Joint Alignment From Pairwise Differences with a Noisy Oracle

In this work we consider the problem of recovering $n$ discrete random variables $x_i\in \{0,\dots ,k-1\},1\le i\le n$ (where $k$ is constant) with the smallest possible number of queries to a noisy oracle that returns for a given query pair $(x_i,x_j)$ a noisy measurement of their modulo $k$ pairwise difference, i.e., $y_{ij}=(x_i-x_j)\mathrm{m}\mathrm{o}\mathrm{d}k$. This is a joint discrete alignment problem with important applications in computer vision, graph mining, and spectroscopy imaging. Our main result is a polynomial time algorithm that learns exactly with high probability the alignment (up to some unrecoverable offset) using $O(n^{1+o(1)})$ queries.