Shedding a PAC-Bayesian Light on Adaptive Sliced-Wasserstein Distances

• URL: http://arxiv.org/abs/2206.03230v3
• Date: Wed, 31 May 2023 14:32:40 GMT
• Title: Shedding a PAC-Bayesian Light on Adaptive Sliced-Wasserstein Distances
• Abstract summary: The Sliced-Wasserstein distance is a theoretically grounded alternative to the Wasserstein distance. We leverage the PAC-Bayesian theory and a central observation that SW may be interpreted as an average risk.
• Score: 19.717598118374056
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: The Sliced-Wasserstein distance (SW) is a computationally efficient and theoretically grounded alternative to the Wasserstein distance. Yet, the literature on its statistical properties -- or, more accurately, its generalization properties -- with respect to the distribution of slices, beyond the uniform measure, is scarce. To bring new contributions to this line of research, we leverage the PAC-Bayesian theory and a central observation that SW may be interpreted as an average risk, the quantity PAC-Bayesian bounds have been designed to characterize. We provide three types of results: i) PAC-Bayesian generalization bounds that hold on what we refer as adaptive Sliced-Wasserstein distances, i.e. SW defined with respect to arbitrary distributions of slices (among which data-dependent distributions), ii) a principled procedure to learn the distribution of slices that yields maximally discriminative SW, by optimizing our theoretical bounds, and iii) empirical illustrations of our theoretical findings.

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