Sparse Component Analysis: a New Tool for Data Mining

Abstract

In many practical problems for data mining the data X under consideration (given as (m x A^)-matrix) is of the form X = AS, where the matrices A and S with dimensions mxn and nx N respectively (often called mixing matrix or dictionary and source matrix) are unknown (m < n < N). We formulate conditions (SCA-conditions) under which we can recover A and S uniquely (up to scaling and permutation), such that S is sparse in the sense that each column of S has at least one zero element. We call this the Sparse Component Analysis problem (SCA). We present new algorithms for identification of the mixing matrix (under SCAconditions), and for source recovery (under identifiability conditions). The methods are illustrated with examples showing good performance of the algorithms. Typical examples are EEC and fMRI data sets, in which the SCA algorithm allows us to detect some features of the brain signals. Special attention is given to the application of our method to the transposed system X^ = S^ A^ utilizing the sparseness of the mixing matrix A in appropriate situations. We note that the sparseness conditions could be obtained with some preprocessing methods and no independence conditions for the source signals are imposed (in contrast to Independent Component Analysis). We applied our method to fMRI data sets with dimension (128 x 128 x 98) and to EEC data sets from a 256-channels EEC machine.

Local Dimension-Reduced Dynamical Spatio-Temporal Models for Resting State Network Estimation

“…LDSTM analysis identified thirty nine well-localized spatial components comprising cortical (18), subcortical (2) and cerebellar (19) regions. Cortical and subcortical spatial components (a k 's) are shown in Fig.…”

“…Local dimension-reduced modelling (LDSTM) as presented here addresses an approach to source estimation and localization in resting state fMRI data analysis that dispenses with artificial stochastic model assumptions, such as those used in classical blind source separation (principal component analysis (PCA), independent component analysis (ICA) and non-negative matrix factorization (NMF) [3,18,19,21]). In addition to being sparse, the columns of the observation matrix act as point-spreading functions that allow system sources and their observation matrix to be identified via LSCA [32] of the whole fMRI dataset.…”

A two-step super-Gaussian independent component analysis approach for fMRI data

“…Within the signal processing community, besides independence, sparsity is another commonly imposed assumption that arises from the principle of parsimony (Babaie-Zadeh et al, 2006;Bronstein et al, 2005;Chen et al, 1998;Georgiev et al, 2007;Mairal et al, 2010;Wright et al, 2010). For neural coding, the sparseness means that the number of activated neurons is much less than the total number of neurons in the population in one time period.…”

“…The assumption underlying sparse representation is that each voxel's fMRI signal is linearly composed of sparse components. Georgiev et al (2007) proposed a sparse component analysis algorithm to identify the sources in fMRI data without the independent assumption. Li et al (2009Li et al ( , 2012 applied sparse representation for detection of voxels in fMRI data with task relevant information (Li et al, 2009) and proposed a sparse representation-based multivariate pattern analysis algorithm to localize brain activation patterns corresponding to different stimulus classes/brain states (Li et al, 2012).…”

Error bounds for suboptimal solutions to kernel principal component analysis

“…Various approaches have been proposed to cope with these drawbacks by considering sparse suboptimal solutions to KPCA; see, e.g., [1,9], [19,Chapter 14], and the references therein. For other methods that employ sparsity constraints in data mining applications, such as sparse component analysis, see, e.g., [6,7,16].…”