Monday, December 2, 2019
12/02/2019 - 4:00pm
The art of using quantum field theory to derive mathematical results often lies in a mysterious transition between infinite dimensional geometry and finite dimensional geometry. In this talk we describe a general mathematical framework to study the quantum geometry of \sigma-models when they are effectively localized to small fluctuations around constant maps. We illustrate how to turn the physics idea of exact semi-classical approximation into a geometric set-up in this framework, using Gauss-Manin connection. This leads to a theory of “counting constant maps” in a nontrivial way. We explain this program by a concrete example of topological quantum mechanics and show how “counting constant loops” leads to a simple proof of the algebraic index theorem.
12/02/2019 - 4:00pm
Abstract: Symmetric tensors are multi-dimensional arrays invariant to permutation of indices. Symmetric tensors arise in machine learning and statistics when applying the method of moments, as higher-dimensional analogs of the sample covariance matrix. In tasks from demixing Gaussian mixture models to blind source separation, it is informative to decompose a symmetric tensor as a sum of symmetric outer products of vectors.
This talk presents a novel algorithm for computing low-rank symmetric tensor decompositions, based on a modified tensor power method. Numerical experiments demonstrate that our algorithm significantly outperforms state-of-the-art methods, per standard performance metrics. We provide supporting theoretical guarantees, through connections to optimization theory, algebraic geometry and dynamical systems.
We also extend the algorithm to compute a certain generalization of symmetric tensor decompositions. By applying the method of moments, this enables estimation of a union of linear subspaces from noisy point samples, i.e., robust subspace clustering. Applications to motion segmentation and image segmentation are discussed.