Blind Separation of Post-nonlinear Mixtures using Linearizing Transformations and Temporal Decorrelation

We propose two methods that reduce the post-nonlinear blind source separation problem (PNL-BSS) to a linear BSS problem. The first method is based on the concept of maximal correlation: we apply the alternating conditional expectation (ACE) algorithm---a powerful technique from non-parametric statistics---to approximately invert the componentwise non-linear functions.The second method is a Gaussianizing transformation, which is motivated by the fact that linearly mixed signals before nonlinear transformation are approximately Gaussian distributed. This heuristic, but simple and efficient procedure works as good as the ACE method.Using the framework provided by ACE, convergence can be proven. The optimal transformations obtained by ACE coincide with the sought-after inverse functions of the nonlinearities. After equalizing the nonlinearities, temporal decorrelation separation (TDSEP) allows us to recover the source signals. Numerical simulations testing "ACE-TD" and "Gauss-TD" on realistic examples are performed with excellent results.

[1]  Dinh Tuan Pham,et al.  Joint Approximate Diagonalization of Positive Definite Hermitian Matrices , 2000, SIAM J. Matrix Anal. Appl..

[2]  Gunnar Rätsch,et al.  An introduction to kernel-based learning algorithms , 2001, IEEE Trans. Neural Networks.

[3]  J. Karhunen,et al.  Advances in Nonlinear Blind Source Separation , 2003 .

[4]  D. Chakrabarti,et al.  A fast fixed - point algorithm for independent component analysis , 1997 .

[5]  Bernhard Schölkopf,et al.  Nonlinear Component Analysis as a Kernel Eigenvalue Problem , 1998, Neural Computation.

[6]  Juha Karhunen,et al.  Nonlinear Independent Component Analysis Using Ensemble Learning: Experiments and Discussion , 2000 .

[7]  Motoaki Kawanabe,et al.  Kernel-Based Nonlinear Blind Source Separation , 2003, Neural Computation.

[8]  T. Sejnowski,et al.  Separation of post-nonlinear mixtures using ACE and temporal decorrelation , 2001 .

[9]  Philippe Garat,et al.  Blind separation of mixture of independent sources through a quasi-maximum likelihood approach , 1997, IEEE Trans. Signal Process..

[10]  Lawrence E. Larson,et al.  Radio frequency integrated circuit technology for low-power wireless communications , 1998, IEEE Wirel. Commun..

[11]  P. Philips,et al.  JADETD : COMBINING HIGHER-ORDER STATISTICS AND TEMPORALINFORMATION FOR BLIND SOURCE SEPARATION ( WITH NOISE ) , 1999 .

[12]  Andreas Ziehe,et al.  An approach to blind source separation based on temporal structure of speech signals , 2001, Neurocomputing.

[13]  Andrzej Cichocki,et al.  Information-theoretic approach to blind separation of sources in non-linear mixture , 1998, Signal Process..

[14]  Juan K. Lin,et al.  Faithful Representation of Separable Distributions , 1997, Neural Computation.

[15]  Christian Jutten,et al.  Nonlinear source separation: the post-nonlinear mixtures , 1997, ESANN.

[16]  Antoine Souloumiac,et al.  Jacobi Angles for Simultaneous Diagonalization , 1996, SIAM J. Matrix Anal. Appl..

[17]  Ali Mansour,et al.  Blind Separation of Sources , 1999 .

[18]  B. Pompe.  Measuring statistical dependences in a time series , 1993 .

[19]  Andreas Ziehe,et al.  Combining time-delayed decorrelation and ICA: towards solving the cocktail party problem , 1998, Proceedings of the 1998 IEEE International Conference on Acoustics, Speech and Signal Processing, ICASSP '98 (Cat. No.98CH36181).

[20]  Vladimir Vapnik,et al.  Statistical learning theory , 1998 .

[21]  Juha Karhunen,et al.  A Maximum Likelihood Approach to Nonlinear Blind Source Separation , 1997, ICANN.

[22]  Dinh-Tuan Pham,et al.  Improving algorithm speed in PNL mixture separation and Wiener system inversion , 2003 .

[23]  Pierre Comon.  Independent component analysis - a new concept? signal processing , 1994 .

