Georgia Institute of Technology College of Sciences

The Lovasz Local Lemma is a powerful tool to prove existence of combinatorial structures satisfying certain properties. In a constructive proof of the LLL, Moser and Tardos introduced a proof technique that is now referred to as the entropy compression method. In this talk I will describe the main idea of the method and apply it to a problem easily solved using the LLL. I will also describe recent applications of the idea to various graph coloring problems.

Dense kernel matrices resulting from pairwise evaluations of a kernel function arise naturally in machine learning and statistics. Previous work in constructing sparse transport maps or sparse approximate inverse Cholesky factors of such matrices by minimizing Kullback-Leibler divergence recovers the Vecchia approximation for Gaussian processes. However, these methods often rely only on geometry to construct the sparsity pattern, ignoring the conditional effect of adding an entry. In this work, we construct the sparsity pattern by leveraging a greedy selection algorithm that maximizes mutual information with target points, conditional on all points previously selected. For selecting k points out of N, the naive time complexity is O(N k^4), but by maintaining a partial Cholesky factor we reduce this to O(N k^2). Furthermore, for multiple (m) targets we achieve a time complexity of O(N k^2 + N m^2 + m^3) which is maintained in the setting of aggregated Cholesky factorization where a selected point need not condition every target. We directly apply the selection algorithm to image classification and recovery of sparse Cholesky factors. By minimizing Kullback-Leibler divergence, we apply the algorithm to Cholesky factorization, Gaussian process regression, and preconditioning with the conjugate gradient, improving over k-nearest neighbors particularly in high dimensional, unusual, or otherwise messy geometries with non-isotropic kernel functions.

The mathematical theory of the recovery of quantum states stored in a quantum memory, is intimately related to the subadditivity property of the entropy function, and the class of states known as quantum Markov chains. In this talk we will introduce some of the basic ideas of this area of quantum information theory. We discuss a theorem regarding recovery of a widely studied class of quantum states, the matrix product states, and its implication for the mutual information stored over separated regions of a one dimensional quantum memory. This is joint work with Pavel Svetlichnyy and Shivan Mittal.