Structured Factorizations in Scalar Product Spaces

An Analysis of Tensor Models for Learning on Structured Data

While tensor factorizations have become increasingly popular for learning on various forms of structured data, only very few theoretical results exist on the generalization abilities of these methods. Here, we discuss the tensor product as a principled way to represent structured data in vector spaces for machine learning tasks. To derive generalization error bounds for tensor factorizations, w...

Some Properties of the Exchange Operator with respect to Structured Matrices defined by Indefinite Scalar Product Spaces

Perturbation Bounds for Hyperbolic Matrix Factorizations

Several matrix factorizations depend on orthogonal factors, matrices that preserve the Euclidean scalar product. Some of these factorizations can be extended and generalized to (J, J̃)-orthogonal factors, that is, matrices that satisfy H JH = J̃ , where J and J̃ are diagonal with diagonal elements ±1. The purpose of this work is to analyze the perturbation of matrix factorizations that have a (J, ...

Pursuits in Structured Non-Convex Matrix Factorizations

Efficiently representing real world data in a succinct and parsimonious manner is of central importance in many fields. We present a generalized greedy pursuit framework, allowing us to efficiently solve structured matrix factorization problems, where the factors are allowed to be from arbitrary sets of structured vectors. Such structure may include sparsity, non-negativeness, order, or a combi...

Structured Condition Numbers and Backward Errors in Scalar Product Spaces

We investigate the effect of structure-preserving perturbations on the solution to a linear system, matrix inversion, and distance to singularity. Particular attention is paid to linear and nonlinear structures that form Lie algebras, Jordan algebras and automorphism groups of a scalar product. These include complex symmetric, pseudo-symmetric, persymmetric, skewsymmetric, Hamiltonian, unitary,...