Mathematics in Independent Component Analysis

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160 Chapter 10. IEEE TNN 16(4):992-996, 2005 996 IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 16, NO. 4, JULY 2005 the excellent performance of the another two our algorithms in the un- can be described by a linear weighted sum of the inputs, followed by derdetermined BSS problem, for separation of artificially created sig- some nonlinear transfer function [1], [2]. In this letter, however, the nals with sufficient level of sparseness. neural computing models studied are based on artificial neurons which often have binary inputs and outputs, and no adjustable weight between REFERENCES nodes. Neuron functions are stored in lookup tables, which can be [1] F. Abrard, Y. Deville, and P. White, “From blind source separation to blind source cancellation in the underdetermined case: A new approach based on time-frequency analysis,” in Proc. 3rd Int. Conf. Independent implemented using commercially available random access memories (RAMs). These systems and the nodes that they are composed of will be described, respectively, as weightless neural networks (WNNs) and Component Analysis and Signal Separation (ICA’2001), San Diego, CA, weightless nodes [1], [2]. They differ from other models, such as the Dec. 9–13, 2001, pp. 734–739. [2] A. Cichocki and S. Amari, Adaptive Blind Signal and Image Processing. New York: Wiley, 2002. [3] S. Chen, D. Donoho, and M. Saunders, “Atomic decomposition by basis pursuit,” SIAM J. Sci. Comput., vol. 20, no. 1, pp. 33–61, 1998. weighted neural networks, whose training is accomplished by means of adjustments of weights. In the literature, the terms “RAM-based” and “N-tuple based” have been used to refer to WNNs. In this letter, the computability (computational power) of a class of [4] D. Donoho and M. Elad, “Optimally sparse representation in general (nonorthogonal) dictionaries via � minimization,” Proc. Nat. Acad. Sci., vol. 100, no. 5, pp. 2197–2202, 2003. [5] A. Hyvärinen, J. Karhunen, and E. Oja, Independent Component Analysis. New York: Wiley, 2001. WNNs, called general single-layer sequential weightless neural networks (GSSWNNs), is investigated. Such a class is an important representative of the research on temporal pattern processing in (WNNs) [3]–[8]. As oneof thecontributions, an algorithm (constructiveproof) [6] A. Jourjine, S. Rickard, and O. Yilmaz, “Blind separation of disjoint to map any probabilistic automaton (PA) into a GSSWNN is presented. orthogonal signals: Demixing N sources from 2 mixtures,” in Proc. 2000 IEEE Conf. Acoustics, Speech, and Signal Processing (ICASSP’00), vol. 5, Istanbul, Turkey, Jun. 2000, pp. 2985–2988. [7] T.-W. Lee, M. S. Lewicki, M. Girolami, and T. J. Sejnowski, “Blind sourse separation of more sources than mixtures using overcomplete rep- In fact, the proposed method not only allows the construction of any PA, but also increases the class of functions that can be computed by such networks. For instance, at a theoretical level, these networks are not restricted to finite-state languages (regular languages) and can now resentaitons,” IEEE Signal Process. Lett., vol. 6, no. 4, pp. 87–90, 1999. [8] D. D. Lee and H. S. Seung, “Learning the parts of objects by nonnegative matrix factorization,” Nature, vol. 40, pp. 788–791, 1999. [9] F. J. Theis, E. W. Lang, and C. G. Puntonet, “A geometric algorithm for overcomplete linear ICA,” Neurocomput., vol. 56, pp. 381–398, 2004. [10] K. Waheed and F. Salem, “Algebraic overcomplete independent com- deal with some context-free languages. Practical motivations for investigating probabilistic automata and GSSWNNs arefound in their possible application to, among others things, syntactic pattern recognition, multimodal search, and learning control [9]. ponent analysis,” in Proc. Int. Conf. Independent Component Analysis (ICA’03), Nara, Japan, pp. 1077–1082. II. DEFINITIONS [11] M. Zibulevsky and B. A. Pearlmutter, “Blind source separation by sparse decomposition in a signal dictionary,” Neural Comput., vol. 13, no. 4, pp. A. Probabilistic Automata 863–882, 2001. Probabilistic automata are a generalization of ordinary deterministic finitestateautomata (DFA) for which an input symbol could takethe Equivalence Between RAM-Based Neural Networks and Probabilistic Automata automaton into any of its states with a certain probability [9]. Definition 2.1: A PA is a 5-tuple e€ a@6Y Y rY �sYpA, where • 6a�'IY'PY FFFY'�6�� is