Improved Accuracy and Parallelism for MRRR-based Eigensolvers -- A Mixed Precision Approach

The real symmetric tridiagonal eigenproblem is of outstanding importance in numerical computations; it arises frequently as part of eigensolvers for standard and generalized dense Hermitian eigenproblems that are based on a reduction to tridiagonal form. For its solution, the algorithm of Multiple Relatively Robust Representations (MRRR) is among the fastest methods. Although fast, the solvers based on MRRR do not deliver the same accuracy as competing methods like Divide & Conquer or the QR algorithm. In this paper, we demonstrate that the use of mixed precisions leads to improved accuracy of MRRR-based eigensolvers with limited or no performance penalty. As a result, we obtain eigensolvers that are not only equally or more accurate than the best available methods, but also -in most circumstances- faster and more scalable than the competition

Symmetric Submodular Function Minimization Under Hereditary Family Constraints

We present an efficient algorithm to find non-empty minimizers of a symmetric submodular function over any family of sets closed under inclusion. This for example includes families defined by a cardinality constraint, a knapsack constraint, a matroid independence constraint, or any combination of such constraints. Our algorithm make $O(n^3)$ oracle calls to the submodular function where $n$ is the cardinality of the ground set. In contrast, the problem of minimizing a general submodular function under a cardinality constraint is known to be inapproximable within $o(\sqrt{n/\log n})$ (Svitkina and Fleischer [2008]). The algorithm is similar to an algorithm of Nagamochi and Ibaraki [1998] to find all nontrivial inclusionwise minimal minimizers of a symmetric submodular function over a set of cardinality $n$ using $O(n^3)$ oracle calls. Their procedure in turn is based on Queyranne's algorithm [1998] to minimize a symmetric submodularComment: 13 pages, Submitted to SODA 201

Heavy-tailed Independent Component Analysis

Independent component analysis (ICA) is the problem of efficiently recovering a matrix $A \in \mathbb{R}^{n\times n}$ from i.i.d. observations of $X=AS$ where $S \in \mathbb{R}^n$ is a random vector with mutually independent coordinates. This problem has been intensively studied, but all existing efficient algorithms with provable guarantees require that the coordinates $S_i$ have finite fourth moments. We consider the heavy-tailed ICA problem where we do not make this assumption, about the second moment. This problem also has received considerable attention in the applied literature. In the present work, we first give a provably efficient algorithm that works under the assumption that for constant $\gamma > 0$, each $S_i$ has finite $(1+\gamma)$-moment, thus substantially weakening the moment requirement condition for the ICA problem to be solvable. We then give an algorithm that works under the assumption that matrix $A$ has orthogonal columns but requires no moment assumptions. Our techniques draw ideas from convex geometry and exploit standard properties of the multivariate spherical Gaussian distribution in a novel way.Comment: 30 page