Applied Math Colloquium

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For more information, contact Philippe Rigollet and Laurent Demanet

Spring 2024

Most talks are at 4:30-5:30 in 2-190 unless otherwise noted.

Date Speaker Abstract
February 22

Hoon Cho
(Yale University)

Computational solutions for privacy challenges in biomedicine

Abstract: The sensitive nature of biomedical data presents significant challenges for data sharing. Traditional safeguards, such as access controls and privacy regulations, have resulted in the fragmentation of biomedical data across silos. In this talk, I will demonstrate the crucial role mathematical techniques play in overcoming privacy challenges in biomedicine. I will first describe how joint probabilistic modeling of genomic and transcriptomic data can improve our understanding of privacy risks. Following this, I will discuss how our novel algorithm design, which integrates ideas from cryptography, distributed optimization, and statistical genetics, has led to a suite of secure federated (SF) tools. These tools facilitate large-scale, privacy-preserving joint analysis of biomedical data across silos. Finally, I will share our recent efforts to deploy these tools across biobanks and discuss future research directions.

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April 4

Vladimir Koltchinskii
(Georgia Tech)

Estimation of functionals of covariance operators in high-dimensional and infinite-dimensional models.

Abstract: In many problems of high-dimensional statistics and machine learning, it is of importance to estimate some low-dimensional features of unknown high-dimensional or infinite-dimensional covariance operators. In particular, the features of interest include various spectral characteristics of the covariance operator such as its eigenvalues, linear forms of its eigenvectors, bilinear forms of its spectral projections, etc. More generally, the features could be represented as locally smooth functionals of the covariance. Naive plug-in estimators based on sample covariance are usually sub-optimal due to their large bias and higher order bias reduction methods are of crucial importance in these problem. We study functional estimation problem in a dimension-free framework with its complexity characterized by so called effective rank of the covariance operator. In this framework, we developed new estimators of a given functional of unknown covariance based on linear aggregation of several plug-in estimators with different sample sizes. We show that these estimators provide higher order bias reduction and achieve the minimax optimal error rates in broad classes of H\"older smooth functionals.

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April 18

Gabriel Peyré
(Ecole Normale Supérieure, Paris)

Conservation Laws for Gradient Flows

Abstract: Understanding the geometric properties of gradient descent dynamics is a key ingredient in deciphering the recent success of very large machine learning models. A striking observation is that trained over-parameterized models retain some properties of the optimization initialization. This "implicit bias" is believed to be responsible for some favorable properties of the trained models and could explain their good generalization properties. In this talk I will first rigorously expose the definition and basic properties of "conservation laws", which are maximal sets of independent quantities conserved during gradient flows of a given model (e.g. of a ReLU network with a given architecture) with any training data and any loss. Then I will explain how to find the exact number of these quantities by performing finite-dimensional algebraic manipulations on the Lie algebra generated by the Jacobian of the model. In the specific case of linear and ReLu networks, this procedure recovers the conservation laws known in the literature, and prove that there are no other laws. The associated paper can be found here and the open source code is here This is a joint work with Sibylle Marcotte and Rémi Gribonval.

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May 9

Qin Li

Multiscale inverse problem, from Maxwell to Boltzmann to Calderon

Abstract: Inverse problems, spanning from nanometer to meter scales, such as inverse Maxwell, inverse Boltzmann, and inverse diffusion (Calderon), pervade diverse scientific domains. Hilbert’s 6th problem hints at a unified mathematical framework for these disparate challenges. In this talk, we explore the interconnections of these inverse problems, and provide the argument that they fundamentally represent the same mathematical essence across scales. We also discuss the implications of this unification on computational methodologies. It will be a light-hearted talk.