Programs
Fall 2021

Geometric Methods in Optimization and Sampling

Aug. 18Dec. 17, 2021

Optimization and sampling are two of the most important mathematical topics at the interface of data science and computation. The two questions are, in fact, connected mathematically through a powerful framework articulated around the geometry of probability distributions. The geometric toolbox that underlies optimization and sampling was initiated in the study of partial differential equations (PDEs) and has evolved into different mathematical disciplines: probability, calculus of variations, analysis and geometry. While connections are slowly beginning to percolate across disciplines, this program is aimed to be a catalyst for new and interdisciplinary ideas using a principled and unified approach to optimization and sampling.

A central goal of this program is to develop and promote a geometric approach to various computational problems in sampling, optimization, and PDEs. For example, the geometry of Optimal Transport has been instrumental to establish fruitful connections between diffusion processes, gradient flows, and diffusive PDEs by eliciting hidden convexity. This success calls for a versatile toolbox to tackle algorithmic questions arising in sampling, optimization, and particle methods for solving PDEs by leveraging the hidden geometric structure of each problem in a systematic way. Moreover, in a large class of problems this geometric structure is supplemented by additional symmetries or other algebraic structures that can be exploited to design better algorithms. 

These recent connections between sampling, optimization, and PDEs have placed the fields in a unique position for mutual impact. This program aims at bringing together researchers from various backgrounds to tackle these challenging problems using a unified approach by focusing on the following aspects:

  • Sampling as an optimization problem
  • Geometry and optimal transport
  • The PDE perspective on sampling and optimization
  • Eliciting convexity via geometry in sampling and optimization
  • The interplay of algebra and geometry in optimization

gm2021announcements [at] lists.simons.berkeley.edu (body: subscribe%20gm2021announcements%40lists.simons.berkeley.edu) (Click here to subscribe to our announcements email list for this program.)

Organizers:
Philippe Rigollet (MIT, co-chair), Martin Wainwright (UC Berkeley, co-chair), Katy Craig (UC Santa Barbara), Simone Di Marino (University of Genova), Nisheeth Vishnoi (Yale), Ashia Wilson (MIT)

List of participants (tentative list, including organizers):
Peter Bartlett (UC Berkeley), Yann Brenier (CNRS/ENS Paris), Katy Craig (UC Santa Barbara), Simone Di Marino (University of Genova), Jelena Diakonikolas (University of Wisconsin, Madison), Wilfrid Gangbo (UCLA), Michael Jordan (UC Berkeley), Jan Maas (IST Austria), Carola-Bibiane Schönlieb (University of Cambridge), Suvrit Sra (MIT), Nisheeth Vishnoi (Yale), Andre Wibisono (Yale), Ashia Wilson (MIT), Martin Wainwright (UC Berkeley), Jose A. Carillo (University of Oxford), Mikaela Iacobelli (ETH Zurich), Yin Tat Lee (University of Washington), Andrea Montanari (Stanford), Dejan Slepčev (CMU), Prasad Tetali (Georgia Institute of Technology)

Those interested in participating in this program should send an email to the organizers at this gm2021 [at] lists.simons.berkeley.edu (at this address)