### Geometry and topology in statistical inference

LECTURE SERIES by SAYAN MUKHERJEE, DUKE UNIVERSITY, on

## GEOMETRY AND TOPOLOGY IN STATISTICAL INFERENCE

This event is supported by the Statistics Research Training Group

(http://www.stat.ucdavis.edu/research/nsf-rtg/index.html)

Sayan Mukherjee, Duke, will visit us Feb. 1-3, and he will give a lecture series on

**Geometry and topology in statistical inference**

I. Geometry in Inference Monday, Feb. 1, 3-5pm

II. Topology in inference Tuesday, Feb. 2, 10am - 12pm

III. Stochastic topology Wednesday, Feb. 3, 9 - 11am

Location: MSB 1147

While the lectures are prepared with a Statistics/Applied Math audience in mind, some of the topics will be of interest for a more general audience. For more details on the contents of the lectures see below.

Sayan has appointments in Statistical Science, Computer Science, and Mathematics, and he is affiliated with the Information Initiative at Duke. Here is a link to his homepage for more information: https://stat.duke.edu/~sayan/

He will also give a joint Statistics/Biostatistics seminar on Tuesday, Feb. 2 at 4:10pm with the title: Inference and dynamics. The abstract can be found here.

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Here are some more details on the contents of the lectures:

**I. Geometry in Inference (**Monday, Feb. 1, 3-5pm)**: **

The use of geometry in a variety of statistical problems will be developed. Topics include

supervised dimension reduction, learning mixtures of subspaces, inference of conditional

dependence, and stochastic gradient based methods based on proximal functions.

**II. Topology in inference (**Tuesday, Feb. 2, 10am - 12pm)

The use of topology in statistical inference. Problems regarding modeling surfaces and

shapes will be discussed. A central idea of Persistence Homology will be discussed and

a statistical perspective will be given. Relations to extrema of Gaussian random fields

will be discussed.

**III. Stochastic topology (**Wednesday, Feb. 3, 9 - 11am)

I will describe recent efforts to define random walks on simplicial complexes with stationary

distributions related to the combinatorial (Hodge) Laplacian. This work will touch on

higher-order Cheeger inequalities, an extension of label propagation to edges or

higher-order complexes, and a generalization of results for near linear time solutions

for linear systems.

I will also discuss ideas of percolation on stochastic processes from a topological perspective.

For example, given n points down from a point process on a manifold, consider the random

set which consists of the union of balls of radius r around the points. As n goes to infinity, r is

sent to zero at varying rates. For this stochastic process, I will provide scaling limits and

phase transitions on the counts of Betti numbers and critical points.