Adaptive Neural Algorithms for PCA and ICA

Adaptive Neural Algorithms for PCA and ICA

Radu Mutihac (University of Bucharest, Romania)
Copyright: © 2009 |Pages: 9
DOI: 10.4018/978-1-59904-849-9.ch004
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Abstract

Artificial neural networks (ANNs) (McCulloch & Pitts, 1943) (Haykin, 1999) were developed as models of their biological counterparts aiming to emulate the real neural systems and mimic the structural organization and function of the human brain. Their applications were based on the ability of self-designing to solve a problem by learning the solution from data. A comparative study of neural implementations running principal component analysis (PCA) and independent component analysis (ICA) was carried out. Artificially generated data additively corrupted with white noise in order to enforce randomness were employed to critically evaluate and assess the reliability of data projections. Analysis in both time and frequency domains showed the superiority of the estimated independent components (ICs) relative to principal components (PCs) in faithful retrieval of the genuine (latent) source signals. Neural computation belongs to information processing dealing with adaptive, parallel, and distributed (localized) signal processing. In data analysis, a common task consists in finding an adequate subspace of multivariate data for subsequent processing and interpretation. Linear transforms are frequently employed in data model selection due to their computational and conceptual simplicity. Some common linear transforms are PCA, factor analysis (FA), projection pursuit (PP), and, more recently, ICA (Comon, 1994). The latter emerged as an extension of nonlinear PCA (Hotelling, 1993) and developed in the context of blind source separation (BSS) (Cardoso, 1998) in signal and array processing. ICA is also related to recent theories of the visual brain (Barlow, 1991), which assume that consecutive processing steps lead to a progressive reduction in the redundancy of representation (Olshausen and Field, 1996). This contribution is an overview of the PCA and ICA neuromorphic architectures and their associated algorithmic implementations increasingly used as exploratory techniques. The discussion is conducted on artificially generated sub- and super-Gaussian source signals.
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Background

In neural computation, transforming methods amount to unsupervised learning, since the representation is only learned from data without any external control. Irrespective of the nature of learning, the neural adaptation may be formally conceived as an optimization problem: an objective function describes the task to be performed by the network and a numerical optimization procedure allows adapting network parameters (e.g., connection weights, biases, internal parameters). This process amounts to search or nonlinear programming in a quite large parameter space. However, any prior knowledge available on the solution might be efficiently exploited to narrow the search space. In supervised learning, the additional knowledge is incorporated in the net architecture or learning rules (Gold, 1996). A less extensive research was focused on unsupervised learning. In this respect, the mathematical methods usually employed are drawn from classical constrained multivariate nonlinear optimization and rely on the Lagrange multipliers method, the penalty or barrier techniques, and the classical numerical algebra techniques, such as deflation/renormalization (Fiori, 2000), the Gram-Schmidt orthogonalization procedure, or the projection over the orthogonal group (Yang, 1995).

Key Terms in this Chapter

Learning Rule: Weight change strategy in a connectionist system aiming to optimize a certain objective function. Learning rules are iteratively applied to the training set inputs with error gradually reduced as the weights are adapting.

Independent Component Analysis (ICA): An exploratory method for separating a linear mixture of latent signal sources into independent components as optimal estimates of the original sources on the basis of their mutual statistical independence and non-Gaussianity.

Blind Source Separation (BSS): Separation of latent nonredundant (e.g., mutually statistically independent or decorrelated) source signals from a set of linear mixtures, such that the regularity of each resulting signal is maximized, and the regularity between the signals is minimized (i.e. statistical independence is maximized) without (almost) any information on the sources.

Confirmatory Data Analysis (CDA): An approach which, subsequent to data acquisition, proceeds with the imposition of a prior model and analysis, estimation, and testing model parameters.

Exploratory Data Analysis (EDA): An approach based on allowing the data itself to reveal its underlying structure and model heavily using the collection of techniques known as statistical graphics.

Principal Component Analysis (PCA): An orthogonal linear transform based on singular value decomposition that projects data to a subspace that preserves maximum variance.

Artificial Neural Networks (ANNs): An information-processing synthetic system made up of several simple nonlinear processing units connected by elements that have information storage and programming functions adapting and learning from patterns, which mimics a biological neural network.

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