Hertrich, Johannes;
Wald, Christian;
Altekrüger, Fabian;
Hagemann, Paul;
(2024)
Generative Sliced MMD Flows with Riesz Kernels.
In:
Proceedings of the Twelfth International Conference on Learning Representations.
ICLR: Vienna, Austria.
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Abstract
Maximum mean discrepancy (MMD) flows suffer from high computational costs in large scale computations. In this paper, we show that MMD flows with Riesz kernels K(x, y) = −∥x − y∥ r , r ∈ (0, 2) have exceptional properties which allow their efficient computation. We prove that the MMD of Riesz kernels, which is also known as energy distance, coincides with the MMD of their sliced version. As a consequence, the computation of gradients of MMDs can be performed in the one-dimensional setting. Here, for r = 1, a simple sorting algorithm can be applied to reduce the complexity from O(MN +N2 ) to O((M +N) log(M +N)) for two measures with M and N support points. As another interesting follow-up result, the MMD of compactly supported measures can be estimated from above and below by the Wasserstein-1 distance. For the implementations we approximate the gradient of the sliced MMD by using only a finite number P of slices. We show that the resulting error has complexity O( p d/P), where d is the data dimension. These results enable us to train generative models by approximating MMD gradient flows by neural networks even for image applications. We demonstrate the efficiency of our model by image generation on MNIST, FashionMNIST and CIFAR10.
| Type: | Proceedings paper |
|---|---|
| Title: | Generative Sliced MMD Flows with Riesz Kernels |
| Event: | The Twelfth International Conference on Learning Representations |
| Location: | Vienna |
| Dates: | 7 May 2024 - 11 May 2024 |
| Open access status: | An open access version is available from UCL Discovery |
| Publisher version: | https://openreview.net/forum?id=VdkGRV1vcf |
| Language: | English |
| Additional information: | This version is the author accepted manuscript. For information on re-use, please refer to the publisher’s terms and conditions. |
| Keywords: | sliced maximum mean discrepancy, energy distance, gradient flows, Riesz kernels, generative modelling |
| UCL classification: | UCL UCL > Provost and Vice Provost Offices > UCL BEAMS UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Engineering Science > Dept of Computer Science |
| URI: | https://discovery.ucl.ac.uk/id/eprint/10194631 |
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