Independent Component Analysis for Brain fMRI Does Indeed Select for Maximal Independence Vince D. Calhoun 1,2,3 *, Vamsi K. Potluru 1,3 , Ronald Phlypo 5 , Rogers F. Silva 1,2 , Barak A. Pearlmutter 4 , Arvind Caprihan 1 , Sergey M. Plis 1 , Tu ¨ lay Adalı 5 1 Medical Image Analysis Lab, The Mind Research Network, Albuquerque, New Mexico, United States of America, 2 Department of Electrical and Computer Engineering, University of New Mexico, Albuquerque, New Mexico, United States of America, 3 Department of Computer Science, University of New Mexico, Albuquerque, New Mexico, United States of America, 4 Hamilton Institute and Department of Computer Science, National University of Ireland Maynooth, Co. Kildare, Ireland, 5 Department of Computer Science and Electrical Engineering, University of Maryland Baltimore County, Baltimore, Maryland, United States of America Abstract A recent paper by Daubechies et al. claims that two independent component analysis (ICA) algorithms, Infomax and FastICA, which are widely used for functional magnetic resonance imaging (fMRI) analysis, select for sparsity rather than independence. The argument was supported by a series of experiments on synthetic data. We show that these experiments fall short of proving this claim and that the ICA algorithms are indeed doing what they are designed to do: identify maximally independent sources. Citation: Calhoun VD, Potluru VK, Phlypo R, Silva RF, Pearlmutter BA, et al. (2013) Independent Component Analysis for Brain fMRI Does Indeed Select for Maximal Independence. PLoS ONE 8(8): e73309. doi:10.1371/journal.pone.0073309 Editor: Dante R. Chialvo, National Research & Technology Council, Argentina Received April 7, 2013; Accepted July 12, 2013; Published August 29, 2013 Copyright: ß 2013 Calhoun et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Funding: This work was funded by National Institutes of Health grants 2R01EB000840 & 5P20RR021938. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript. Competing Interests: The authors have declared that no competing interests exist. * E-mail: vcalhoun@unm.edu Introduction Independent component analysis (ICA) [1–4] is a widely used signal processing approach that has been applied to areas including speech separation, communications, and functional magnetic resonance (fMRI) data analysis. Given a set of linearly mixed observations, recovering the underlying components is an ill-defined problem. However, the assumption of independence among the sources turns out to be surprisingly powerful and effective for a wide range of problems in various practical domains. Sparsity is another commonly imposed assumption that arises naturally from the principle of parsimony: the simplest explanation is preferred. Sparsity is also motivated by evidence of neuronal coding efficiency and sparse coding in the nervous system. Sparse representations can help avoid the problem of overfitting while also leading to solutions that are easier to interpret. Applications of sparse signal processing methods include dictionary learning [5], speech separation [6], and feature learning [7]. Daubechies et al. [8] claims that ICA for fMRI optimizes for sparsity rather than independence. This is established by first noting that Infomax and FastICA are two algorithms widely used for fMRI analysis and then showing that they separate sparse components better than independent ones on a synthetic dataset. Recreating the synthetic dataset and conducting additional experiments shows that the FastICA and Infomax algorithms indeed do what they are designed to do. Both ICA algorithms can separate sources with either high or low degrees of sparsity, as long as the distributional assumptions of the algorithms are approxi- mately met. To understand the conditions under which these algorithms work requires correct interpretation of what the sources are in an ICA formulation. We examine exactly what the sources are in the examples given in Daubechies et al. [8] and show that there is an important mismatch between the concept of source therein and what an ICA source actually is, which is ultimately at the heart of the unsupported conclusions presented in Daubechies et al. [8]. Review and Critique of the Presented Evidence We now briefly review the evidence presented in Daubechies et al. [8] to support the claim that Infomax [3] and FastICA [9] select for sparsity and not independence. Following Daubechies et al., we refer to the versions of the two algorithms with their default nonlinearities, sigmoid for Infomax, which is a good match for sources with super-Gaussian distributions, and the high kurtosis nonlinearity for FastICA. Daubechies et al. [8] exhibits experi- mental results in which 1) ICA algorithm performance suffers when the assumptions on the sources are violated, and 2) ICA algorithms can separate sources in certain cases even if the sources are not strictly independent. The two points above, both of which were already widely known in the ICA community at the time, are not sufficient evidence to support the claim that ICA selects for sparsity and not independence. In addition, Daubechies et al. [8] presents a case in which the sources are somewhat dependent but also very sparse, and Infomax and FastICA do well. This result is used to claim that it is sparsity rather than independence that matters. We augment this experiment with new evidence which shows that the same ICA algorithms perform equally well in the case of both minimum and maximum sparsity (using the definition of sparsity in Daubechies et al. [8]), suggesting that the role of sparsity (if any) is minor in the separation performance. PLOS ONE | www.plosone.org 1 August 2013 | Volume 8 | Issue 8 | e73309