Identifiability of an X-rank decomposition of polynomial maps

TL;DR

This paper investigates the identifiability of a polynomial decomposition model related to X-rank, providing new algebraic geometry results on rank and uniqueness, with accessible explanations for broader audiences.

Contribution

It introduces the polynomial decomposition as a special case of X-rank and proves new results on generic rank and identifiability, expanding understanding in algebraic geometry applications.

Findings

New results on generic and maximal rank of polynomial decompositions

Proofs of identifiability conditions for specific polynomial models

Accessible presentation of algebraic geometry tools for broader audience

Abstract

In this paper, we study a polynomial decomposition model that arises in problems of system identification, signal processing and machine learning. We show that this decomposition is a special case of the X-rank decomposition --- a powerful novel concept in algebraic geometry that generalizes the tensor CP decomposition. We prove new results on generic/maximal rank and on identifiability of a particular polynomial decomposition model. In the paper, we try to make results and basic tools accessible for general audience (assuming no knowledge of algebraic geometry or its prerequisites).