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Fast tensorial JADE

Joni Virta, Niko Lietzén, Pauliina Ilmonen, Klaus Nordhausen

Abstract

In this work, we propose a novel method for tensorial independent component analysis. Our approach is based on TJADE and $ k $-JADE, two recently proposed generalizations of the classical JADE algorithm. Our novel method achieves the consistency and the limiting distribution of TJADE under mild assumptions, and at the same time offers notable improvement in computational speed. Detailed mathematical proofs of the statistical properties of our method are given and, as a special case, a conjecture on the properties of $ k $-JADE is resolved. Simulations and timing comparisons demonstrate remarkable gain in speed. Moreover, the desired efficiency is obtained approximately for finite samples. The method is applied successfully to large-scale video data, for which neither TJADE nor $ k $-JADE is feasible. Finally, an experimental procedure is proposed to select the values of a set of tuning parameters.

Fast tensorial JADE

Abstract

In this work, we propose a novel method for tensorial independent component analysis. Our approach is based on TJADE and -JADE, two recently proposed generalizations of the classical JADE algorithm. Our novel method achieves the consistency and the limiting distribution of TJADE under mild assumptions, and at the same time offers notable improvement in computational speed. Detailed mathematical proofs of the statistical properties of our method are given and, as a special case, a conjecture on the properties of -JADE is resolved. Simulations and timing comparisons demonstrate remarkable gain in speed. Moreover, the desired efficiency is obtained approximately for finite samples. The method is applied successfully to large-scale video data, for which neither TJADE nor -JADE is feasible. Finally, an experimental procedure is proposed to select the values of a set of tuning parameters.

Paper Structure

This paper contains 13 sections, 6 theorems, 106 equations, 11 figures.

Table of Contents

  1. Tensorial independent component analysis
  2. Tensorial JADE
  3. Tensorial $k$-JADE
  4. A note on tensorial ICA
  5. Simulations and examples
  6. Finite-sample efficiency, setting 1
  7. Finite-sample efficiency, setting 2
  8. Finite-sample efficiency, setting 3
  9. Timing comparison
  10. Video example
  11. Estimation of the maximal kurtosis multiplicity $\nu$
  12. Discussion
  13. Proofs

Key Result

Theorem \oldthetheorem

Let Assumptions assu:JADE and assu:FOBI$(v)$ hold for some fixed $v$. Then the $k$-TJADE functional $\boldsymbol{\Gamma}^k$ is a matrix IC functional for all $k \geq v$.

Figures (11)

  • Figure 1: The 1-mode, 2-mode and 3-mode vectors of a 3-dimensional tensor. Elina Vartiainen©.
  • Figure 2: The 1-mode, 2-mode and 3-mode faces of a 3-dimensional tensor. Elina Vartiainen©.
  • Figure 3: Means of the transformed MD indices. The solid black line is the limiting value of TJADE (under orthogonal mixing) and the dashed black line is the limiting value of VJADE. The solid line is missing from the third panel as the limiting value of TJADE is not known under normal mixing.
  • Figure 4: Means of the transformed MD indices. The solid black line is the limiting value of TJADE (under orthogonal mixing) and the dashed black line is the limiting value of VJADE. The solid line is missing from the third panel as the limiting value of TJADE is not known under normal mixing.
  • Figure 5: Means of the transformed MD indices. The solid black line is the limiting value of TJADE (under orthogonal mixing).
  • ...and 6 more figures

Theorems & Definitions (15)

  • Definition 1
  • Definition 2
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Lemma 1
  • proof : Proof of Lemma \ref{['lemma:diag']}
  • Lemma 2
  • proof : Proof of Lemma \ref{['lem:H_simplification']}
  • Lemma 3
  • ...and 5 more