[Paper Review] X-rank and identifiability for a polynomial decomposition model

Pierre Comon, Qi Yang+1 more|arXiv (Cornell University)|Mar 4, 2016

Tensor decomposition and applications28 references3 citations

TL;DR

This paper introduces X-rank decomposition as a generalization of tensor CP decomposition for polynomial models in system identification, signal processing, and machine learning. It establishes generic and maximal rank properties and proves identifiability conditions, making advanced algebraic geometry tools accessible to non-experts.

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 the polynomial decomposition model. In the paper, we try to make results and basic tools accessible for a general audience (assuming no knowledge of algebraic geometry or its prerequisites).

Motivation & Objective

• To develop a unified framework for polynomial decomposition using the novel X-rank concept from algebraic geometry.
• To establish generic and maximal rank properties for the polynomial decomposition model.
• To prove identifiability conditions under which the decomposition is unique.
• To make advanced algebraic geometry concepts accessible to researchers without prior expertise in the field.

Proposed method

• The paper formulates the polynomial decomposition model as a special case of X-rank decomposition, generalizing tensor CP decomposition.
• It leverages tools from algebraic geometry to analyze the structure and rank of the decomposition.
• The authors derive conditions under which the decomposition is generically identifiable by studying the dimension of the secant varieties.
• They introduce and apply the concept of X-rank to characterize the minimal number of terms needed in the decomposition.
• Theoretical results are developed using geometric reasoning, with an emphasis on clarity and accessibility for non-specialists.
• The framework is applied to real-world problems in system identification, signal processing, and machine learning to demonstrate practical relevance.

Experimental results

Research questions

• RQ1 What is the generic rank of the polynomial decomposition model under the X-rank framework?
• RQ2 What is the maximal rank of the decomposition, and how does it relate to the model's dimensionality?
• RQ3 Under what conditions is the decomposition uniquely identifiable?
• RQ4 How can algebraic geometry concepts like X-rank be made accessible to researchers outside the field?

Key findings

• The X-rank decomposition generalizes the tensor CP decomposition and provides a new algebraic framework for polynomial models.
• The paper establishes the generic rank of the decomposition model, showing the typical number of terms required for representation.
• It proves that the maximal rank is bounded by the dimension of the ambient space and the structure of the variety.
• Identifiability is achieved when the number of terms is below a critical threshold related to the dimension of the secant variety.
• The results are derived using geometric arguments that are systematically explained to be accessible without prior algebraic geometry knowledge.
• The framework enables new insights into polynomial decomposition in applications such as system identification and signal processing.