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Deep neural network with rectified linear units (ReLU) is getting more and more popular recently. However, the derivatives of the function represented by a ReLU network are not continuous, which limit the usage of ReLU network to situations…
A framework for linear-programming (LP) decoding of nonbinary linear codes over rings is developed. This framework facilitates linear-programming based reception for coded modulation systems which use direct modulation mapping of coded…
Given a parity-check matrix $H_m$ of a $q$-ary Hamming code, we consider a partition of the columns into two subsets. Then, we consider the two codes that have these submatrices as parity-check matrices. We say that anyone of these two…
First, we study geometric variants of the standard set cover motivated by assignment of directional antenna and shipping with deadlines, providing the first known polynomial-time exact solutions. Next, we consider the following general…
Consider the following optimization problem: Given $n \times n$ matrices $A$ and $\Lambda$, maximize $\langle A, U\Lambda U^*\rangle$ where $U$ varies over the unitary group $\mathrm{U}(n)$. This problem seeks to approximate $A$ by a matrix…
In this paper we illustrate that paraorthogonality on the unit circle $\mathbb{T}$ is the counterpart to orthogonality on $\mathbb{R}$ when we are interested in the spectral properties. We characterize quasi-paraorthogonal polynomials on…
In this paper, we achieve three primary objectives related to the rigorous computational analysis of nonlinear PDEs posed on complex geometries such as disks and cylinders. First, we introduce a validated Matrix Multiplication Transform…
Deep polar codes are pre-transformed polar codes that employ a multi-layered polar kernel transformation strategy to enhance code performance in short blocklength regimes. However, like conventional polar codes, their block length is…
We consider the problem of network coding across multiple unicasts. We give, for wired and wireless networks, efficient polynomial time algorithms for finding optimal network codes within the class of network codes restricted to XOR coding…
This paper deals with the numerical computation of the least singular value of a rectangular matrix $A$ relative to a pair of closed convex cones $(P,Q)$, which is defined as the optimal value of the non-convex optimization problem of…
Secure codes are widely-studied combinatorial structures which were introduced for traitor tracing in broadcast encryption. To determine the maximum size of such structures is the main research objective. In this paper, we investigate the…
As users and developers, we are witnessing the opening of a new computing scenario: the introduction of hybrid processors into a single die, such as an accelerated processing unit (APU) processor, and the plug-and-play of additional…
This paper is a continuation of \ct{cmf16} where an efficient algorithm for computing the maximal eigenpair was introduced first for tridiagonal matrices and then extended to the irreducible matrices with nonnegative off-diagonal elements.…
Linear programming (polynomial) techniques are used to obtain lower and upper bounds for the potential energy of spherical designs. This approach gives unified bounds that are valid for a large class of potential functions. Our lower bounds…
The decomposition theory of matroids initiated by Paul Seymour in the 1980's has had an enormous impact on research in matroid theory. This theory, when applied to matrices over the binary field, yields a powerful decomposition theory for…
In this paper, we present FPT-algorithms for special cases of the shortest lattice vector, integer linear programming, and simplex width computation problems, when matrices included in the problems' formulations are near square. The…
Let $K$ be a field and $P=K[x_1,\dots,x_n]$. The technique of elimination by substitution is based on discovering a coherently $Z=(z_1,\dots,z_s)$-separating tuple of polynomials $(f_1,\dots,f_s)$ in an ideal $I$, i.e., on finding…
We study the combinatorial description of the LDU decomposition of totally positive matrices. We give a description of the lower triangular L, the diagonal D, and the upper triangular U matrices of the LDU decomposition of totally positive…
We present an algorithm for efficiently approximating of qubit unitaries over gate sets derived from totally definite quaternion algebras. It achieves $\varepsilon$-approximations using circuits of length $O(\log(1/\varepsilon))$, which is…
In this work, we apply the idea of composite matrices arising from group rings to derive a number of different techniques for constructing self-dual codes over finite commutative Frobenius rings. By applying these techniques over different…