学术报告

【online】On the flow of the (regularized) sliced Wasserstein distance and its applications

发布人：发布时间： 2020-08-31

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Series Talks on Optimal Transportation Problems, PDEs and Their Applications in Image Processing

“最优运输问题、PDE和在图像处理中的应用”系列学术报告

**题目：**On the flow of the (regularized) sliced Wasserstein distance and its applications

**报告人：**Prof. Filippo Santambrogio（Université Claude Bernard - Lyon 1）

**时间：**2020年9月4日 下午 4：00

**地点：**Zoom会议室 ID: 674 5009 3281 密码：770491

**摘要：**The sliced Wasserstein distance SW_2 is a distance similar to the usual W_2 distance, but much easier to compute. The gradient flow of SW_2^2 w.r.t. W_2 was already studied years ago by M. Bernot as a way to produce a flow map pushing a given measure to another, with similar features than the optimal transport map. Partial results about the PDE arising from this flow were present in N. Bonnotte’s PhD thesis (Orsay, 2013), but many questions are still open, in particular the convergence for long time to the steady state. Recently, a paper by Liutkus et al. proposes a regularized version of this flow, adding an entropy to the functional (i.e. a Laplacian to the equation, or Brownian diffusion to the particles). In this case it is possible to obtain long-time convergence, and this idea allows to obtain a non-parametric implicit generative

model, with some interesting results on real data in imaging sciences. I will present few results from the mathematical analysis of the PDE and of the corresponding JKO scheme, and few ideas on how to optimize the choice of the flow map. This comes from an ongoing work in collaboration with N. Bonneel, J. Digne and our student E. Ciuperca.

**邀请人：**杨孝平 老师