Scalable unsupervised alignment of general metric and non-metric structures

• URL: http://arxiv.org/abs/2406.13507v1
• Date: Wed, 19 Jun 2024 12:54:03 GMT
• Title: Scalable unsupervised alignment of general metric and non-metric structures
• Authors: Sanketh Vedula, Valentino Maiorca, Lorenzo Basile, Francesco Locatello, Alex Bronstein,
• Abstract summary: Aligning data from different domains is a fundamental problem in machine learning with broad applications across very different areas. We learn a related well-scalable linear assignment problem (LAP) whose solution is also a minimizer of the quadratic assignment problem (QAP) We evaluate our approach on synthetic and real datasets from single-cell multiomics and neural latent spaces.
• Score: 21.29255788365408
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Aligning data from different domains is a fundamental problem in machine learning with broad applications across very different areas, most notably aligning experimental readouts in single-cell multiomics. Mathematically, this problem can be formulated as the minimization of disagreement of pair-wise quantities such as distances and is related to the Gromov-Hausdorff and Gromov-Wasserstein distances. Computationally, it is a quadratic assignment problem (QAP) that is known to be NP-hard. Prior works attempted to solve the QAP directly with entropic or low-rank regularization on the permutation, which is computationally tractable only for modestly-sized inputs, and encode only limited inductive bias related to the domains being aligned. We consider the alignment of metric structures formulated as a discrete Gromov-Wasserstein problem and instead of solving the QAP directly, we propose to learn a related well-scalable linear assignment problem (LAP) whose solution is also a minimizer of the QAP. We also show a flexible extension of the proposed framework to general non-metric dissimilarities through differentiable ranks. We extensively evaluate our approach on synthetic and real datasets from single-cell multiomics and neural latent spaces, achieving state-of-the-art performance while being conceptually and computationally simple.

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