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Sparse Coding and Automatic Relevance Determination for Multi-way models

Sparse Coding and Automatic Relevance Determination for Multi-way models. Morten Mørup DTU Informatics Intelligent Signal Processing Technical University of Denmark. Joint work with Lars Kai Hansen DTU Informatics Intelligent Signal Processing Technical University of Denmark.

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Sparse Coding and Automatic Relevance Determination for Multi-way models

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  1. Sparse Coding and Automatic Relevance Determination for Multi-way models Morten Mørup DTU Informatics Intelligent Signal Processing Technical University of Denmark Joint work with Lars Kai Hansen DTU Informatics Intelligent Signal Processing Technical University of Denmark Sparse’09 8 April 2009

  2. Multi-way modeling has become an important tool for Unsupervised Learning and are in frequent use today in a variety of fields including Vector: 1-way array/ 1st order tensor, Matrix: 2-way array/ 2nd order tensor, 3-way array/3rd order tensor • Psychometrics (Subject x Task x Time) • Chemometrics (Sample x Emission x Absorption) • NeuroImaging (Channel x Time x Trial) • Textmining (Term x Document x Time/User) • Signal Processing (ICA, i.e. diagonalization of Cummulants) Sparse’09 8 April 2009

  3. I x JK Matrix J x KI Matrix K x IJ Matrix Some Tensor algebra XI x J x K • Matricizing X  X(n) • N-mode multiplication i.e. for matrix multiplication CA=A x1C , ABT=A x2B X(1) X(2) X(3) Sparse’09 8 April 2009

  4. Tensor Decomposition The two most commonly used tensor decomposition models are the CandeComp/PARAFAC (CP) model and the Tucker model Sparse’09 8 April 2009

  5. CP-model unique • Tucker model not unique • What constitutes an adequate number of components, i.e. determining J for CP and J1,J2,…,JN for Tucker is an open problem, particularly difficult for the Tucker model as the number of components are specified for each modality separately Sparse’09 8 April 2009

  6. Agenda • To use sparse coding to Simplify the Tucker core forming a unique representation as well as enable interpolation between the Tucker (full core) and CP (diagonal core) model • To use sparse coding to turn off excess components in the CP and Tucker model and thereby select the model order. • To tune the pruning strength from data by Automatic Relevance Determination (ARD). Sparse’09 8 April 2009

  7. Nature, 1996 Sparse Coding Bruno A. Olshausen David J. Field Preserve Information Preserve Sparsity (Simplicity) Tradeoff parameter Sparse’09 8 April 2009

  8. Sparse Coding (cont.) Image patch 1 to N … Patch image Vectorize patches  A corresponds to Gaborlike features resemblingsimple cell behavior inV1 of the brain! Sparse coding attempts to both learn dictionary and encoding at the same time (This problem is not convex!) Sparse’09 8 April 2009

  9. Automatic Relevance Determination (ARD) • Automatic Relevance Determination (ARD) is a hierarchical Bayesian approach widely used for model selection • In ARD hyper-parameters explicitly represents the relevance of different features by defining the range of variation. (i.e., Range of variation0  Feature removed) Sparse’09 8 April 2009

  10. A Bayesian formulation of the Lasso / BPD problem Sparse’09 8 April 2009

  11. ARD in reality a L0-norm optimization scheme. As such ARD based on Laplace prior corresponds to L0-norm optimization by re-weighted L1-norm L0 norm by re-weighted L2 follows by imposing Gaussian prior instead of Laplace Sparse’09 8 April 2009

  12. Sparse Tucker decomposition by ARD Brakes into standard Lasso/BPD sub-problems of the form Sparse’09 8 April 2009

  13. Solving efficiently the Lasso/BPD sub-problems Notice, in general the alternating sub-problems for Tucker estimation have J<I! Sparse’09 8 April 2009

  14. The Tucker ARD Algorithm Sparse’09 8 April 2009

  15. Results Fluorescene data: emission x excitation x samples Tucker(10,10,10) models were fitted to the data, given are below the extracted cores Synthetic Tucker(3,4,5) Tucker(3,6,4) Tucker(?,4,4) Tucker(4,4,4) Tucker(3,3,3) Sparse’09 8 April 2009

  16. Sparse Coding in fact 2-way case of Tucker model • The presented ARD framework extracts a 20 component model when trained on the natural image data of (Olshausen and Field, Nature 1996) Image patch 1 to N … Patch image Vectorize patches  Sparse’09 8 April 2009

  17. Conclusion • Imposing sparseness on the core enable to interpolate between Tucker and CP model • ARD based on MAP estimation form a simple framework to tune the pruning in sparse coding models giving the model order. • ARD framework especially useful for the Tucker model where the order is specified for each mode separately which makes exhaustive evaluation of all potential models expensive. • ARD-estimation based on MAP is closely related to L0 norm estimation based on reweighted L1 and L2 norm. Thus, ARD form a principled framework for learning the sparsity pattern in CS. Sparse’09 8 April 2009

  18. http://mmds.imm.dtu.dk Sparse’09 8 April 2009

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