A fast semi-direct least squares algorithm for hierarchically block separable matrices

We present a fast algorithm for linear least squares problems governed by hierarchically block separable (HBS) matrices. Such matrices are generally dense but data-sparse and can describe many important operators including those derived from asymptotically smooth radial kernels that are not too oscillatory. The algorithm is based on a recursive skeletonization procedure that exposes this sparsity and solves the dense least squares problem as a larger, equality-constrained, sparse one. It relies on a sparse QR factorization coupled with iterative weighted least squares methods. In essence, our scheme consists of a direct component, comprised of matrix compression and factorization, followed by an iterative component to enforce certain equality constraints. At most two iterations are typically required for problems that are not too ill-conditioned. For an $M \times N$ HBS matrix with $M \geq N$ having bounded off-diagonal block rank, the algorithm has optimal $\mathcal{O} (M + N)$ complexity. If the rank increases with the spatial dimension as is common for operators that are singular at the origin, then this becomes $\mathcal{O} (M + N)$ in 1D, $\mathcal{O} (M + N^{3/2})$ in 2D, and $\mathcal{O} (M + N^{2})$ in 3D. We illustrate the performance of the method on both over- and underdetermined systems in a variety of settings, with an emphasis on radial basis function approximation and efficient updating and downdating.Comment: 24 pages, 8 figures, 6 tables; to appear in SIAM J. Matrix Anal. App

A fast approach for overcomplete sparse decomposition based on smoothed L0 norm

In this paper, a fast algorithm for overcomplete sparse decomposition, called SL0, is proposed. The algorithm is essentially a method for obtaining sparse solutions of underdetermined systems of linear equations, and its applications include underdetermined Sparse Component Analysis (SCA), atomic decomposition on overcomplete dictionaries, compressed sensing, and decoding real field codes. Contrary to previous methods, which usually solve this problem by minimizing the L1 norm using Linear Programming (LP) techniques, our algorithm tries to directly minimize the L0 norm. It is experimentally shown that the proposed algorithm is about two to three orders of magnitude faster than the state-of-the-art interior-point LP solvers, while providing the same (or better) accuracy.Comment: Accepted in IEEE Transactions on Signal Processing. For MATLAB codes, see (http://ee.sharif.ir/~SLzero). File replaced, because Fig. 5 was missing erroneousl

An improved multi-parametric programming algorithm for flux balance analysis of metabolic networks

Flux balance analysis has proven an effective tool for analyzing metabolic networks. In flux balance analysis, reaction rates and optimal pathways are ascertained by solving a linear program, in which the growth rate is maximized subject to mass-balance constraints. A variety of cell functions in response to environmental stimuli can be quantified using flux balance analysis by parameterizing the linear program with respect to extracellular conditions. However, for most large, genome-scale metabolic networks of practical interest, the resulting parametric problem has multiple and highly degenerate optimal solutions, which are computationally challenging to handle. An improved multi-parametric programming algorithm based on active-set methods is introduced in this paper to overcome these computational difficulties. Degeneracy and multiplicity are handled, respectively, by introducing generalized inverses and auxiliary objective functions into the formulation of the optimality conditions. These improvements are especially effective for metabolic networks because their stoichiometry matrices are generally sparse; thus, fast and efficient algorithms from sparse linear algebra can be leveraged to compute generalized inverses and null-space bases. We illustrate the application of our algorithm to flux balance analysis of metabolic networks by studying a reduced metabolic model of Corynebacterium glutamicum and a genome-scale model of Escherichia coli. We then demonstrate how the critical regions resulting from these studies can be associated with optimal metabolic modes and discuss the physical relevance of optimal pathways arising from various auxiliary objective functions. Achieving more than five-fold improvement in computational speed over existing multi-parametric programming tools, the proposed algorithm proves promising in handling genome-scale metabolic models.Comment: Accepted in J. Optim. Theory Appl. First draft was submitted on August 4th, 201

Underdetermined-order recursive least-squares adaptive filtering: The concept and algorithms

Published versio

Further Results on Performance Analysis for Compressive Sensing Using Expander Graphs

Compressive sensing is an emerging technology which can recover a sparse signal vector of dimension n via a much smaller number of measurements than n. In this paper, we will give further results on the performance bounds of compressive sensing. We consider the newly proposed expander graph based compressive sensing schemes and show that, similar to the l_1 minimization case, we can exactly recover any k-sparse signal using only O(k log(n)) measurements, where k is the number of nonzero elements. The number of computational iterations is of order O(k log(n)), while each iteration involves very simple computational steps