Matrix Factorizations Based on Induced Norms

TL;DR

This paper introduces a novel matrix decomposition method based on induced norms, generalizing duality diagrams and enabling new Euclidean multidimensional scaling models for data analysis.

Contribution

It presents a stepwise matrix decomposition framework using induced norms and extends duality diagrams to Banach spaces, broadening analytical tools in data analysis.

Findings

Generalized duality diagrams for Banach spaces

New family of Euclidean multidimensional scaling models

Enhanced matrix decomposition techniques

Abstract

We decompose a matrix Y into a sum of bilinear terms in a stepwise manner, by considering Y as a mapping from a finite dimensional Banach space into another finite dimensional Banach space. We provide transition formulas, and represent them in a duality diagram, thus generalizing the well known duality diagram in the french school of data analysis. As an application, we introduce a family of Euclidean multidimensional scaling models.