Related papers: On the bit-complexity of sparse polynomial multipl…
Gr{\"o}bner bases is one the most powerful tools in algorithmic non-linear algebra. Their computation is an intrinsically hard problem with a complexity at least single exponential in the number of variables. However, in most of the cases,…
To preserve data privacy, multi-party computation (MPC) enables executing Machine Learning (ML) algorithms on private data. However, MPC frameworks do not include optimized operations on sparse data. This absence makes them unsuitable for…
We study the cost of multiplication modulo triangular families of polynomials. Following previous work by Li, Moreno Maza and Schost, we propose an algorithm that relies on homotopy and fast evaluation-interpolation techniques. We obtain a…
Sparse signal reconstruction algorithms have attracted research attention due to their wide applications in various fields. In this paper, we present a simple Bayesian approach that utilizes the sparsity constraint and a priori statistical…
This paper proposes new factorizations for computing the Neumann series. The factorizations are based on fast algorithms for small prime sizes series and the splitting of large sizes into several smaller ones. We propose a different basis…
Specialized function gradient computing hardware could greatly improve the performance of state-of-the-art optimization algorithms, e.g., based on gradient descent or conjugate gradient methods that are at the core of control, machine…
We present a non-commutative algorithm for the multiplication of a 2 x 2 block-matrix by its adjoint, defined by a matrix ring anti-homomorphism. This algorithm uses 5 block products (3 recursive calls and 2 general products)over C or in…
In this paper, we analyze in depth a simplicial decomposition like algorithmic framework for large scale convex quadratic programming. In particular, we first propose two tailored strategies for handling the master problem. Then, we…
We introduce a fast algorithm for computing sparse Fourier transforms supported on smooth curves or surfaces. This problem appear naturally in several important problems in wave scattering and reflection seismology. The main observation is…
Recently, a class of algorithms combining classical fixed point iterations with repeated random sparsification of approximate solution vectors has been successfully applied to eigenproblems with matrices as large as $10^{108} \times…
We describe a method to evaluate multivariate polynomials over a finite field and discuss its multiplicative complexity.
Based on the observation that $\mathbb{Q}^{(p-1) \times (p-1)}$ is isomorphic to a quotient skew polynomial ring, we propose a new method for $(p-1)\times (p-1)$ matrix multiplication over $\mathbb{Q}$, where $p$ is a prime number. The main…
Recently there has been much interest in "sparsifying" sums of rank one matrices: modifying the coefficients such that only a few are nonzero, while approximately preserving the matrix that results from the sum. Results of this sort have…
We propose a new approximate factorization for solving linear systems with symmetric positive definite sparse matrices. In a nutshell the algorithm is to apply hierarchically block Gaussian elimination and additionally compress the fill-in.…
Multiplication of polynomials is among key operations in computer algebra which plays important roles in developing techniques for other commonly used polynomial operations such as division, evaluation/interpolation, and factorization. In…
We show that assuming the availability of the processor with variable precision arithmetic, we can compute matrix-by-matrix multiplications in $O(N^2log_2N)$ computational complexity. We replace the standard matrix-by-matrix multiplications…
Traditional numerical methods for calculating matrix eigenvalues are prohibitively expensive for high-dimensional problems. Iterative random sparsification methods allow for the estimation of a single dominant eigenvalue at reduced cost by…
Univariate polynomial root-finding is a classical subject, still important for modern computing. Frequently one seeks just the real roots of a polynomial with real coefficients. They can be approximated at a low computational cost if the…
A polynomial matrix inequality is a formula asserting that a polynomial matrix is positive semidefinite. Polynomial matrix optimization concerns minimizing the smallest eigenvalue of a symmetric polynomial matrix subject to a tuple of…
We present an implementation of Pagh's compressed matrix multiplication algorithm, a randomized algorithm that constructs sketches of matrices to compute an unbiased estimate of their product. By leveraging fast polynomial multiplication…