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Cumulant Tensors in Partitioned Independent Component Analysis

Marina Garrote-López, Monroe Stephenson

TL;DR

This work explores Partitioned Independent Component Analysis (PICA), an extension of the well-established Independent Component Analysis framework, and assumes alternative independence conditions, in particular, the PICA case, where only partitions of the sources are mutually independent.

Abstract

In this work, we explore Partitioned Independent Component Analysis (PICA), an extension of the well-established Independent Component Analysis (ICA) framework. Traditionally, ICA focuses on extracting a vector of independent source signals from a linear combination of them defined by a mixing matrix. We aim to provide a comprehensive understanding of the identifiability of this mixing matrix in ICA. Significant to our investigation, recent developments by Mesters and Zwiernik relax these strict independence requirements, studying the identifiability of the mixing matrix from zero restrictions on cumulant tensors. In this paper, we assume alternative independence conditions, in particular, the PICA case, where only partitions of the sources are mutually independent. We study this case from an algebraic perspective, and our primary result generalizes previous results on the identifiability of the mixing matrix.

Cumulant Tensors in Partitioned Independent Component Analysis

TL;DR

This work explores Partitioned Independent Component Analysis (PICA), an extension of the well-established Independent Component Analysis framework, and assumes alternative independence conditions, in particular, the PICA case, where only partitions of the sources are mutually independent.

Abstract

In this work, we explore Partitioned Independent Component Analysis (PICA), an extension of the well-established Independent Component Analysis (ICA) framework. Traditionally, ICA focuses on extracting a vector of independent source signals from a linear combination of them defined by a mixing matrix. We aim to provide a comprehensive understanding of the identifiability of this mixing matrix in ICA. Significant to our investigation, recent developments by Mesters and Zwiernik relax these strict independence requirements, studying the identifiability of the mixing matrix from zero restrictions on cumulant tensors. In this paper, we assume alternative independence conditions, in particular, the PICA case, where only partitions of the sources are mutually independent. We study this case from an algebraic perspective, and our primary result generalizes previous results on the identifiability of the mixing matrix.
Paper Structure (19 sections, 37 theorems, 81 equations, 1 figure)

This paper contains 19 sections, 37 theorems, 81 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Independent Component Analysis
  3. The ICA model
  4. Identifiability of ICA
  5. Cumulant Tensors
  6. Symmetric tensors
  7. Moment and cumulant tensors
  8. Properties of Cumulant Tensors
  9. Cumulant tensors and independence
  10. Partitioned independence
  11. Mean Independence
  12. Independence induced by a graph
  13. Partitioned Independence Component Analysis
  14. The PICA model
  15. Algebraic approach in Component Analysis
  16. ...and 4 more sections

Key Result

Theorem 2.1

Let $\mathbf{s} \in \mathbb{R}^d$ be a vector with mutually independent components, of which at most one is Gaussian. Let $A$ be an orthogonal $d\times d$ matrix and ${\mathbf{y}}$ the white vector ${\mathbf{y}} = A \mathbf{s}.$ The following are equivalent:

Figures (1)

  • Figure 1: (L) Graph with 5 vertices and two connected components. (M) Star tree $S_d$ with $d-1$ leaves. (R) Chain graph with $d$ vertices.

Theorems & Definitions (68)

  • Theorem 2.1: Comon comon
  • Definition 2.1
  • Definition 2.2
  • Remark 2.1: normal
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3: Cramér cramer_1970
  • proof
  • ...and 58 more