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The heart of modern machine learning (ML) is the approximation of high-dimensional functions. Traditional approaches, such as approximation by piecewise polynomials, wavelets, or other linear combinations of fixed basis functions, suffer from the curse of dimensionality (CoD). This does not seem to be the case for the neural network-based ML models. To quantify this, we need to develop the corresponding mathematical framework. At the same time, we might be able to use ML to solve problems in computational science that we could not solve before due to CoD. In this talk, I will report the progress made so far at the theoretical front, and highlight the main remaining challenges. I will also discuss some examples along the lines of “AI for Science”.

https://harvard.zoom.us/j/779283357?pwd=MitXVm1pYUlJVzZqT3lwV2pCT1ZUQT09

The n-queens problem asks how many ways there are to place n queens on an n x n chessboard so that no two queens can attack one another, and the toroidal n-queens problem asks the same question where the board is considered on the surface of a torus. Let Q(n) denote the number of n-queens configurations on the classical board and T(n) the number of toroidal n-queens configurations. The toroidal problem was first studied in 1918 by Pólya who showed that T(n)>0 if and only if n is not divisible by 2 or 3. Much more recently Luria showed that T(n) is at most ((1+o(1))ne^{-3})^n and conjectured equality when n is not divisible by 2 or 3. We prove this conjecture, prior to which no non-trivial lower bounds were known to hold for all (sufficiently large) n not divisible by 2 or 3. We also show that Q(n) is at least ((1+o(1))ne^{-3})^n for all natural numbers n which was independently proved by Luria and Simkin and, combined with our toroidal result, completely settles a conjecture of Rivin, Vardi and Zimmerman regarding both Q(n) and T(n).

In this talk we’ll discuss our methods used to prove these results. A crucial element of this is translating the problem to one of counting matchings in a 4-partite 4-uniform hypergraph. Our strategy combines a random greedy algorithm to count `almost’ configurations with a complex absorbing strategy that uses ideas from the methods of randomised algebraic construction and iterative absorption.

This is joint work with Peter Keevash.

Password: 1251442

https://harvard.zoom.us/j/99651364593?pwd=Q1R0RTMrZ2NZQjg1U1ZOaUYzSE02QT09

This is a work in progress joint with Daniel Disegni. We formulate a p-adic analogue of the Arithmetic Gan–Gross–Prasad conjecture for unitary groups, relating the p-adic height pairing of the arithmetic diagonal cycles to the first central derivative (along the cyclotomic direction) of a p-adic Rankin—Selberg L-function associated to cuspidal automorphic representations. In the good ordinary case we are able to prove the conjecture, at least when the ramifications are mild at inert primes. We deduce some application to the p-adic version of the Bloch-Kato conjecture.

This is a work in progress joint with Daniel Disegni. We formulate a p-adic analogue of the Arithmetic Gan–Gross–Prasad conjecture for unitary groups, relating the p-adic height pairing of the arithmetic diagonal cycles to the first central derivative (along the cyclotomic direction) of a p-adic Rankin—Selberg L-function associated to cuspidal automorphic representations. In the good ordinary case we are able to prove the conjecture, at least when the ramifications are mild at inert primes. We deduce some application to the p-adic version of the Bloch-Kato conjecture.

Despite their impressive performance, machine learning systems remain prohibitively unreliable in safety-, trust-, and ethically sensitive domains. Recent discussions in different sub-fields of AI have reached the consensus of knowledge need in machine learning; few discussions have touched upon the diagnosis of what knowledge is needed. In this talk, I will present our ongoing work on ARCH, a knowledge-driven, human-centered, and reasoning-based tool, for diagnosing the unknowns of a machine learning system. ARCH leverages human intelligence to create domain knowledge required for a given task and to describe the internal behavior of a machine learning system; it infers the missing or incorrect knowledge of the system with the built-in probabilistic, abductive reasoning engine. ARCH is a generic tool that can be applied to machine learning in different contexts. In the talk, I will present several applications in which ARCH is currently being developed and tested, including health, finance, and smart buildings.

Zoom ID: 950 2372 5230 (Password: cmsa)

Zoom link: https://harvard.zoom.us/j/96657833341

Password: cmsa

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*Note special day and time*

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Subscribe to Harvard CMSA seminar videos (more to be uploaded):
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https://www.youtube.com/channel/UCM06KiUOw1vRrmvD8U274Ww