Stochastic trapping in a solvable model of on-line independent component analysis

Previous analytical studies of on-line Independent Component Analysis (ICA) learning rules have focussed on asymptotic stability and efficiency. In practice the transient stages of learning will often be more significant in determining the success of an algorithm. This is demonstrated here with an analysis of a Hebbian ICA algorithm which can find a small number of non-Gaussian components given data composed of a linear mixture of independent source signals. An idealised data model is considered in which the sources comprise a number of non-Gaussian and Gaussian sources and a solution to the dynamics is obtained in the limit where the number of Gaussian sources is infinite. Previous stability results are confirmed by expanding around optimal fixed points, where a closed form solution to the learning dynamics is obtained. However, stochastic effects are shown to stabilise otherwise unstable sub-optimal fixed points. Conditions required to destabilise one such fixed point are obtained for the case of a single non-Gaussian component, indicating that the initial learning rate \eta required to successfully escape is very low (\eta = O(N^{-2}) where N is the data dimension) resulting in very slow learning typically requiring O(N^3) iterations. Simulations confirm that this picture holds for a finite system.Comment: 17 pages, 3 figures. To appear in Neural Computatio

Non-negative mixtures

This is the author's accepted pre-print of the article, first published as M. D. Plumbley, A. Cichocki and R. Bro. Non-negative mixtures. In P. Comon and C. Jutten (Ed), Handbook of Blind Source Separation: Independent Component Analysis and Applications. Chapter 13, pp. 515-547. Academic Press, Feb 2010. ISBN 978-0-12-374726-6 DOI: 10.1016/B978-0-12-374726-6.00018-7file: Proof:p\PlumbleyCichockiBro10-non-negative.pdf:PDF owner: markp timestamp: 2011.04.26file: Proof:p\PlumbleyCichockiBro10-non-negative.pdf:PDF owner: markp timestamp: 2011.04.2

Role of homeostasis in learning sparse representations

Neurons in the input layer of primary visual cortex in primates develop edge-like receptive fields. One approach to understanding the emergence of this response is to state that neural activity has to efficiently represent sensory data with respect to the statistics of natural scenes. Furthermore, it is believed that such an efficient coding is achieved using a competition across neurons so as to generate a sparse representation, that is, where a relatively small number of neurons are simultaneously active. Indeed, different models of sparse coding, coupled with Hebbian learning and homeostasis, have been proposed that successfully match the observed emergent response. However, the specific role of homeostasis in learning such sparse representations is still largely unknown. By quantitatively assessing the efficiency of the neural representation during learning, we derive a cooperative homeostasis mechanism that optimally tunes the competition between neurons within the sparse coding algorithm. We apply this homeostasis while learning small patches taken from natural images and compare its efficiency with state-of-the-art algorithms. Results show that while different sparse coding algorithms give similar coding results, the homeostasis provides an optimal balance for the representation of natural images within the population of neurons. Competition in sparse coding is optimized when it is fair. By contributing to optimizing statistical competition across neurons, homeostasis is crucial in providing a more efficient solution to the emergence of independent components