive measurements to existing minimax bounds. The results hold practical significance for applications where obtaining full-dimension measurements is costly or impractical, such as radar, MRI, genomics, and large-scale recommender systems, enabling scalable, adaptive PCA with provable performance. Overall, the work advances theory and practice of online, resource-efficient subspace tracking under realistic measurement constraints.

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 . Our theoretical analysis demonstrates that the algorithm experiences two phases: (1) a warmup phase requiring 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 for iterations . The guarantee aligns with existing minimax lower bounds with an added factor of 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.
Paper Structure (33 sections, 2 theorems, 124 equations, 2 figures, 1 algorithm)

This paper contains 33 sections, 2 theorems, 124 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Principal Component Analysis and Subspace Tracking
  4. Existing Convergence Bounds
  5. Compressive Measurements and Adaptive Sensing
  6. Problem Formulation
  7. Problem Statement
  8. Methods
  9. Self-term bound
  10. Cross-term bound
  11. Combining Bounds and Constructing Recurrence
  12. Warmup Phase Proof Sketch
  13. Local Phase Proof Sketch
  14. Results
  15. Tracking a Moving Eigenvector
  16. ...and 18 more sections

Key Result

Theorem 1

(informal) Let $\boldsymbol{u}_0$ be a random unit vector in $\mathbb{R}^d$ and consider measurements drawn as $\boldsymbol{v}_t\sim \mathcal{N}(\boldsymbol{0}, \boldsymbol{\Sigma})$. Let the covariance matrix $\boldsymbol{\Sigma}$ have first and second largest eigenvalues $\lambda_1$ and $\lambda_2

Figures (2)

  • Figure 1: Empirical results of applying Algorithm \ref{['alg:oja_comp']} compared to the convergence bound in Theorem \ref{['thm:formal']}. We also compare against the fully sampled version of Oja's algorithm which does not use compressive measurements. The transition between the warmup and local convergence phase is well-characterized, and our theorem shows a conservative upper bound to the performance of the adaptive sensing algorithm. We simulate data with an eigengap of $\Delta=1$ in dimension $d=10$ with the learning rate schedule defined by Theorem \ref{['thm:formal']}. Error bars show the 20th and 80th percentiles of the 20 trials.
  • Figure 2: A comparison between the predicted fixed point (Equation \ref{['eq:fixed']}) and the empirically observed fixed point when running Algorithm \ref{['alg:oja_comp']}. Our fixed point estimate is slightly conservative compared to the empirical fixed point. We simulate data with an eigengap of $\Delta=1$ in dimension $d=10$ with the learning rate defined by Equation \ref{['eq:fixed_lr']}. Error bars show the 20th and 80th percentiles.

Theorems & Definitions (2)

  • Theorem 1
  • Theorem 2