Correspondence retrieval

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Alexandr Andoni, Daniel Hsu, Kevin Shi, Xiaorui Sun ;
Proceedings of the 2017 Conference on Learning Theory, PMLR 65:105-126, 2017.

Abstract

This article studies the correspondence retrieval problem: a set of $k$ distinct but unknown points $\mathbf{x}_1, \mathbf{x}_2, \dotsc, \mathbf{x}_k ∈\mathbb{R}^d$ are to be recovered from the unordered collection of projection values $⟨\mathbf{w}_i,\mathbf{x}_1 ⟩, ⟨\mathbf{w}_i,\mathbf{x}_2 ⟩, \dotsc, ⟨\mathbf{w}_i,\mathbf{x}_k ⟩$ onto $n$ known measurement vectors $\mathbf{w}_1, \mathbf{w}_2, \dotsc, \mathbf{w}_n$. Importantly, the correspondence of the $k$ projections ${⟨\mathbf{w}_i,\mathbf{x}_j ⟩}_j=1^k$ across different measurements is unknown. A special case of this problem is the well-studied problem of (real-valued) phase retrieval. In the case of independent standard Gaussian measurement vectors, the main algorithm proposed in this work requires $n = d+1$ measurements to correctly return the $k$ unknown points with high probability. This number of measurements is optimal, and it is smaller than the number of measurements required for a stronger “for all” guarantee even in the phase retrieval setting. The algorithm is based on reductions to the Shortest Vector Problem on certain random lattices, and employs the Lenstra, Lenstra, and Lovász (1982) basis reduction algorithm in a manner similar to the Lagarias & Odlyzko (1985) algorithm for solving random instances of Subset Sum. Another algorithm, essentially due to Yi, Caramanis, & Sanghavi (2016), based on higher-order moments and tensor decompositions is shown to work in a setting where the projection values are corrupted by additive Gaussian noise, but it requires a significantly larger number of measurements.

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