a finite set of ordered symbols called the input alphabet; Marcilio C. P. de Souto, Teresa B. Ludermir, and Wilson R. de Oliveira • • a ��HY�PY FFFY�� r is a mapping of �� is a finite set of states; 2 6 into theset of � 2 � stochastic state transition matrices (where � is the number of states in ). The Abstract—In this letter, the computational power of a class of random interpretation of r@—�AY—� P 6, can bestated as follows. access memory (RAM)-based neural networks, called general single-layer sequential weightless neural networks (GSSWNNs), is analyzed. The theoretical results presented, besides helping the understanding of the temporal behavior of these networks, could also provide useful insights for the developing of new learning algorithms. r@—�A a‘�� @—�A“, where ��@—�A ! H is theprobability of � entering state �� from state �� under input —�, and �� a �aI I, for all � aIYFFFY�. Thedomain of r can be extended from 6 to 6 Index Terms—Automata theory, computability, random access memory (RAM) node, probabilistic automata, RAM-based neural networks, weightless neural networks (WNNs). I. INTRODUCTION The neuron model used in the great majority of work involving neural networks is related to variations of the McCulloch–Pitts neuron, which will becalled theweighted neuron. A typical weighted neuron Manuscript received July 6, 2003; revised October 26, 2004. M. C. P. de Souto is with the Department of Informatics and Applied Mathematics, Federal University of Rio Grande do Norte, Campus Universitario, Natal-RN 59072-970, Brazil (e-mail: marcilio@dimap.ufrn.br). T. B. Ludermir is with the Center of Informatics, Federal University of Pernambuco, Recife 50732-970, Brazil. W. R. de Oliveira is with the Department of Physics and Mathematics, Rural Federal University of Pernambuco, Recife 52171-900, Brazil. Digital Object Identifier 10.1109/TNN.2005.849838 3 by defining the following: 1) 2) r@ Aas�, where is theempty string and s� is an � 2 � identity matrix; r@—� Y—� Y FFFY—� A a r@—� Ar@—� AY FFFYr@—� A, where � ! P and • —� �s P P 6Y� aIYFFFY�. is theinitial statein which themachineis found before the first symbol of the input string is processed; • p is the set of final states @p A. The language accepted by a PA e€ is „ @e€ Aa�@3Y �@3AA�3 P 6 3 Y�@3A a%Hr@3A%p b H� where: 1) %H is a �-dimensional row vector, in which the �th component is equal to one if �� a �s, and 0 otherwise and 2) %p is an �-dimensional column vector, in which the �th component is equal to 1 if �� P p and 0 otherwise. The language accepted by e€ with cut-point(threshold) !, such that H !`I,isv@e€ Y!Aa�3�3 P 6 3 and %Hr@3A%p b!�. Probabilistic automata recognize exactly the class of weighted regular languages (WRLs) [9]. Such a class of languages includes properly 1045-9227/$20.00 © 2005 IEEE
Chapter 11 EURASIP JASP, 2007 Paper F.J. Theis, P. Georgiev, and A. Cichocki. Robust sparse component analysis based on a generalized hough transform. EURASIP Journal on Applied Signal Processing, 2007 Reference (Theis et al., 2007a) Summary in section 1.4.1 161
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Statistical machine learning of bio
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to Jakob
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vi Preface
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viii CONTENTS 3 Signal Processing 8
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Chapter 1 Statistical machine learn
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1.1. Introduction 5 auditory cortex
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1.2. Uniqueness issues in independe
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S U M M A R Y T his �le contains
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1.3. Dependent component analysis 1
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Table 1.1: BSS algorithms based on
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1.4. Sparseness 29 R 3 R 3 A BSRA R
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1.4. Sparseness 31 Sparse projectio
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1.7. Outlook 51 noise data signal i
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298 BIBLIOGRAPHY Barber, C., Dobkin
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300 BIBLIOGRAPHY Georgiev, P. (2001
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302 BIBLIOGRAPHY Keck, I., Theis, F
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