Sivan Toledo

Haim Avron, Christos Boutsidis, Sivan Toledo et al.

We present a fast algorithm for approximate Canonical Correlation Analysis (CCA). Given a pair of tall-and-thin matrices, the proposed algorithm first employs a randomized dimensionality reduction transform to reduce the size of the input matrices, and then applies any CCA algorithm to the new pair of matrices. The algorithm computes an approximate CCA to the original pair of matrices with provable guarantees, while requiring asymptotically less operations than the state-of-the-art exact algorithms.

Haim Avron, Sivan Toledo

We define the notion of effective stiffness and show that it can used to build sparsifiers, algorithms that sparsify linear systems arising from finite-element discretizations of PDEs. In particular, we show that sampling $O(n\log n)$ elements according to probabilities derived from effective stiffnesses yields a high quality preconditioner that can be used to solve the linear system in a small number of iterations. Effective stiffness generalizes the notion of effective resistance, a key ingredient of recent progress in developing nearly linear symmetric diagonally dominant (SDD) linear solvers. Solving finite elements problems is of considerably more interest than the solution of SDD linear systems, since the finite element method is frequently used to numerically solve PDEs arising in scientific and engineering applications. Unlike SDD systems, which are relatively easy to solve, there has been limited success in designing fast solvers for finite element systems, and previous algorithms usually target discretization of limited class of PDEs like scalar elliptic or 2D trusses. Our sparsifier is general; it applies to a wide range of finite-element discretizations. A sparsifier does not constitute a complete linear solver. To construct a solver, one needs additional components (e.g., an efficient elimination or multilevel scheme for the sparsified system). Still, sparsifiers have been a critical tools in efficient SDD solvers, and we believe that our sparsifier will become a key ingredient in future fast finite-element solvers.

Haim Avron, Alex Druinsky, Sivan Toledo

We describe a randomized Krylov-subspace method for estimating the spectral condition number of a real matrix A or indicating that it is numerically rank deficient. The main difficulty in estimating the condition number is the estimation of the smallest singular value σ_{\min} of A. Our method estimates this value by solving a consistent linear least-squares problem with a known solution using a specific Krylov-subspace method called LSQR. In this method, the forward error tends to concentrate in the direction of a right singular vector corresponding to σ_{\min}. Extensive experiments show that the method is able to estimate well the condition number of a wide array of matrices. It can sometimes estimate the condition number when running a dense SVD would be impractical due to the computational cost or the memory requirements. The method uses very little memory (it inherits this property from LSQR) and it works equally well on square and rectangular matrices.

Alex Druinsky, Sivan Toledo

Several widely-used textbooks lead the reader to believe that solving a linear system of equations Ax = b by multiplying the vector b by a computed inverse inv(A) is inaccurate. Virtually all other textbooks on numerical analysis and numerical linear algebra advise against using computed inverses without stating whether this is accurate or not. In fact, under reasonable assumptions on how the inverse is computed, x = inv(A)*b is as accurate as the solution computed by the best backward-stable solvers. This fact is not new, but obviously obscure. We review the literature on the accuracy of this computation and present a self-contained numerical analysis of it.

Alex Druinsky, Eyal Carlebach, Sivan Toledo

We propose a new inertia-revealing factorization for sparse symmetric matrices. The factorization scheme and the method for extracting the inertia from it were proposed in the 1960s for dense, banded, or tridiagonal matrices, but they have been abandoned in favor of faster methods. We show that this scheme can be applied to any sparse symmetric matrix and that the fill in the factorization is bounded by the fill in the sparse QR factorization of the same matrix (but is usually much smaller). We describe our serial proof-of-concept implementation, and present experimental results, studying the method's numerical stability and performance.

Vadim Stotland, Oded Schwartz, Sivan Toledo

Algorithms and implementations for computing the sign function of a triangular matrix are fundamental building blocks in algorithms for computing the sign of arbitrary square real or complex matrices. We present novel recursive and cache efficient algorithms that are based on Higham's stabilized specialization of Parlett's substitution algorithm for computing the sign of a triangular matrix. We show that the new recursive algorithms are asymptotically optimal in terms of the number of cache misses that they generate. One of the novel algorithms that we present performs more arithmetic than the non-recursive version, but this allows it to benefit from calling highly-optimized matrix-multiplication routines; the other performs the same number of operations as the non-recursive version, but it uses custom computational kernels instead. We present implementations of both, as well as a cache-efficient implementation of a block version of Parlett's algorithm. Our experiments show that the blocked and recursive versions are much faster than the previous algorithms, and that the inertia strongly influences their relative performance, as predicted by our analysis.

Haim Avron, Gil Shklarski, Sivan Toledo

This short communication shows that in some cases scalar elliptic finite element matrices cannot be approximated well by an SDD matrix. We also give a theoretical analysis of a simple heuristic method for approximating an element by an SDD matrix.