Joint Alignment From Pairwise Differences with a Noisy Oracle

03/13/2020

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by   Michael Mitzenmacher, et al.

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In this work we consider the problem of recovering n discrete random variables x_i∈{0,...,k-1}, 1 ≤ i ≤ 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) 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.