“最优运输问题、PDE及其在图像处理中的应用”系列学术报告(三)

发布者:朱亚宾发布时间:2020-08-31浏览次数:12

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

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

    报告时间(Time):2020年9月4日16:00(北京时间)

    方式(Online):Zoom  ID:  674 5009 3281 (密码:770491)

    摘要(Abstract):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.

    报告人简介:Filippo Santambrogio,2006年在意大利比萨高等师范大学获得博士学位,现为法国里昂第一大学(里昂克洛德·贝尔那大学)Camille Jordan 学院教授。Santambrogio教授长期从事变分法、最优运输、退化椭圆偏微分方程、发展方程、梯度流、种群动力学、均衡问题、博弈论、经济学应用、数值实现和近似的研究。2007被意大利 Dei Lincei 科学院授予意大利青年学者(数学分析方向)称号,2017年入选法国大学机构委员。2015年出版专著《Optimal Transport for Applied Mathematicians. Calculations of Variations, PDEs, and Modeling》。

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 数学与统计学院

2020年9月2日