Spiked Covariance Estimation from ModuloReduced Measurements
Abstract
Consider the rank1 spiked model: $\bf{X}=\sqrt{\nu}\xi \bf{u}+ \bf{Z}$, where $\nu$ is the spike intensity, $\bf{u}\in\mathbb{S}^{k1}$ is an unknown direction and $\xi\sim \mathcal{N}(0,1),\bf{Z}\sim \mathcal{N}(\bf{0},\bf{I})$. Motivated by recent advances in analogtodigital conversion, we study the problem of recovering $\bf{u}\in \mathbb{S}^{k1}$ from $n$ i.i.d. moduloreduced measurements $\bf{Y}=[\bf{X}]\mod \Delta$, focusing on the highdimensional regime ($k\gg 1$). We develop and analyze an algorithm that, for most directions $\bf{u}$ and $\nu=\mathrm{poly}(k)$, estimates $\bf{u}$ to high accuracy using $n=\mathrm{poly}(k)$ measurements, provided that $\Delta\gtrsim \sqrt{\log k}$. Up to constants, our algorithm accurately estimates $\bf{u}$ at the smallest possible $\Delta$ that allows (in an informationtheoretic sense) to recover $\bf{X}$ from $\bf{Y}$. A key step in our analysis involves estimating the probability that a line segment of length $\approx\sqrt{\nu}$ in a random direction $\bf{u}$ passes near a point in the lattice $\Delta \mathbb{Z}^k$. Numerical experiments show that the developed algorithm performs well even in a nonasymptotic setting.
 Publication:

arXiv eprints
 Pub Date:
 October 2021
 arXiv:
 arXiv:2110.01150
 Bibcode:
 2021arXiv211001150R
 Keywords:

 Computer Science  Information Theory;
 Mathematics  Statistics Theory;
 Statistics  Machine Learning
 EPrint:
 AISTATS, 2022