The Complexity of Random Functions of Many Variables, Gérard Ben Arous

From Katie Gentilello September 13th, 2016

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A function of many variables, when chosen at random, is typically very complex. It has an exponentially large number of local minima or maxima, or critical points. It defines a very complex landscape, the topology of its level lines (for instance their Euler characteristic) is surprisingly complex. This complex picture is valid even in very simple cases, for random homogeneous polynomials of degree p larger than 2. This has important consequences. For instance trying to find the minimum value of such a function may thus be very difficult.

The mathematical tool suited to understand this complexity is the spectral theory of large random matrices. The classification of the different types of complexity has been understood for a few decades in the statistical physics of disordered media, and in particular spin-glasses, where the random functions may define the energy landscapes. It is also relevant in many other fields, including computer science and Machine learning. I will review recent work with collaborators in mathematics (A. Auffinger, J. Cerny) , statistical physics (C. Cammarota, G. Biroli, Y. Fyodorov, B. Khoruzenko), and computer science (Y. LeCun and his team at Facebook, A. Choromanska, L. Sagun among others), as well as recent work of E. Subag and E.Subag and O.Zeitouni.

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Tags: stelson lectureschool of mathematicsgeorgia techrecorded by gtlibrary

name: Gérard Ben Arous
biography: A specialist of probability theory and its applications, Gérard Ben Arous arrived to NYU's Courant Institute as a Professor of Mathematics in 2002. He was appointed Director of the Courant Institute and Vice Provost for Science and Engineering Development in September 2011.
Appears In: Open Scholarship

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The Complexity of Random Functions of Many Variables, Gérard Ben Arous

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