Statistically Optimal Generative Modeling with Maximum Deviation from the Empirical Distribution

Elen Vardanyan, Sona Hunanyan, Tigran Galstyan, Arshak Minasyan, Arnak S. Dalalyan
Proceedings of the 41st International Conference on Machine Learning, PMLR 235:49203-49225, 2024.

Abstract

This paper explores the problem of generative modeling, aiming to simulate diverse examples from an unknown distribution based on observed examples. While recent studies have focused on quantifying the statistical precision of popular algorithms, there is a lack of mathematical evaluation regarding the non-replication of observed examples and the creativity of the generative model. We present theoretical insights into this aspect, demonstrating that the Wasserstein GAN, constrained to left-invertible push-forward maps, generates distributions that not only avoid replication but also significantly deviate from the empirical distribution. Importantly, we show that left-invertibility achieves this without compromising the statistical optimality of the resulting generator. Our most important contribution provides a finite-sample lower bound on the Wasserstein-1 distance between the generative distribution and the empirical one. We also establish a finite-sample upper bound on the distance between the generative distribution and the true data-generating one. Both bounds are explicit and show the impact of key parameters such as sample size, dimensions of the ambient and latent spaces, noise level, and smoothness measured by the Lipschitz constant.

Cite this Paper

BibTeX
@InProceedings{pmlr-v235-vardanyan24a, title = {Statistically Optimal Generative Modeling with Maximum Deviation from the Empirical Distribution}, author = {Vardanyan, Elen and Hunanyan, Sona and Galstyan, Tigran and Minasyan, Arshak and Dalalyan, Arnak S.}, booktitle = {Proceedings of the 41st International Conference on Machine Learning}, pages = {49203--49225}, year = {2024}, editor = {Salakhutdinov, Ruslan and Kolter, Zico and Heller, Katherine and Weller, Adrian and Oliver, Nuria and Scarlett, Jonathan and Berkenkamp, Felix}, volume = {235}, series = {Proceedings of Machine Learning Research}, month = {21--27 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v235/main/assets/vardanyan24a/vardanyan24a.pdf}, url = {https://proceedings.mlr.press/v235/vardanyan24a.html}, abstract = {This paper explores the problem of generative modeling, aiming to simulate diverse examples from an unknown distribution based on observed examples. While recent studies have focused on quantifying the statistical precision of popular algorithms, there is a lack of mathematical evaluation regarding the non-replication of observed examples and the creativity of the generative model. We present theoretical insights into this aspect, demonstrating that the Wasserstein GAN, constrained to left-invertible push-forward maps, generates distributions that not only avoid replication but also significantly deviate from the empirical distribution. Importantly, we show that left-invertibility achieves this without compromising the statistical optimality of the resulting generator. Our most important contribution provides a finite-sample lower bound on the Wasserstein-1 distance between the generative distribution and the empirical one. We also establish a finite-sample upper bound on the distance between the generative distribution and the true data-generating one. Both bounds are explicit and show the impact of key parameters such as sample size, dimensions of the ambient and latent spaces, noise level, and smoothness measured by the Lipschitz constant.} }
Endnote
%0 Conference Paper %T Statistically Optimal Generative Modeling with Maximum Deviation from the Empirical Distribution %A Elen Vardanyan %A Sona Hunanyan %A Tigran Galstyan %A Arshak Minasyan %A Arnak S. Dalalyan %B Proceedings of the 41st International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2024 %E Ruslan Salakhutdinov %E Zico Kolter %E Katherine Heller %E Adrian Weller %E Nuria Oliver %E Jonathan Scarlett %E Felix Berkenkamp %F pmlr-v235-vardanyan24a %I PMLR %P 49203--49225 %U https://proceedings.mlr.press/v235/vardanyan24a.html %V 235 %X This paper explores the problem of generative modeling, aiming to simulate diverse examples from an unknown distribution based on observed examples. While recent studies have focused on quantifying the statistical precision of popular algorithms, there is a lack of mathematical evaluation regarding the non-replication of observed examples and the creativity of the generative model. We present theoretical insights into this aspect, demonstrating that the Wasserstein GAN, constrained to left-invertible push-forward maps, generates distributions that not only avoid replication but also significantly deviate from the empirical distribution. Importantly, we show that left-invertibility achieves this without compromising the statistical optimality of the resulting generator. Our most important contribution provides a finite-sample lower bound on the Wasserstein-1 distance between the generative distribution and the empirical one. We also establish a finite-sample upper bound on the distance between the generative distribution and the true data-generating one. Both bounds are explicit and show the impact of key parameters such as sample size, dimensions of the ambient and latent spaces, noise level, and smoothness measured by the Lipschitz constant.
APA
Vardanyan, E., Hunanyan, S., Galstyan, T., Minasyan, A. & Dalalyan, A.S.. (2024). Statistically Optimal Generative Modeling with Maximum Deviation from the Empirical Distribution. Proceedings of the 41st International Conference on Machine Learning, in Proceedings of Machine Learning Research 235:49203-49225 Available from https://proceedings.mlr.press/v235/vardanyan24a.html.

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