## Electronic Journal of Statistics

- Electron. J. Statist.
- Volume 8, Number 1 (2014), 1438-1459.

### Quantifying identifiability in independent component analysis

Alexander Sokol, Marloes H. Maathuis, and Benjamin Falkeborg

#### Abstract

We are interested in consistent estimation of the mixing matrix in the ICA model, when the error distribution is close to (but different from) Gaussian. In particular, we consider $n$ independent samples from the ICA model $X=A\epsilon$, where we assume that the coordinates of $\epsilon$ are independent and identically distributed according to a contaminated Gaussian distribution, and the amount of contamination is allowed to depend on $n$. We then investigate how the ability to consistently estimate the mixing matrix depends on the amount of contamination. Our results suggest that in an asymptotic sense, if the amount of contamination decreases at rate $1/\sqrt{n}$ or faster, then the mixing matrix is only identifiable up to transpose products. These results also have implications for causal inference from linear structural equation models with near-Gaussian additive noise.

#### Article information

**Source**

Electron. J. Statist., Volume 8, Number 1 (2014), 1438-1459.

**Dates**

First available in Project Euclid: 20 August 2014

**Permanent link to this document**

https://projecteuclid.org/euclid.ejs/1408540293

**Digital Object Identifier**

doi:10.1214/14-EJS932

**Mathematical Reviews number (MathSciNet)**

MR3263128

**Zentralblatt MATH identifier**

1298.62045

**Subjects**

Primary: 62F12: Asymptotic properties of estimators

Secondary: 62F35: Robustness and adaptive procedures

**Keywords**

Independent component analysis LiNGAM identifiability Kolmogorov norm contaminated distribution asymptotic statistics empirical process linear structural equation model

#### Citation

Sokol, Alexander; H. Maathuis, Marloes; Falkeborg, Benjamin. Quantifying identifiability in independent component analysis. Electron. J. Statist. 8 (2014), no. 1, 1438--1459. doi:10.1214/14-EJS932. https://projecteuclid.org/euclid.ejs/1408540293