Spiked Covariance Estimation from Modulo-Reduced Measurements

Abstract

Consider the rank-1 spiked model: X = √ u + Z, where is the spike intensity, u ∈ −1 is an unknown direction and ∼ (0, 1), Z ∼ (0, I). Motivated by recent advances in analog-to-digital conversion, we study the problem of recovering u ∈focusing on the high-dimensional regime ( 1). We develop and analyze an algorithm that, for most directions u and = poly( ), estimates u to high accuracy using = poly( ) measurements, provided that Δ log . Up to constants, our algorithm accurately estimates u at the smallest possible Δ that allows (in an information-theoretic sense) to recover X from Y. A key step in our analysis involves estimating the probability that a line segment of length ≈ √ in a random direction u passes near a point in the lattice Δℤ . Numerical experiments show that the developed algorithm performs well even in a non-asymptotic setting.