Optimal Learning of Joint Alignments with a Faulty Oracle

TL;DR

This paper introduces an efficient, simple algorithm for recovering joint alignments from noisy pairwise difference observations, requiring fewer queries and matching the theoretical lower bounds for non-adaptive methods.

Contribution

It presents a novel, easy-to-implement algorithm for joint alignment with noisy data, improving on prior methods in simplicity and efficiency, and proves its optimality among non-adaptive algorithms.

Findings

Algorithm performs O(n log n / (k δ^2)) queries with high probability

Proven lower bound matches the algorithm's query complexity

Simplifies previous approaches based on power methods

Abstract

We consider the following problem, which is useful in applications such as joint image and shape alignment. The goal is to recover $n$ discrete variables $g_i \in \{0, \ldots, k-1\}$ (up to some global offset) given noisy observations of a set of their pairwise differences $\{(g_i - g_j) \bmod k\}$; specifically, with probability $\frac{1}{k}+\delta$ for some $\delta > 0$ one obtains the correct answer, and with the remaining probability one obtains a uniformly random incorrect answer. We consider a learning-based formulation where one can perform a query to observe a pairwise difference, and the goal is to perform as few queries as possible while obtaining the exact joint alignment. We provide an easy-to-implement, time efficient algorithm that performs $O\big(\frac{n \lg n}{k \delta^2}\big)$ queries, and recovers the joint alignment with high probability. We also show that our algorithm is optimal by proving a general lower bound that holds for all non-adaptive algorithms. Our work improves significantly recent work by Chen and Cand\'{e}s \cite{chen2016projected}, who view the problem as a constrained principal components analysis problem that can be solved using the power method. Specifically, our approach is simpler both in the algorithm and the analysis, and provides additional insights into the problem structure.