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 unknown direction 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 the problem 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 the high-dimensional regime ($k\gg 1$). We develop and analyze an algorithm that, for most directions $\bf{u}$ and $\nu=\mathrm{poly}(k)$, estimates $\bf{u}$ 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 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 $\approx\sqrt{\nu}$ 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 non-asymptotic setting.

翻译：考虑一级 - 1 的升降模式 : $\ bf{ X\ qqrt} 美元 \ bf{ little{ { little{ little{ little{ u\ f} $, 其中美元是峰值的峰值, $\ fr} S\ k-1} 是一个未知的方向, $\x\ sml= mathcr{ N}, 美元是=bf{, 1, 1, \bf\ smal} 。 由模拟到数字转换的最新进展驱动, 我们研究的是 $\ b{ littlex} 美元 美元和 美元\ 美元=\ 美元=\ kr} 以 美元=xlttal_ 美元为最低值的恢复问题 。