Paul Breiding

The Condition Number of Join Decompositions

The join set of a finite collection of smooth embedded submanifolds of a mutual vector space is defined as their Minkowski sum. Join decompositions generalize some ubiquitous decompositions in multilinear algebra, namely tensor rank, Waring, partially symmetric rank and block term decompositions. This paper examines the numerical sensitivity of join decompositions to perturbations; specifically, we consider the condition number for general join decompositions. It is characterized as a distance to a set of ill-posed points in a supplementary product of Grassmannians. We prove that this condition number can be computed efficiently as the smallest singular value of an auxiliary matrix. For some special join sets, we characterized the behavior of sequences in the join set converging to the latters boundary points. Finally, we specialize our discussion to the tensor rank and Waring decompositions and provide several numerical experiments confirming the key results.

The Expected Number of Eigenvalues of a Real Gaussian Tensor

A real number

Convergence analysis of Riemannian Gauss–Newton methods and its connection with the geometric condition number

\lambda

HomotopyContinuation.jl: A Package for Homotopy Continuation in Julia

is called a Z-eigenvalue of a tensor

Distribution of the eigenvalues of a random system of homogeneous polynomials

A

HomotopyContinuation.jl - a package for solving systems of polynomial equations in Julia

, if

A Riemannian Trust Region Method for the Canonical Tensor Rank Approximation Problem

\lambda

The average number of critical rank-one-approximations to a symmetric tensor

is an eigenvalue of

Numerical and statistical aspects of tensor decompositions

A

Towards a condition number theorem for the tensor rank decomposition

and the corresponding eigenvector