Aligning Embeddings and Geometric Random Graphs: Informational Results and Computational Approaches for the Procrustes-Wasserstein Problem

TL;DR

This paper investigates the theoretical limits and computational methods for aligning high-dimensional point clouds via the Procrustes-Wasserstein problem, introducing a new algorithm and analyzing its performance in different regimes.

Contribution

It provides new information-theoretic insights and proposes the Ping-Pong algorithm for efficient alignment, with conditions for exact recovery after a single iteration.

Findings

Established bounds in high and low-dimensional regimes.

Proposed the Ping-Pong algorithm for alignment.

Demonstrated competitive performance against state-of-the-art methods.

Abstract

The Procrustes-Wasserstein problem consists in matching two high-dimensional point clouds in an unsupervised setting, and has many applications in natural language processing and computer vision. We consider a planted model with two datasets $X,Y$ that consist of $n$ datapoints in $\mathbb{R}^d$, where $Y$ is a noisy version of $X$, up to an orthogonal transformation and a relabeling of the data points. This setting is related to the graph alignment problem in geometric models. In this work, we focus on the euclidean transport cost between the point clouds as a measure of performance for the alignment. We first establish information-theoretic results, in the high ($d \gg \log n$) and low ($d \ll \log n$) dimensional regimes. We then study computational aspects and propose the Ping-Pong algorithm, alternatively estimating the orthogonal transformation and the relabeling, initialized via a Franke-Wolfe convex relaxation. We give sufficient conditions for the method to retrieve the planted signal after one single step. We provide experimental results to compare the proposed approach with the state-of-the-art method of Grave et al. (2019).