Related papers: A faster hafnian formula for complex matrices and …
Matrix completion is the problem of recovering a low rank matrix by observing a small fraction of its entries. A series of recent works [KOM12,JNS13,HW14] have proposed fast non-convex optimization based iterative algorithms to solve this…
We propose a novel stochastic algorithm that randomly samples entire rows and columns of the matrix as a way to approximate an arbitrary matrix function using the power series expansion. This contrasts with existing Monte Carlo methods,…
We propose an improved algorithm for counting the number of Hamiltonian cycles in a directed graph. The basic idea of the method is sequential acceptance/rejection, which is successfully used in approximating the number of perfect matchings…
The maximum bipartite matching problem is among the most fundamental and well-studied problems in combinatorial optimization. A beautiful and celebrated combinatorial algorithm of Hopcroft and Karp (1973) shows that maximum bipartite…
Matrix factorization methods are important tools in data mining and analysis. They can be used for many tasks, ranging from dimensionality reduction to visualization. In this paper we concentrate on the use of matrix factorizations for…
Computing all-pairs shortest paths is a fundamental and much-studied problem with many applications. Unfortunately, despite intense study, there are still no significantly faster algorithms for it than the $\mathcal{O}(n^3)$ time algorithm…
The efficient implementation of matrix arithmetic operations underpins the speedups of many quantum algorithms. We develop a suite of methods to perform matrix arithmetics -- with the result encoded in the off-diagonal blocks of a…
Matrix powering is a fundamental computational primitive in linear algebra. It has widespread applications in scientific computing and engineering, and underlies the solution of time-homogeneous linear ordinary differential equations,…
Kernel matrices appear in machine learning and non-parametric statistics. Given $N$ points in $d$ dimensions and a kernel function that requires $\mathcal{O}(d)$ work to evaluate, we present an $\mathcal{O}(dN\log N)$-work algorithm for the…
We give an approximation algorithm for packing and covering linear programs (linear programs with non-negative coefficients). Given a constraint matrix with n non-zeros, r rows, and c columns, the algorithm computes feasible primal and dual…
The Hilbert-space Gaussian Process (HGP) approach offers a hyperparameter-independent basis function approximation for speeding up Gaussian Process (GP) inference by projecting the GP onto M basis functions. These properties result in a…
We present a simple and convenient analytical formula for efficient exact computation of the hafnian of Toeplitz matrices of a special type. An interpretation of the obtained results is given in the language of perfect matchings and Bessel…
The biggest cost of computing with large matrices in any modern computer is related to memory latency and bandwidth. The average latency of modern RAM reads is 150 times greater than a clock step of the processor. Throughput is a little…
We give an explicit algorithm and source code for computing optimal weights for combining a large number N of alphas. This algorithm does not cost O(N^3) or even O(N^2) operations but is much cheaper, in fact, the number of required…
Let {\alpha} be the maximal value such that the product of an n x n^{\alpha} matrix by an n^{\alpha} x n matrix can be computed with n^{2+o(1)} arithmetic operations. In this paper we show that \alpha>0.30298, which improves the previous…
Multipoint polynomial evaluation and interpolation are fundamental for modern symbolic and numerical computing. The known algorithms solve both problems over any field of constants in nearly linear arithmetic time, but the cost grows to…
Graph isomorphism problem is a known hard problem. In this paper, a novel randomized algorithm is proposed for this problem which is very simple and fast. It solves the graph isomorphism problem with running time O(n^2.373) for any pair of…
Cohn and Umans proposed a framework for developing fast matrix multiplication algorithms based on the embedding computation in certain groups algebras. In subsequent work with Kleinberg and Szegedy, they connected this to the search for…
Understanding the singular value spectrum of a matrix $A \in \mathbb{R}^{n \times n}$ is a fundamental task in countless applications. In matrix multiplication time, it is possible to perform a full SVD and directly compute the singular…
A graph $H$ is single-crossing if it can be drawn in the plane with at most one crossing. For any single-crossing graph $H$, we give an $O(n^4)$ time algorithm for counting perfect matchings in graphs excluding $H$ as a minor. The runtime…