Research Interests:
Research Interests:
Research Interests:
Mathematics, Computer Science, Statistics, Data Mining, Data Analysis, and 15 moreMonte Carlo, Approximation Algorithms, Numerical Analysis, Clustering Algorithms, Algorithm, Monte Carlo Methods, Data Structure, Singular value decomposition, APPROXIMATION ALGORITHM, Data storage, Matrix Decomposition, Data Consistency, Monte Carlo Method, Application Software, and Best approximation
Research Interests:
Research Interests:
Abstract We give sufficient conditions for the description of the isometry group of a tensor product in terms of the isometry groups of the factors. A slightly more general theory involving tensor products and saturated linear groups is... more
Abstract We give sufficient conditions for the description of the isometry group of a tensor product in terms of the isometry groups of the factors. A slightly more general theory involving tensor products and saturated linear groups is developed for this purpose.
Research Interests:
Research Interests:
Research Interests:
Research Interests:
Research Interests:
Research Interests:
Research Interests:
In this paper, we consider an extension and rigorous justification of Karhunen-Loeve transform (KLT) which is an optimal technique for data compression. We propose and study the generic KLT which is treated as the best weighted linear... more
In this paper, we consider an extension and rigorous justification of Karhunen-Loeve transform (KLT) which is an optimal technique for data compression. We propose and study the generic KLT which is treated as the best weighted linear estimator of a given rank under the condition that the associated covariance matrix is singular. As a result, the generic KLT is constructed in terms of the pseudo-inverse matrices that imply a development of the special technique. In particular, we give a solution of the new low-rank matrix approximation problem that provides a basis for the generic KLT. Theoretical aspects of the generic KLT are carefully studied.
Research Interests:
Research Interests:
Research Interests:
Research Interests:
Research Interests:
Research Interests:
Research Interests:
Research Interests:
Research Interests:
Let be the field of invariant functions induced by the conjugacy action of the general linear group on in m+1 tuples of square matrices such that the first one is singular. It is shown here that is rational.
Research Interests:
Research Interests:
Research Interests:
Research Interests:
Research Interests:
Research Interests:
Research Interests:
Research Interests:
... Anal., 8 (1988), 141-148. [2], S. Bellavia, C. Cartis, NIM Gould, B. Morini and Ph. ... [18], GH Golub and CF Van Loan, "Matrix Computations,'' 3rd... more
... Anal., 8 (1988), 141-148. [2], S. Bellavia, C. Cartis, NIM Gould, B. Morini and Ph. ... [18], GH Golub and CF Van Loan, "Matrix Computations,'' 3rd Edition, Johns Hopkins University Press, Baltimore and London, 1996. [19], NIM Gould, D. Orban and Ph. ...
Research Interests:
Research Interests:
Research Interests:
Research Interests:
Research Interests:
Research Interests:
Research Interests:
Research Interests:
Research Interests:
Research Interests:
Research Interests:
We discuss here analogs of van der Waerden and Tverberg permanent conjectures for haffnians on the convex set of matrices whose extreme points are symmetric permutation matrices with zero diagonal.
We study theoretical and computational properties of the pressure function for subshifts of finite type on the integer lattice ^d, multidimensional SOFT, which are called Potts models in mathematical physics. We show that the pressure is... more
We study theoretical and computational properties of the pressure function for subshifts of finite type on the integer lattice ^d, multidimensional SOFT, which are called Potts models in mathematical physics. We show that the pressure is Lipschitz and convex. We use the properties of convex functions to show rigorously that the phase transition of the first order correspond exactly to the points where the pressure is not differentiable. We give computable upper and lower bounds for the pressure, which can be arbitrary close the values of the pressure given a sufficient computational power. We apply our numerical methods to confirm Baxter's heuristic computations for two dimensional monomer-dimer model, and to compute the pressure and the density entropy as functions of two variables for the two dimensional monomer-dimer model.
