Georgia Institute of Technology College of Sciences

Grid homology is a purely combinatorial description of knot Floer homology in which the counting of psuedo-holomorphic disks is replaced with a counting of polygons in grid diagrams. This talk will provide an introduction to this theory, and is aimed at an audience with little to no experience with Heegaard Floer homology. 

For computational-intensive mixed integer non-linear optimization problems, a major challenge is to verify/guarantee the quality of any feasible solution under mild assumptions in a tractable fashion. In this talk, we focus on tackling this challenge by constructing tight relaxations and designing approximation algorithms for two different mixed integer non-linear optimization problems.

In the first part, we focus on the (row) sparse principal component analysis (rsPCA) problem. Solving rsPCA is the problem of finding the top-r leading principal components of a covariance matrix such that all these principal components are linear combinations of a subset of k variables. The rsPCA problem is a widely used dimensionality reduction tool with an additional sparsity constraint to enhance its interpretability. We propose: (a) a convex integer programming relaxation of rsPCA that gives upper (dual) bounds for rsPCA, and; (b) a new local search algorithm for finding primal feasible solutions for rsPCA. We also show that, in the worst-case, the dual bounds provided by the convex IP are within an affine function of the global optimal value. We demonstrate our techniques applied to large-scale covariance matrices.

In the second part, we focus on improving the execution speed of compute-intensive numerical code. The compute-intensive numerical code, especially of the variety encountered in deep neural network inference and training, is often written using nested for-loops. One of the main bottlenecks that significantly influence the nested for-loops' execution speed is the so-called memory latency. Iteration space tiling is a common memory management technique used to deal with memory latency. We study the problem of automatically optimizing the implementation of these nested loops by formulating the iteration space tiling problem into an integer geometric programming (IGP) problem. We show how to design an efficient approximation algorithm for this problem and how to use the so-called "non-uniform tiling" technique to improve the execution speed.

The first part of the talk is joint work with Santanu S. Dey, Rahul Mazumder, Macro Molinaro, and the second part of the talk is joint work with Ofer Dekel.