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Newton Method for the ICA Mixture Model

This paper introduces the Newton method for modeling sensor array data with multiple independent sources using the ICA mixture model. The method is computationally efficient and can handle non-stationary source activity.

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Newton Method for the ICA Mixture Model

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  1. Newton Method for theICA Mixture Model Jason A. Palmer1 Scott Makeig1Ken Kreutz-Delgado2 Bhaskar D. Rao2 1 Swartz Center for Computational Neuroscience2 Dept of Electrical and Computer EngineeringUniversity of California San Diego, La Jolla, CA

  2. Introduction • Want to model sensor array data with multiple independent sources — ICA • Non-stationary source activity — mixture model • Want the adaptation to be computationally efficient — Newton method

  3. Outline • ICA mixture model • Basic Newton method • Positive definiteness of Hessian when model source densities are true source densities • Newton for ICA mixture model • Example applications to analysis of EEG

  4. ICA Mixture Model—toy example • 3 models in two dimensions, 500 points per model • Newton method converges < 200 iterations, natural gradient fails to converge, has difficulty on poorly conditioned models

  5. ICA Mixture Model • Want to model observations x(t), t = 1,…,N, different models “active” at different times • Bayesian linear mixture model, h = 1, . . . , M : • Conditionally linear given the model, : • Samples are modeled as independent in time:

  6. Source Density Mixture Model • Each source density mixture component has unknown location, scale, and shape: • Generalizes Gaussian mixture model, more peaked, heavier tails

  7. ICA Mixture Model—Invariances • The complete set of parameters to be estimated is: h = 1, . . ., M, i = 1, . . ., n, j = 1, . . ., m • Invariances: W row norm/source density scale and model centers/source density locations:

  8. Basic ICA Newton Method • Transform gradient (1st derivative) of cost function using inverse Hessian (2nd derivative) • Cost function is data log likelihood: • Gradient: • Natural gradient (positive definite transform):

  9. Newton Method – Hessian • Take derivative of (i,j)th element of gradient with respect to (k,l)th element of W : • This defines a linear transform : • In matrix form, this is:

  10. Newton Method – Hessian • To invert: rewrite the Hessian transformation in terms of the source estimates: • Define , , : • Want to solve linear equation :

  11. Newton Method – Hessian • The Hessian transformation can be simplified using source independence and zero mean: • This leads to 2x2 block diagonal form:

  12. Newton Direction • Invert Hessian transformation, evaluate at gradient: • Leads to the following equations: • Calculate the Newton direction:

  13. Positive Definiteness of Hessian • Conditions for positive definiteness: • Always true for true when model source densities match true densities: 1) 2) 3)

  14. Newton for ICA Mixture Model • Similar derivation applies to ICA mixture model:

  15. Convergence Rates • Convergence is really much faster than natural gradient. Works with step size 1! • Need correct source density model log likelihood iteration iteration

  16. Segmentation of EEG experiment trials 3 models 4 models trial trial time time log likelihood log likelihood iteration iteration

  17. Applications to EEG—Epilepsy 1 model 5 models log likelihood time time log likelihood difference from single model time

  18. Conclusion • We applied method of Amari, Cardoso and Laheld, to formulate a Newton method for the ICA mixture model • Arbitrary source densities modeled with non-gaussian source mixture model • Non-stationarity modeled with ICA mixture model (multiple mixing matrices learned) • It works! Newton method is substantially faster (superlinear). Also Newton can converge when Natural Gradient fails

  19. Code • There is Matlab code available!! • Generate toy mixture model data for testing • Full method implemented: mixture sources, mixture ICA, Newton • Extended version of paper in preparation, with derivation of mixture model Newton updates • Download from: http://sccn.ucsd.edu/~jason

  20. Acknowledgements • Thanks to Scott Makeig, Howard Poizner, Julie Onton, Ruey-Song Hwang, Rey Ramirez, Diane Whitmer, and Allen Gruber for collecting and consulting on EEG data • Thanks to Jerry Swartz for founding and providing ongoing support the Swartz Center for Computational Neuroscience • Thanks for your attention!

  21. Newton for ICA Mixture Model

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