Research Interests:
We establish several mathematical and computational properties of the nuclear norm for higher-order tensors. We show that like tensor rank, tensor nuclear norm is dependent on the choice of base field --- the value of the nuclear norm of... more
We establish several mathematical and computational properties of the nuclear norm for higher-order tensors. We show that like tensor rank, tensor nuclear norm is dependent on the choice of base field --- the value of the nuclear norm of a real 3-tensor depends on whether we regard it as a real 3-tensor or a complex 3-tensor with real entries. We show that every tensor has a nuclear norm attaining decomposition and every symmetric tensor has a symmetric nuclear norm attaining decomposition. There is a corresponding notion of nuclear rank that, unlike tensor rank, is upper semicontinuous. We establish an analogue of Banach's theorem for tensor spectral norm and Comon's conjecture for tensor rank --- for a symmetric tensor, its symmetric nuclear norm always equals its nuclear norm. We show that computing tensor nuclear norm is NP-hard in several sense. Deciding weak membership in the nuclear norm unit ball of 3-tensors is NP-hard, as is finding an ε-approximation of nuclear no...
Research Interests:
We establish several mathematical and computational properties of the nuclear norm for higher-order tensors. We show that like tensor rank, tensor nuclear norm is dependent on the choice of base field --- the value of the nuclear norm of... more
We establish several mathematical and computational properties of the nuclear norm for higher-order tensors. We show that like tensor rank, tensor nuclear norm is dependent on the choice of base field --- the value of the nuclear norm of a real 3-tensor depends on whether we regard it as a real 3-tensor or a complex 3-tensor with real entries. We show that every tensor has a nuclear norm attaining decomposition and every symmetric tensor has a symmetric nuclear norm attaining decomposition. There is a corresponding notion of nuclear rank that, unlike tensor rank, is upper semicontinuous. We establish an analogue of Banach's theorem for tensor spectral norm and Comon's conjecture for tensor rank --- for a symmetric tensor, its symmetric nuclear norm always equals its nuclear norm. We show that computing tensor nuclear norm is NP-hard in several sense. Deciding weak membership in the nuclear norm unit ball of 3-tensors is NP-hard, as is finding an ε-approximation of nuclear no...
Research Interests:
We show that the irreducible variety of 4 x 4 x 4 complex valued tensors of border rank at most 4 is the zero set of polynomial equations of degree 5 (the Strassen commutative conditions), of degree 6 (the Landsberg-Manivel polynomials),... more
We show that the irreducible variety of 4 x 4 x 4 complex valued tensors of border rank at most 4 is the zero set of polynomial equations of degree 5 (the Strassen commutative conditions), of degree 6 (the Landsberg-Manivel polynomials), and of degree 9 (the symmetrization conditions).
Research Interests:
We study tensors in ^m× n× l whose border rank is l. We characterize the tensors in ^3× 3× 4 and in ^4× 4× 4 of border rank 4 at most.
Let (lambda_d)(p) be the p monomer-dimer entropy on the d-dimensional integer lattice Z^d, where p in [0,1] is the dimer density. We give upper and lower bounds for (lambda_d)(p) in terms of expressions involving (lambda_(d-1))(q). The... more
Let (lambda_d)(p) be the p monomer-dimer entropy on the d-dimensional integer lattice Z^d, where p in [0,1] is the dimer density. We give upper and lower bounds for (lambda_d)(p) in terms of expressions involving (lambda_(d-1))(q). The upper bound is based on a conjecture claiming that the p monomer-dimer entropy of an infinite subset of Z^d is bounded above by (lambda_d)(p). We compute the first three terms in the formal asymptotic expansion of (lambda_d)(p) in powers of 1/d. We prove that the lower asymptotic matching conjecture is satisfied for (lambda_d)(p).
Research Interests:
In this paper we consider a linear homogeneous system of m equations in n unknowns with integer coefficients over the reals. Assume that the sum of the absolute values of the coefficients of each equation does not exceed k+1 for some... more
In this paper we consider a linear homogeneous system of m equations in n unknowns with integer coefficients over the reals. Assume that the sum of the absolute values of the coefficients of each equation does not exceed k+1 for some positive integer k. We show that if the system has a nontrivial solution then there exists a nontrivial solution =(x_1,...,x_n) such that |x_j|/|x_i|< k^n-1 for each i,j satisfying x_ix_j 0. This inequality is sharp. We also prove a conjecture of A. Tyszka related to our results.