Stochastics and Statistics Seminar

Abstract: I will describe recent results that (a) show nearly optimal hardness of learning Gaussian mixtures, and (b) give evidence of average-case hardness of sparse linear regression w.r.t. all efficient algorithms, assuming the worst-case hardness of lattice problems. The talk is based on the following papers with Aparna Gupte and Neekon Vafa. https://arxiv.org/pdf/2204.02550.pdf https://arxiv.org/pdf/2402.14645.pdf Bio: Vinod Vaikuntanathan is a professor of computer science at MIT and the chief cryptographer at Duality Technologies. His research is in the foundations of cryptography…

Abstract:  It has been empirically observed that the spectrum of neural network Hessians after training have a bulk concentrated near zero, and a few outlier eigenvalues. Moreover, the eigenspaces associated to these outliers have been associated to a low-dimensional subspace in which most of the training occurs, and this implicit low-dimensional structure has been used as a heuristic for the success of high-dimensional classification. We will describe recent rigorous results in this direction for the Hessian spectrum over the course…

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Abstract: The space of probability measures endowed with the optimal transport metric has a rich structure with applications in probability, analysis, and geometry. The notion of (displacement) convexity in this space was discovered by McCann, and forms the backbone of this theory.  I will introduce a new, and stronger, notion of displacement convexity which operates on the matrix level. The motivation behind this definition is to capture the intrinsic dimensionality of probability measures which could have very different behaviors along…

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