Speaker: Joshua S. Agterberg, Assistant Professor of Statistics, University of Illinois

Title: Statistically and Computationally Optimal Estimation and Inference in the Common Subspace Model

Abstract: Given multiple data matrices, many problems in statistics and data science rely on estimating a common subspace that captures certain structure shared by all the data matrices. In this paper we investigate the statistical and computational limits for the common subspace model in which one observes a collection of symmetric low-rank matrices perturbed by noise, where each low-rank matrix shares the same common subspace. Our main results identify several regimes of the signal-to-noise ratio (SNR) such that estimation and inference is statistically or computationally optimal, and we refer to these regimes as weak SNR, moderate SNR, strong estimation SNR, and strong inference SNR. First, we propose an estimator based on projected gradient descent initialized via spectral sum of squares and show that it achieves the optimal $\sin\Theta$ error rate under strong inference SNR, further attaining the asymptotically optimal constant under Gaussian noise. These results are complemented by both statistical and computational lower bounds identifying the weak and moderate SNR regimes. Next, we turn to statistical inference for the $\sin\Theta$ distance itself, and we show that our estimator has an asymptotically Gaussian distribution under strong inference SNR. Based on this limiting result we propose confidence intervals and show that they are minimax optimal in all regimes, though the resulting confidence intervals require knowledge of the SNR. However, we then show that in SNR-adaptive confidence intervals are information-theoretically impossible below the strong inference SNR. Consequently, our results unveil a novel phenomenon: despite the SNR being "above'' the computational limit for estimation, adaptive statistical inference may still be information-theoretically impossible.