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We consider the "partial information decomposition" (PID) problem, which aims to decompose the information that a set of source random variables provide about a target random variable into separate redundant, synergistic, union, and unique…
Boolean tensor decomposition approximates data of multi-way binary relationships as product of interpretable low-rank binary factors, following the rules of Boolean algebra. Here, we present its first probabilistic treatment. We facilitate…
In this paper, it is shown that every polynomial function is mixed monotone globally with a polynomial decomposition function. For univariate polynomials, the decomposition functions can be constructed from the Gram matrix representation of…
This paper studies how to recover parameters in diagonal Gaussian mixture models using tensors. High-order moments of the Gaussian mixture model are estimated from samples. They form incomplete symmetric tensors generated by hidden…
Physiological signals are often organized in the form of multiple dimensions (e.g., channel, time, task, and 3D voxel), so it is better to preserve original organization structure when processing. Unlike vector-based methods that destroy…
In some fields such as Mathematics Mechanization, automated reasoning and Trustworthy Computing etc., exact results are needed. Symbolic computations are used to obtain the exact results. Symbolic computations are of high complexity. In…
In the present paper we derive complicated families of orthogonal polynomials in one variable from scratch using the known ones as building blocks. We recall the basics of operational formalism and introduce the notations we use throughout…
Given a system of n homogeneous polynomials in n variables which is equivariant with respect to the canonical actions of the symmetric group of n symbols on the variables and on the polynomials, it is proved that its resultant can be…
In a multiple linear regression model, the algebraic formula of the decomposition theorem explains the relationship between the univariate regression coefficient and partial regression coefficient using geometry. It was found that…
We introduce some basic notions and results for quaternionic linear operators analogous to those for complex linear operators. Our main result is to prove the additive and multiplicative Jordan-Chevalley decompositions for quaternionic…
It was recently shown [7, 9] that "properly built" linear and polyhedral estimates nearly attain minimax accuracy bounds in the problem of recovery of unknown signal from noisy observations of linear images of the signal when the signal set…
The modular decomposition is a technique that applies but is not restricted to graphs. The notion of module naturally appears in the proofs of many graph theoretical theorems. Computing the modular decomposition tree is an important…
Multi-view learning is frequently used in data science. The pairwise correlation maximization is a classical approach for exploring the consensus of multiple views. Since the pairwise correlation is inherent for two views, the extensions to…
We consider an approximation scheme for multivariate information assuming that synergistic information only appearing in higher order joint distributions is suppressed, which may hold in large classes of systems. Our approximation scheme…
This work is divided into three parts. The first part concerns polynomials in one variable with all real roots. We consider linear transformations that preserve real rootedness, as well as matrices that preserve interlacing. The second part…
An interpolation-based decoding scheme for interleaved subspace codes is presented. The scheme can be used as a (not necessarily polynomial-time) list decoder as well as a probabilistic unique decoder. Both interpretations allow to decode…
The widespread use of multisensor technology and the emergence of big datasets have created the need to develop tools to reduce, approximate, and classify large and multimodal data such as higher-order tensors. While early approaches…
We present an exact and complete algorithm to isolate the real solutions of a zero-dimensional bivariate polynomial system. The proposed algorithm constitutes an elimination method which improves upon existing approaches in a number of…
This paper analyses a Waring type decomposition of a noncommuting (NC) polynomial $p$ with respect to the goal of evaluating $p$ efficiently on tuples of matrices. Such a decomposition can reduce the number of matrix multiplications needed…
The purpose of this note is to survey a methodology to solve systems of polynomial equations and inequalities. The techniques we discuss use the algebra of multivariate polynomials with coefficients over a field to create large-scale linear…