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

Spiked Covariance Estimation from Modulo-Reduced Measurements