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Spiked Covariance Estimation from Modulo-Reduced Measurements

Consider the rank-1 spiked model: $\bf{X}=\sqrtνξ\bf{u}+ \bf{Z}$, where $ν$ is the spike intensity, $\bf{u}\in\mathbb{S}^{k-1}$ is an unknown direction and $ξ\sim \mathcal{N}(0,1),\bf{Z}\sim \mathcal{N}(\bf{0},\bf{I})$. Motivated by recent advances in analog-to-digital conversion, we study the problem of recovering $\bf{u}\in \mathbb{S}^{k-1}$ from $n$ i.i.d. modulo-reduced measurements $\bf{Y}=[\bf{X}]\mod Δ$, focusing on the high-dimensional regime ($k\gg 1$). We develop and analyze an algorithm that, for most directions $\bf{u}$ and $ν=\mathrm{poly}(k)$, estimates $\bf{u}$ to high accuracy using $n=\mathrm{poly}(k)$ measurements, provided that $Δ\gtrsim \sqrt{\log k}$. Up to constants, our algorithm accurately estimates $\bf{u}$ at the smallest possible $Δ$ that allows (in an information-theoretic sense) to recover $\bf{X}$ from $\bf{Y}$. A key step in our analysis involves estimating the probability that a line segment of length $\approx\sqrtν$ in a random direction $\bf{u}$ passes near a point in the lattice $Δ\mathbb{Z}^k$. Numerical experiments show that the developed algorithm performs well even in a non-asymptotic setting.