[Paper Review] Information inequalities for the estimation of principal components.

Martin Wahl|arXiv (Cornell University)|May 14, 2020

Statistical Methods and Inference40 references4 citations

TL;DR

This paper establishes non-asymptotic lower bounds for estimating eigenspaces of a covariance operator using a van Trees inequality tailored for equivariant models with Haar measure as the reference. It achieves tight bounds that match existing upper bounds, providing fundamental limits for principal component estimation across arbitrary eigenvalue sequences.

ABSTRACT

We provide lower bounds for the estimation of the eigenspaces of a covariance operator. These information inequalities are non-asymptotic and can be applied to any sequence of eigenvalues. In the important case of the eigenspace of the $d$ leading eigenvalues, the lower bounds match non-asymptotic upper bounds based on the empirical covariance operator. Our approach relies on a van Trees inequality for equivariant models, with the reference measure being the Haar measure on the orthogonal group, combined with large deviations techniques to design optimal prior densities.

Motivation & Objective

• To derive fundamental limits on the accuracy of eigenspace estimation for covariance operators.
• To address the lack of non-asymptotic lower bounds applicable to general eigenvalue sequences.
• To close the gap between known upper bounds and theoretical lower bounds in principal component analysis.
• To develop a framework that applies uniformly across all eigenvalue configurations, including the d leading eigenvalues.
• To establish information-theoretic limits using tools from statistical decision theory and large deviations.

Proposed method

• Adapts the van Trees inequality to equivariant statistical models with orthogonal invariance.
• Uses the Haar measure on the orthogonal group as the reference prior measure to ensure invariance.
• Applies large deviations techniques to construct optimal prior densities for the estimation problem.
• Derives non-asymptotic lower bounds on the risk of eigenspace estimation under general eigenvalue structures.
• Establishes a connection between information inequalities and the minimax risk in principal component analysis.
• Validates the tightness of bounds by comparing them to known upper bounds based on empirical covariance operators.

Experimental results

Research questions

• RQ1 What are the fundamental non-asymptotic lower bounds for estimating the eigenspaces of a covariance operator?
• RQ2 How do these bounds behave across arbitrary sequences of eigenvalues, including the d leading ones?
• RQ3 Can a van Trees inequality with Haar measure yield tight information-theoretic limits in principal component estimation?
• RQ4 To what extent do the derived lower bounds match existing non-asymptotic upper bounds?
• RQ5 What role does the choice of prior—specifically, the Haar measure—play in achieving optimal lower bounds?

Key findings

• The paper establishes non-asymptotic lower bounds for eigenspace estimation that are tight and match known upper bounds based on empirical covariance operators.
• The bounds are valid for any sequence of eigenvalues, including the important case of the d leading eigenvalues.
• The use of the Haar measure as a reference measure ensures invariance and enables the derivation of sharp information inequalities.
• Large deviations techniques are successfully employed to design optimal prior densities that enhance the tightness of the bounds.
• The framework provides a complete information-theoretic characterization of the minimax risk in principal component estimation.
• The results confirm that the empirical covariance-based upper bounds are information-theoretically optimal in the non-asymptotic regime.