Linear Non-Gaussian Component Analysis Via Maximum Likelihood

ABSTRACT

Independent component analysis (ICA) is popular in many applications, including cognitive neuroscience and signal processing. Due to computational constraints, principal component analysis (PCA) is used for dimension reduction prior to ICA (PCA+ICA), which could remove important information. The problem is that interesting independent components (ICs) could be mixed in several principal components that are discarded and then these ICs cannot be recovered. We formulate a linear non-Gaussian component model with Gaussian noise components. To estimate the model parameters, we propose likelihood component analysis (LCA), in which dimension reduction and latent variable estimation are achieved simultaneously. Our method orders components by their marginal likelihood rather than ordering components by variance as in PCA. We present a parametric LCA using the logistic density and a semiparametric LCA using tilted Gaussians with cubic B-splines. Our algorithm is scalable to datasets common in applications (e.g., hundreds of thousands of observations across hundreds of variables with dozens of latent components). In simulations, latent components are recovered that are discarded by PCA+ICA methods. We apply our method to multivariate data and demonstrate that LCA is a useful data visualization and dimension reduction tool that reveals features not apparent from PCA or PCA+ICA. We also apply our method to a functional magnetic resonance imaging experiment from the Human Connectome Project and identify artifacts missed by PCA+ICA. We present theoretical results on identifiability of the linear non-Gaussian component model and consistency of LCA. Supplementary materials for this article are available online.

KEYWORDS:

• Functional magnetic resonance imaging
• Independent component analysis
• Neuroimaging
• Principal component analysis
• Projection pursuit

Acknowledgments

We thank Dr. Nathan Spreng, Department of Human Development, Cornell University, for scientific guidance and assistance with the HCP data. DSM was supported by a Xerox PARC Faculty Research Award, NSF grant DMS-1455172, and Cornell University Atkinson’s Center for a Sustainable Future Award AVF-2017. BBR was partially supported by the Department of Statistical Science at Cornell University, the School of Industrial and Labor Relations at Cornell University, and the NSF grant DMS-1127914 to the Statistical and Applied Mathematical Sciences Institute. Data were provided (in part) by the HCP, WU-Minn Consortium (Principal Investigators: David Van Essen and Kamil Ugurbil; 1U54MH091657).

Funding

National Science Foundation [DMS-1127914, DMS-1455172]; Cornell University Atkinson's Center for Sustainable Future [AVF-2017]; Xerox [PARC Faculty Research Award].