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Log-Euclidean Signatures for Intrinsic Distances Between Unaligned Datasets
Tal Shnitzer · Mikhail Yurochkin · Kristjan Greenewald · Justin Solomon

Wed Jul 20 10:20 AM -- 10:25 AM (PDT) @ Hall G
in MISC: Representation Learning/Causality »

The need for efficiently comparing and representing datasets with unknown alignment spans various fields, from model analysis and comparison in machine learning to trend discovery in collections of medical datasets. We use manifold learning to compare the intrinsic geometric structures of different datasets by comparing their diffusion operators, symmetric positive-definite (SPD) matrices that relate to approximations of the continuous Laplace-Beltrami operator from discrete samples. Existing methods typically assume known data alignment and compare such operators in a pointwise manner. Instead, we exploit the Riemannian geometry of SPD matrices to compare these operators and define a new theoretically-motivated distance based on a lower bound of the log-Euclidean metric. Our framework facilitates comparison of data manifolds expressed in datasets with different sizes, numbers of features, and measurement modalities. Our log-Euclidean signature (LES) distance recovers meaningful structural differences, outperforming competing methods in various application domains.

Keywords: PM: Spectral Methods · MISC: Representation Learning · MISC: Unsupervised and Semi-supervised Learning · MISC: Transfer, Multitask and Meta-learning

Author Information

Tal Shnitzer (MIT)
Mikhail Yurochkin (IBM Research, MIT-IBM Watson AI Lab)
Kristjan Greenewald (IBM)
Justin Solomon (MIT)

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