School of Mathematics

Generative Modeling by Estimating Gradients of the Data Distribution

Stefano Ermon
Stanford University
May 12, 2020
Existing generative models are typically based on explicit representations of probability distributions (e.g., autoregressive or VAEs) or implicit sampling procedures (e.g., GANs). We propose an alternative approach based on modeling directly the vector field of gradients of the data distribution (scores). Our framework allows flexible energy-based model architectures, requires no sampling during training or the use of adversarial training methods.

Using discrepancy theory to improve the design of randomized controlled trials

Daniel Spielman
Yale University
May 11, 2020
In randomized experiments, such as a medical trials, we randomly assign the treatment, such as a drug or a placebo, that each experimental subject receives. Randomization can help us accurately estimate the difference in treatment effects with high probability. We also know that we want the two groups to be similar: ideally the two groups would be similar in every statistic we can measure beforehand. Recent advances in algorithmic discrepancy theory allow us to divide subjects into groups with similar statistics.

Spectral characterizations of Besse and Zoll Reeb flows

Marco Mazzucchelli
École normale supérieure de Lyon
May 8, 2020
In this talk, I will address a geometric inverse problem from contact geometry: is it possible to recognize whether all orbits of a given Reeb flow are closed from the knowledge of the action spectrum? Borrowing the terminology from Riemannian geometry, Reeb flows all of whose orbits are closed are sometimes called Besse, and Besse Reeb flows all of whose orbits have the same minimal period are sometimes called Zoll.

Learning probability distributions; What can, What can't be done

Shai Ben-David
University of Waterloo
May 7, 2020
A possible high level description of statistical learning is that it aims to learn about some unknown probability distribution ("environment”) from samples it generates ("training data”). In its most general form, assuming no prior knowledge and asking to find accurate approximations to the data generating distributions, there can be no success guarantee. In this talk I will discuss two major directions of relaxing that too hard problem.

On triple product L functions

Jayce Robert Getz
Duke University
May 7, 2020
Establishing the conjectured analytic properties of triple product L-functions is a crucial case of Langlands functoriality. However, little is known. I will present work in progress on the case of triples of automorphic representations on GL_3; in some sense this is the smallest case that appears out of reach via standard techniques. The approach is based on the beautiful fibration method of Braverman and Kazhdan for constructing Schwartz spaces and proving analogues of the Poisson summation formula.

Cutting Planes Proofs of Tseitin and Random Formulas

Noah Fleming
University of Toronto
May 5, 2020
Proof Complexity studies the length of proofs of propositional tautologies in various restricted proof systems. One of the most well-studied is the Cutting Planes proof system, which captures reasoning which can be expressed using linear inequalities. A series of papers proved lower bounds on the length of Cutting Planes using the method of feasible interpolation whereby proving lower bounds on the size of Cutting Planes lower bounds proofs of a certain restricted class of formulas is reduced to monotone circuit lower bounds.

Boosting Simple Learners

Shay Moran
May 5, 2020
We study boosting algorithms under the assumption that the given weak learner outputs hypotheses from a class of bounded capacity. This assumption is inspired by the common convention that weak hypotheses are “rules-of-thumbs” from an “easy-to-learn class”. Formally, we assume the class of weak hypotheses has a bounded VC dimension.

We focus on two main questions:

Local Statistics, Semidefinite Programming, and Community Detection

Prasad Raghavendra
University of California, Berkeley
May 4, 2020
We propose a new hierarchy of semidefinite programming relaxations for inference problems. As test cases, we consider the problem of community detection in block models. The vertices are partitioned into k communities, and a graph is sampled conditional on a prescribed number of inter- and intra-community edges.

Latent Stochastic Differential Equations for Irregularly-Sampled Time Series

David Duvenaud
University of Toronto
April 30, 2020
Much real-world data is sampled at irregular intervals, but most time series models require regularly-sampled data. Continuous-time models address this problem, but until now only deterministic (ODE) models or linear-Gaussian models were efficiently trainable with millions of parameters. We construct a scalable algorithm for computing gradients of samples from stochastic differential equations (SDEs), and for gradient-based stochastic variational inference in function space, all with the use of adaptive black-box SDE solvers.