The ICA model is subject to a constraint that at most one of these components is Gaussian, which is required for model identifiability. Linear non-Gaussian component analysis (LNGCA) generalizes the ICA model to a linear latent factor model with any number of both non-Gaussian components (signals) and Gaussian components (noise), where observations are linear combinations of independent components. Although the individual Gaussian components are not identifiable, the Gaussian subspace is identifiable. We introduce an estimator along with its optimization approach in which non-Gaussian and Gaussian components are estimated simultaneously, maximizing the discrepancy of each non-Gaussian component from Gaussianity while minimizing the discrepancy of each Gaussian component from Gaussianity. When the number of non-Gaussian components is unknown, we develop a statistical test to determine it based on resampling and the discrepancy of estimated components. Through a variety of simulation studies, we demonstrate the improvements of our estimator over competing estimators, and we illustrate the effectiveness of the test to determine the number of non-Gaussian components. Further, we apply our method to real data examples and demonstrate its practical value.

The statistical dependencies which independent component analysis (ICA) cannot remove often provide rich information beyond the linear independent components. It would thus be very useful to estimate the dependency structure from data. While such models have been proposed, they usually concentrated on higher-order correlations such as energy (square) correlations. Yet, linear correlations are a most fundamental and informative form of dependency in many real data sets. Linear correlations are usually completely removed by ICA and related methods, so they can only be analyzed by developing new methods which explicitly allow for linearly correlated components. In this paper, we propose a probabilistic model of linear non-Gaussian components which are allowed to have both linear and energy correlations. The precision matrix of the linear components is assumed to be randomly generated by a higher-order process and explicitly parametrized by a parameter matrix. The estimation of the parameter matrix is shown to be particularly simple because using score matching, the objective function is a quadratic form. Using simulations with artificial data, we demonstrate that the proposed method improves identifiability of non-Gaussian components by simultaneously learning their correlation structure. Applications on simulated complex cells with natural image input, as well as spectrograms of natural audio data show that the method finds new kinds of dependencies between the components.

We propose a new linear method for dimension reduction to identify non-Gaussian components in high dimensional data. Our method, NGCA (non-Gaussian component analysis), uses a very general semi-parametric framework. In contrast to existing projection methods we define what is uninteresting (Gaussian): by projecting out uninterestingness, we can estimate therelevant non-Gaussian subspace. We show that the estimation error of finding the non-Gaussian components tends to zero at a parametric rate.Once NGCA components are identified and extracted, various tasks can be applied in the data analysis process, like data visualization, clustering, denoising or classification. A numerical study demonstrates the usefulness of our method.

We propose a new linear method for dimension reduction to identify non-Gaussian components in high dimensional data. Our method, NGCA (non-Gaussian component analysis), uses a very general semi-parametric framework. In contrast to existing projection methods we define what is uninteresting (Gaussian): by projecting out uninterestingness, we can estimate the relevant non-Gaussian subspace. We show that the estimation error of finding the non-Gaussian components tends to zero at a parametric rate. Once NGCA components are identified and extracted, various tasks can be applied in the data analysis process, like data visualization, clustering, denoising or classification. A numerical study demonstrates the usefulness of our method.

We propose a new linear method for dimension reduction to identify non-Gaussian components in high dimensional data. Our method, NGCA (non-Gaussian component analysis), uses a very general semi-parametric framework. In contrast to existing projection methods we define what is uninteresting (Gaussian): by projecting out uninterestingness, we can estimate the relevant non-Gaussian subspace. We show that the estimation error of finding the non-Gaussian components tends to zero at a parametric rate. Once NGCA components are identified and extracted, various tasks can be applied in the data analysis process, like data visualization, clustering, denoising or classification. A numerical study demonstrates the usefulness of our method.