[24]  Andreas Ziehe,et al.  Artifact Reduction in Magnetoneurography Based on Time-Delayed Second Order Correlations , 1998 .

[25]  Dinh Tuan Pham,et al.  BLIND SOURCE SEPARATION IN POST NONLINEAR MIXTURES , 2001 .

[26]  Andreas Ziehe,et al.  TDSEP — an efficient algorithm for blind separation using time structure , 1998 .

[27]  Christian Jutten,et al.  Blind separation of sources, part I: An adaptive algorithm based on neuromimetic architecture , 1991, Signal Process..

[28]  Terrence J. Sejnowski,et al.  An Information-Maximization Approach to Blind Separation and Blind Deconvolution , 1995, Neural Computation.

[29]  E. Oja,et al.  Independent Component Analysis , 2013 .

[30]  Pierre Comon,et al.  Independent component analysis, A new concept? , 1994, Signal Process..

[31]  Lucas C. Parra,et al.  Convolutive blind separation of non-stationary sources , 2000, IEEE Trans. Speech Audio Process..

[32]  Vladimir N. Vapnik,et al.  The Nature of Statistical Learning Theory , 2000, Statistics for Engineering and Information Science.

[33]  Motoaki Kawanabe,et al.  Kernel Feature Spaces and Nonlinear Blind Souce Separation , 2001, NIPS.

[34]  C. Jutten,et al.  QUADRATIC DEPENDENCE MEASURE FOR NONLINEAR BLIND SOURCES SEPARATION , 2003 .

[35]  Gustavo Deco,et al.  Linear redundancy reduction learning , 1995, Neural Networks.

[36]  P. Spreij.  Probability and Measure , 1996 .

[37]  Anisse Taleb,et al.  A generic framework for blind source separation in structured nonlinear models , 2002, IEEE Trans. Signal Process..

[38]  Schuster,et al.  Separation of a mixture of independent signals using time delayed correlations. , 1994, Physical review letters.

[39]  A. Ziehe,et al.  A LINEAR LEAST-SQUARES ALGORITHM FOR JOINT DIAGONALIZATION , 2003 .

[40]  J. Friedman,et al.  Estimating Optimal Transformations for Multiple Regression and Correlation. , 1985 .

[41]  Athanasios Koutras.  Blind separation of non-linear convolved speech mixtures , 2002, 2002 IEEE International Conference on Acoustics, Speech, and Signal Processing.

[42]  R. Tibshirani,et al.  Generalized additive models for medical research , 1986, Statistical methods in medical research.

[43]  Wolfgang Härdle,et al.  Applied Nonparametric Regression , 1991 .

[44]  J. Cardoso,et al.  Blind beamforming for non-gaussian signals , 1993 .

[45]  Masahiro Oya,et al.  Blind Separation of Sources Using Temporal Correlation of the Observed Signals , 1997 .

[46]  Andreas Ziehe,et al.  Blind separation of post-nonlinear mixtures using gaussianizing transformations and temporal decorrelation , 2003 .

[47]  E. Oja,et al.  Nonlinear Blind Source Separation by Variational Bayesian Learning , 2003, IEICE Trans. Fundam. Electron. Commun. Comput. Sci..

[48]  Christian Jutten,et al.  Source separation in post-nonlinear mixtures , 1999, IEEE Trans. Signal Process..

[49]  Te-Won Lee,et al.  Blind source separation of nonlinear mixing models , 1997, Neural Networks for Signal Processing VII. Proceedings of the 1997 IEEE Signal Processing Society Workshop.

[50]  H. Kantz,et al.  Nonlinear time series analysis , 1997 .

[51]  Eric Moulines,et al.  A blind source separation technique using second-order statistics , 1997, IEEE Trans. Signal Process..

[52]  Andrzej Cichocki,et al.  A New Learning Algorithm for Blind Signal Separation , 1995, NIPS.

[53]  A. Yeredor.  Blind separation of Gaussian sources via second-order statistics with asymptotically optimal weighting , 2000, IEEE Signal Processing Letters.

[54]  A. Hyvärinen,et al.  Nonlinear Blind Source Separation by Self-Organizing Maps , 1996 .

[55]  Alexander J. Smola,et al.  Learning with kernels , 1998 .