Faster Sparse Matrix Inversion and Rank Computation in Finite Fields

We improve the current best running time value to invert sparse matrices over finite fields, lowering it to an expected $O\big(n^{2.2131}\big)$ time for the current values of fast rectangular matrix multiplication. We achieve the same running time for the computation of the rank and nullspace of a sparse matrix over a finite field. This improvement relies on two key techniques. First, we adopt the decomposition of an arbitrary matrix into block Krylov and Hankel matrices from Eberly et al. (ISSAC 2007). Second, we show how to recover the explicit inverse of a block Hankel matrix using low displacement rank techniques for structured matrices and fast rectangular matrix multiplication algorithms. We generalize our inversion method to block structured matrices with other displacement operators and strengthen the best known upper bounds for explicit inversion of block Toeplitz-like and block Hankel-like matrices, as well as for explicit inversion of block Vandermonde-like matrices with structured blocks. As a further application, we improve the complexity of several algorithms in topological data analysis and in finite group theory

Faster Sparse Matrix Inversion and Rank Computation in Finite Fields

We improve the current best running time value to invert sparse matrices over finite fields, lowering it to an expected O(n^{2.2131}) time for the current values of fast rectangular matrix multiplication. We achieve the same running time for the computation of the rank and nullspace of a sparse matrix over a finite field. This improvement relies on two key techniques. First, we adopt the decomposition of an arbitrary matrix into block Krylov and Hankel matrices from Eberly et al. (ISSAC 2007). Second, we show how to recover the explicit inverse of a block Hankel matrix using low displacement rank techniques for structured matrices and fast rectangular matrix multiplication algorithms. We generalize our inversion method to block structured matrices with other displacement operators and strengthen the best known upper bounds for explicit inversion of block Toeplitz-like and block Hankel-like matrices, as well as for explicit inversion of block Vandermonde-like matrices with structured blocks. As a further application, we improve the complexity of several algorithms in topological data analysis and in finite group theory

On Sparse Representation in Fourier and Local Bases

We consider the classical problem of finding the sparse representation of a signal in a pair of bases. When both bases are orthogonal, it is known that the sparse representation is unique when the sparsity $K$ of the signal satisfies $K<1/\mu(D)$, where $\mu(D)$ is the mutual coherence of the dictionary. Furthermore, the sparse representation can be obtained in polynomial time by Basis Pursuit (BP), when $K<0.91/\mu(D)$. Therefore, there is a gap between the unicity condition and the one required to use the polynomial-complexity BP formulation. For the case of general dictionaries, it is also well known that finding the sparse representation under the only constraint of unicity is NP-hard. In this paper, we introduce, for the case of Fourier and canonical bases, a polynomial complexity algorithm that finds all the possible $K$-sparse representations of a signal under the weaker condition that $K<\sqrt{2} /\mu(D)$. Consequently, when $K<1/\mu(D)$, the proposed algorithm solves the unique sparse representation problem for this structured dictionary in polynomial time. We further show that the same method can be extended to many other pairs of bases, one of which must have local atoms. Examples include the union of Fourier and local Fourier bases, the union of discrete cosine transform and canonical bases, and the union of random Gaussian and canonical bases

TR-2008003: Unified Nearly Optimal Algorithms for Structured Integer Matrices and Polynomials

We seek the solution of banded, Toeplitz, Hankel, Vandermonde, Cauchy and other structured linear systems of equations with integer coefficients. By combining Hensel’s symbolic lifting with either divide-and-conquer algorithms or numerical iterative refinement, we unify the solution for all these structures. We yield the solution in nearly optimal randomized Boolean time, which covers both solution and its correctness verification. Our algorithms and nearly optimal time bounds are extended to the computation of the determinant of a structured integer matrix, its rank and a basis for its null space as well as to some fundamental computations with univariate polynomials that have integer coefficients. Furthermore, we allow to perform lifting modulo a properly bounded power of two t

Computational linear algebra over finite fields

We present here algorithms for efficient computation of linear algebra problems over finite fields

Parametrization of Newton's iteration for computations with structured matrices and applications

We apply a new parametrized version of Newton's iteration in order to compute (over any field F of constants) the solution, or at least-squares solution, to linear system Bx = v with an n × n Toeplitz or Toeplitz-like matrix B, as well as the determinant of B and the coefficients of its characteristic polynomial, det(λI − B), dramatically improving the processor efficiency of the known fast parallel algorithms. Our algorithms, together with some previously known and some recent results of [1““5], as well as with our new techniques for computing polynomial god's and lcm's, imply respective improvement of the known estimates for parallel arithmetic complexity of several fundamental computations with polynomials, and with both structured and general matrices

On Sparse Representation in Fourier and Local Bases

We consider the classical problem of finding the sparse representation of a signal in a pair of bases. When both bases are orthogonal, it is known that the sparse representation is unique when the sparsity K of the signal satisfies K <; 1/μ(D), where μ(D) is the mutual coherence of the dictionary. Furthermore, the sparse representation can be obtained in polynomial time by basis pursuit (BP), when K <; 0.91/μ(D). Therefore, there is a gap between the unicity condition and the one required to use the polynomial-complexity BP formulation. For the case of general dictionaries, it is also well known that finding the sparse representation under the only constraint of unicity is NP-hard. In this paper, we introduce, for the case of Fourier and canonical bases, a polynomial complexity algorithm that finds all the possible K-sparse representations of a signal under the weaker condition that K <; √2/μ(D). Consequently, when K <; 1/μ(D), the proposed algorithm solves the unique sparse representation problem for this structured dictionary in polynomial time. We further show that the same method can be extended to many other pairs of bases, one of which must have local atoms. Examples include the union of Fourier and local Fourier bases, the union of discrete cosine transform and canonical bases, and the union of random Gaussian and canonical bases.Engineering and Applied Science