Spiked Covariance Estimation from Modulo-Reduced Measurements

Consider the rank-1 spiked model: X = √νξu + Z, where ν is the spike intensity, u ∈ S^k-1 is an unknown direction and ξ ∼ N(0, 1), Z ∼ N(0, I). Motivated by recent advances in analog-to-digital conversion, we study the problem of recovering u ∈ S^k-1 from n i.i.d. modulo-reduced measurements Y = [X] mod ∆, focusing on the high-dimensional regime (k ≫ 1). We develop and analyze an algorithm that, for most directions u and ν = poly(k), estimates u to high accuracy using n = poly(k) measurements, provided that ∆ & √log k. Up to constants, our algorithm accurately estimates u at the smallest possible ∆ 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 ≈ √ν in a random direction u passes near a point in the lattice ∆Z^k. Numerical experiments show that the developed algorithm performs well even in a non-asymptotic setting.