  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.
