Matrix Recipes for Hard Thresholding Methods

In this paper, we present and analyze a new set of low-rank recovery algorithms for linear inverse problems within the class of hard thresholding methods. We provide strategies on how to set up these algorithms via basic ingredients for different configurations to achieve complexity vs. accuracy tradeoffs. Moreover, we study acceleration schemes via memory-based techniques and randomized, $\epsilon$-approximate matrix projections to decrease the computational costs in the recovery process. For most of the configurations, we present theoretical analysis that guarantees convergence under mild problem conditions. Simulation results demonstrate notable performance improvements as compared to state-of-the-art algorithms both in terms of reconstruction accuracy and computational complexity.Comment: 26 page

Global Thresholding and Multiple Pass Parsing

We present a variation on classic beam thresholding techniques that is up to an order of magnitude faster than the traditional method, at the same performance level. We also present a new thresholding technique, global thresholding, which, combined with the new beam thresholding, gives an additional factor of two improvement, and a novel technique, multiple pass parsing, that can be combined with the others to yield yet another 50% improvement. We use a new search algorithm to simultaneously optimize the thresholding parameters of the various algorithms.Comment: Fixed latex errors; fixed minor errors in published versio

Two-Sample Tests for High Dimensional Means with Thresholding and Data Transformation

We consider testing for two-sample means of high dimensional populations by thresholding. Two tests are investigated, which are designed for better power performance when the two population mean vectors differ only in sparsely populated coordinates. The first test is constructed by carrying out thresholding to remove the non-signal bearing dimensions. The second test combines data transformation via the precision matrix with the thresholding. The benefits of the thresholding and the data transformations are showed by a reduced variance of the test thresholding statistics, the improved power and a wider detection region of the tests. Simulation experiments and an empirical study are performed to confirm the theoretical findings and to demonstrate the practical implementations.Comment: 64 page

Exact Hybrid Covariance Thresholding for Joint Graphical Lasso

This paper considers the problem of estimating multiple related Gaussian graphical models from a $p$-dimensional dataset consisting of different classes. Our work is based upon the formulation of this problem as group graphical lasso. This paper proposes a novel hybrid covariance thresholding algorithm that can effectively identify zero entries in the precision matrices and split a large joint graphical lasso problem into small subproblems. Our hybrid covariance thresholding method is superior to existing uniform thresholding methods in that our method can split the precision matrix of each individual class using different partition schemes and thus split group graphical lasso into much smaller subproblems, each of which can be solved very fast. In addition, this paper establishes necessary and sufficient conditions for our hybrid covariance thresholding algorithm. The superior performance of our thresholding method is thoroughly analyzed and illustrated by a few experiments on simulated data and real gene expression data