# Probability, Geometry, and Computation in High Dimensions

In recent years, the proliferation of learning and statistical inference tasks on complex data has led to renewed focus on the development of theory for the high-dimensional setting. In the existing literature on such problems, it is notable that many fruitful ideas have emerged from the interplay among probability, geometry, and computation in the high-dimensional setting. For instance, the phenomenon of "concentration of measure" lies at the intersection of probability and geometry, and is related to the problem of dimension-free guarantees for many important algorithms. Another example is the contribution of ideas from statistical physics towards both algorithm design and probabilistic analysis, which has led to advances in understanding for some classical problems in computer science.

High-dimensional problems have attracted the attention of researchers across diverse fields of research: computer scientists, mathematicians, physicists, and statisticians have all contributed analytical tools, conceptual advances, and new problem directions. Despite much recent progress, important basic questions remain open, for instance: Which properties of data can be learnt from a small number of samples? Can we characterize general tradeoffs between the quality of data (statistical information) versus the availability of computational resources? For what kinds of problems is it possible to make algorithmic guarantees that are dimension-free, or have minimal dimension-dependence? Beyond the immediate applications, these questions are also related to fundamental questions in approximation theory, convex geometry, and classical models of physical systems.

In advancing our understanding of these problems, it becomes increasingly clear that several traditionally separate fields will have a role to play. Although this sense is widely shared, communication of ideas across different research domains has remained relatively limited, and cultural differences have led different groups to study similar problems but with different framings or in different regimes. This semester program aims to advance the state of research on high-dimensional settings by bringing together a diverse range of research perspectives.

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**Organizers:**

Nike Sun (Massachusetts Institute of Technology; chair), Jian Ding (University of Pennsylvania), Ronen Eldan (Weizmann Institute of Science), Elchanan Mossel (Massachusetts Institute of Technology), Joe Neeman (University of Texas at Austin), Jelani Nelson (University of California at Berkeley), and Tselil Schramm (Stanford University)

**List of Participants (Tentative List, including Organizers):**

Guy Bresler (Massachusetts Institute of Technology), Sébastien Bubeck (Microsoft Research Redmond), Anindya De (University of Pennsylvania), Jian Ding (University of Pennsylvania), Ronen Eldan (Weizmann Institute of Science), David Gamarnik (Massachusetts Institute of Technology), Daniel Kane (University of California at San Diego), Adam Klivans (University of Texas at Austin), James Lee (University of Washington), Yin Tat Lee (University of Washington), Song Mei (University of California at Berkeley), Andrea Montanari (Stanford University), Cristopher Moore (Santa Fe Institute), Elchanan Mossel (Massachusetts Institute of Technology), Joe Neeman (University of Texas at Austin), Jelani Nelson (University of California at Berkeley), Prasad Raghavendra (University of California at Berkeley), Mark Rudelson (University of Michigan), Alistair Sinclair (University of California at Berkeley), Allan Sly (Princeton University), Nike Sun (Massachusetts Institute of Technology), Prasad Tetali (Georgia Institute of Technology), Elisabeth Werner (Case Western Reserve University), Mary Wootters (Stanford University), Tianyi Zheng (University of California at San Diego)

**Research Fellows:**

Ahmed El Alaoui (Stanford University), Aditya Grover (Stanford University), Vishesh Jain (Massachusetts Institute of Technology), Galyna Livshyts (Georgia Institute of Technology), Ashwin Pananjady (University of California at Berkeley), Cindy Rush (Columbia University), Tselil Schramm (Stanford University), Erik Waingarten (Columbia University)

**Visiting Graduate Students and Postdocs:**

Enric Boix (Massachusetts Institute of Technology), Matthew Brennan (Massachusetts Institute of Technology), Michael Celentano (Stanford University), Kabir Chandrasekher (Stanford University), Rishabh Dudeja (Columbia University), Majid Farhadi (Georgia Institute of Technology), Brice Huang (Massachusetts Institute of Technology), Sophie Huiberts (Centrum Wiskunde & Informatica), Haotian Jiang (University of Washington), Eren Kizildag (Massachusetts Institute of Technology), Tomasz Kociumaka (Bar-Ilan University), Frederic Koehler (Massachusetts Institute of Technology), Ben Li (Tel Aviv University), Sidhanth Mohanty (University of California at Berkeley), Dheeraj Nagaraj (Massachusetts Institute of Technology), Ewin Tang (University of Washington), Xinyu Wu (Carnegie Mellon University), Dana Yang (Duke University)

## Workshops

Those interested in participating in this program should send an email to the organizers at this hd2020 [at] lists.simons.berkeley.edu (at this address).