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Generalized sparse matrix-matrix multiplication is a key primitive for many high performance graph algorithms as well as some linear solvers such as multigrid. We present the first parallel algorithms that achieve increasing speedups for an…
This paper proposes a new distributed algorithm for solving linear systems associated with a sparse graph under a generalised diagonal dominance assumption. The algorithm runs iteratively on each node of the graph, with low complexities on…
Graphs arising in statistical problems, signal processing, large networks, combinatorial optimization, and data analysis are often dense, which causes both computational and storage bottlenecks. One way of \textit{sparsifying} a…
Polynomial optimization problems are infinite-dimensional, nonconvex, NP-hard, and are often handled in practice with the moment-sums of squares hierarchy of semidefinite programming bounds. We consider problems where the objective function…
A randomized algorithm for computing a data sparse representation of a given rank structured matrix $A$ (a.k.a. an $H$-matrix) is presented. The algorithm draws on the randomized singular value decomposition (RSVD), and operates under the…
Inversion of sparse matrices with standard direct solve schemes is robust, but computationally expensive. Iterative solvers, on the other hand, demonstrate better scalability; but, need to be used with an appropriate preconditioner (e.g.,…
Every real hyperbolic form in three variables can be realized as the determinant of a linear net of Hermitian matrices containing a positive definite matrix. Such representations are an algebraic certificate for the hyperbolicity of the…
A new algorithm is derived for computing the actions $f(tA)B$ and $f(tA^{1/2})B$, where $f$ is cosine, sinc, sine, hyperbolic cosine, hyperbolic sinc, or hyperbolic sine function. $A$ is an $n\times n$ matrix and $B$ is $n\times n_0$ with…
Snapshot matrices built from solutions to hyperbolic partial differential equations exhibit slow decay in singular values, whereas fast decay is crucial for the success of projection- based model reduction methods. To overcome this problem,…
We consider heuristic algorithm for solving graph isomorphism problem. The algorithm based on a successive splitting of the eigenvalues of the matrices which are modifications (to positive defined) of graphs' adjacency matrices.…
Symmetric submodular maximization is an important class of combinatorial optimization problems, including MAX-CUT on graphs and hyper-graphs. The state-of-the-art algorithm for the problem over general constraints has an approximation ratio…
In many problems in Computational Physics and Chemistry, one finds a special kind of sparse matrices, termed "banded matrices". These matrices, which are defined as having non-zero entries only within a given distance from the main…
We develop a natural variant of Dikin's affine-scaling method, first for semidefinite programming and then for hyperbolic programming in general. We match the best complexity bounds known for interior-point methods. All previous…
We introduce a notion of code sparsification that generalizes the notion of cut sparsification in graphs. For a (linear) code $\mathcal{C} \subseteq \mathbb{F}_q^n$ of dimension $k$ a $(1 \pm \epsilon)$-sparsification of size $s$ is given…
A novel parallel algorithm for matrix multiplication is presented. The hyper-systolic algorithm makes use of a one-dimensional processor abstraction. The procedure can be implemented on all types of parallel systems. It can handle…
Tilings of the hyperbolic plane are of significant interest among many branches of mathematics, physics and computer science. Yet, their construction remains a non-trivial task. Current approaches primarily use tree-based recursive…
We show that one can define and effectively compute Stallings graphs for quasi-convex subgroups of automatic groups (\textit{e.g.} hyperbolic groups or right-angled Artin groups). These Stallings graphs are finite labeled graphs, which are…
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…
Algorithms are presented for evaluating gradients and Hessians of logarithmic barrier functions for two types of convex cones: the cone of positive semidefinite matrices with a given sparsity pattern, and its dual cone, the cone of sparse…
Can linear systems be solved faster than matrix multiplication? While there has been remarkable progress for the special cases of graph structured linear systems, in the general setting, the bit complexity of solving an $n \times n$ linear…