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Independent Component Analysis by Robust Distance Correlation

Sarah Leyder, Jakob Raymaekers, Peter J. Rousseeuw, Tom Van Deuren, Tim Verdonck

TL;DR

Independent Component Analysis (ICA) aims to decompose a $d$-variate observation into mutually independent sources, but classical ICA methods are often fragile in the presence of outliers. The authors propose RICA, a robust ICA framework that minimizes a robust dependence measure by applying a bowl transform to create a bounded, one-to-one, continuous, and redescending mapping before computing distance correlation $\mathrm{dCor}$; whitening is performed with the robust Minimum Covariance Determinant (MCD). The separating matrix is estimated sequentially via Givens rotations, with a derivative-free optimization and optional sweeps to refine the solution, and strong consistency with root-$n$ convergence is established. Empirical studies demonstrate that RICA outperforms competing methods under various contamination scenarios and real-data tasks (image data, cocktail party, and periodic signals), highlighting its practical robustness for blind source separation. Overall, RICA provides a principled, scalable approach to robust ICA that maintains the independence-detection property while mitigating the influence of outliers.

Abstract

Independent component analysis (ICA) is a powerful tool for decomposing a multivariate signal or distribution into fully independent sources, not just uncorrelated ones. Unfortunately, most approaches to ICA are not robust against outliers. Here we propose a robust ICA method called RICA, which estimates the components by minimizing a robust measure of dependence between multivariate random variables. The dependence measure used is the distance correlation (dCor). In order to make it more robust we first apply a new transformation called the bowl transform, which is bounded, one-to-one, continuous, and maps far outliers to points close to the origin. This preserves the crucial property that a zero dCor implies independence. RICA estimates the independent sources sequentially, by looking for the component that has the smallest dCor with the remainder. RICA is strongly consistent and has the usual parametric rate of convergence. Its robustness is investigated by a simulation study, in which it generally outperforms its competitors. The method is illustrated on three applications, including the well-known cocktail party problem.

Independent Component Analysis by Robust Distance Correlation

TL;DR

Independent Component Analysis (ICA) aims to decompose a -variate observation into mutually independent sources, but classical ICA methods are often fragile in the presence of outliers. The authors propose RICA, a robust ICA framework that minimizes a robust dependence measure by applying a bowl transform to create a bounded, one-to-one, continuous, and redescending mapping before computing distance correlation ; whitening is performed with the robust Minimum Covariance Determinant (MCD). The separating matrix is estimated sequentially via Givens rotations, with a derivative-free optimization and optional sweeps to refine the solution, and strong consistency with root- convergence is established. Empirical studies demonstrate that RICA outperforms competing methods under various contamination scenarios and real-data tasks (image data, cocktail party, and periodic signals), highlighting its practical robustness for blind source separation. Overall, RICA provides a principled, scalable approach to robust ICA that maintains the independence-detection property while mitigating the influence of outliers.

Abstract

Independent component analysis (ICA) is a powerful tool for decomposing a multivariate signal or distribution into fully independent sources, not just uncorrelated ones. Unfortunately, most approaches to ICA are not robust against outliers. Here we propose a robust ICA method called RICA, which estimates the components by minimizing a robust measure of dependence between multivariate random variables. The dependence measure used is the distance correlation (dCor). In order to make it more robust we first apply a new transformation called the bowl transform, which is bounded, one-to-one, continuous, and maps far outliers to points close to the origin. This preserves the crucial property that a zero dCor implies independence. RICA estimates the independent sources sequentially, by looking for the component that has the smallest dCor with the remainder. RICA is strongly consistent and has the usual parametric rate of convergence. Its robustness is investigated by a simulation study, in which it generally outperforms its competitors. The method is illustrated on three applications, including the well-known cocktail party problem.
Paper Structure (29 sections, 5 theorems, 46 equations, 12 figures, 9 tables)

This paper contains 29 sections, 5 theorems, 46 equations, 12 figures, 9 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Whitening
  4. Distance covariance and correlation
  5. dCovICA
  6. Robust independent component analysis
  7. Robust distance correlation
  8. RICA
  9. Optimization
  10. Theoretical analysis
  11. Simulation results
  12. Setting
  13. Results
  14. Uncontaminated data
  15. Clustered contamination
  16. ...and 14 more sections

Key Result

Proposition 1

If there exists a unique minimum $\boldsymbol{\theta}_0$ of eq:RICAobjective_angles_pop over $\overline{\Theta}$, and $\boldsymbol{\theta}_0$ is the only $\boldsymbol{\theta}$ in $\Theta$ yielding $\boldsymbol{U}(\boldsymbol{\theta}) = \boldsymbol{U}(\boldsymbol{\theta}_0)$, then we have almost sure

Figures (12)

  • Figure 1: Illustration of the biloop transformation.
  • Figure 2: Illustration of the bowl transform $\psi(\boldsymbol{x}) = (v_1(\boldsymbol{x}),v_2(\boldsymbol{x}))$ for $p=1$.
  • Figure 3: Illustration of the bowl transform $\psi(\boldsymbol{x}) = (v_1(\boldsymbol{x}),v_2(\boldsymbol{x}))$ for $p=2$.
  • Figure 4: The 18 source distributions used in the simulations.
  • Figure 5: Simulation results for uncontaminated data.
  • ...and 7 more figures

Theorems & Definitions (8)

  • Proposition 1
  • Proposition 2
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof