Global Convergence of Adaptive Sensing for Principal Eigenvector Estimation

Abstract

Principal component analysis classically requires full $d$-dimensional samples, yet in various applications hardware limits acquisition to a few scalar measurements per sample. We analyze a compressed variant of Oja's algorithm for estimating the principal eigenvector of the data covariance matrix using only two adaptive measurements per sample. At each iteration, we observe one measurement along the current estimate and one in a random orthogonal direction. We prove that after $t$ iterations, the expected sine-squared error to the true eigenvector is $\mathcal{O}(\lambda_1\lambda_2 d^2 / (\Delta^2 t))$, where $d$ is the ambient dimension, $\lambda_1, \lambda_2$ are the leading eigenvalues, and $\Delta = \lambda_1 - \lambda_2$ is the eigengap. We complement this with a matching information-theoretic lower bound of $\Omega(\lambda_1\lambda_2 d^2 / (\Delta^2 t))$ -- the first for compressed eigenvector estimation -- proving that the $d^2$ factor, an additional factor of $d$ compared to the fully-observed minimax rate $\Theta(\lambda_1\lambda_2 d / (\Delta^2 t))$, is the fundamental cost of compression and cannot be improved. In contrast, any non-adaptive scheme with two measurements per iteration suffers $\Omega(\lambda_2^2 d^3 / (\Delta^2 t))$, an additional power of $d$. This separates fully-observed, adaptive-compressed, and non-adaptive-compressed PCA across three powers of $d$. Our analysis handles the noisy setting where the covariance has nonzero trailing eigenvalues, providing the first convergence guarantee for adaptive compressed subspace tracking beyond the noiseless case.