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Non-negative Tensor Decompositions

Non-negative Tensor Decompositions. Morten Mørup Informatics and Mathematical Modeling Intelligent Signal Processing Technical University of Denmark. Sæby, May 22-2006. Parts of the work done in collaboration with. Sidse M. Arnfred, Dr. Med. PhD Cognitive Research Unit

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Non-negative Tensor Decompositions

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  1. Non-negative Tensor Decompositions Morten Mørup Informatics and Mathematical Modeling Intelligent Signal Processing Technical University of Denmark Morten Mørup

  2. Sæby, May 22-2006 Parts of the work done in collaboration with Sidse M. Arnfred, Dr. Med. PhD Cognitive Research Unit Hvidovre Hospital University Hospital of Copenhagen Mikkel N. Schmidt, Stud. PhDDepartment of Signal Processing Informatics and Mathematical Modeling, Technical University of Denmark Lars Kai Hansen, Professor Department of Signal Processing Informatics and Mathematical Modeling, Technical University of Denmark Morten Mørup

  3. Overview • Non-negativity Matrix Factorization (NMF) • Sparse coding NMF(SNMF) • Sparse Higher Order Non-negative Matrix Factorization (HONMF) • Sparse Non-negative Tensor double deconvolution(SNTF2D) Morten Mørup

  4. Factor Analysis Spearman ~1900 Subjects Int. Subjects Int.  tests tests d VWH Vtests x subjects  Wtests x intelligencesHintelligencesxsubject Non-negative Matrix Factorization (NMF): VWH s.t. Wi,d,Hd,j0 (~1970 Lawson, ~1995 Paatero, ~2000 Lee & Seung) Morten Mørup

  5. The idea behind multiplicative updates Positive term Negative term Morten Mørup

  6. Non-negative matrix factorization (NMF) (Lee & Seung - 2001) NMF gives Part based representation (Lee & Seung – Nature 1999) Morten Mørup

  7. The NMF decomposition is not unique Simplical Cone Positive Orthant Convex Hull z z z y y y x x x NMF only unique when data adequately spans the positive orthant (Donoho & Stodden - 2004) Morten Mørup

  8. Sparse Coding NMF (SNMF) (Mørup & Schmidt, 2006) (Eggert & Körner, 2004) Morten Mørup

  9. Illustration (the swimmer problem) Swimmer Articulations NMF Expressions SNMF Expressions True Expressions Morten Mørup

  10. Why sparseness? • Ensures uniqueness • Eases interpretability (sparse representation  factor effects pertain to fewer dimensions) • Can work as model selection(Sparseness can turn off excess factors by letting them become zero) • Resolves over complete representations (when model has many more free variables than data points) Morten Mørup

  11. Extensions to tensors TUCKER Factor Analysis PARAFAC TUCKER d d = Morten Mørup

  12. Uniqueness • Although PARAFAC in general is unique under mild conditions, the proof of uniqueness by Kruskal is based on k-rank*. However, the k-rank does not apply for non-negativity**. • TUCKER model is not unique, thus no guaranty of uniqueness. Imposing sparseness useful in order to achieve unique decompositions Tensor decompositions known to have problems with degeneracy, however when imposing non-negativity degenerate solutions can’t occur*** *) k-rank: The maximum number of columns chosen by random of a matrix certain to be linearly independent. **) L.-H. Lim and G.H. Golub, 2006. ***) See L.-H. Lim - http://www.etis.ensea.fr/~wtda/Articles/wtda-nnparafac-slides.pdf Morten Mørup

  13. Example why Non-negative PARAFAC isn’t unique Morten Mørup

  14. PARAFAC model estimation d Thus, the PARAFAC model is by the matricizing operation estimated straight forward from regular NMF estimation by interchanging W with A and H with Z. Morten Mørup

  15. TUCKER model estimation TUCKER Morten Mørup

  16. Algorithms for Non-negative TUCKER (PARAFAC follows by setting C=I) (Mørup et al. 2006) Morten Mørup

  17. Application of Non-negative TUCKER and PARAFACNon-negative TUCKER in the following called HONMF(Higher order non-negative matrix factorization)Non-negative PARAFAC called NTF(Non-negative tensor factorization) Morten Mørup

  18. Continuous Wavelet transform Absolute value of wavelet coefficient Complex Morlet wavelet - Real part - Complex part frequency time time   Captures frequency changes through time Morten Mørup

  19. Channel x Time-Frequency x Subjects Subjects channel time-frequency Morten Mørup

  20. Results HONMF with sparseness, above imposed on the core canbe used for model selection -here indicating the PARAFAC model is the appropriate model to the data. Furthermore, the HONMF gives a more part based hence easy interpretable solution than the HOSVD. Morten Mørup

  21. Evaluation of uniqueness Morten Mørup

  22. Data of a Flow Injection Analysis (Nørrgaard, 1994) HONMF with sparse core and mixing captures unsupervised the true mixing and model order! Morten Mørup

  23. Spectroscopy data (Smilde et al. 1999,2004, Andersson & Bro 1998, Nørgard & Ridder 1994) Web mining (Sun et al., 2004) Image Analysis(Vasilescu and Terzopoulos, 2002, Wang and Ahuja, 2003, Jian and Gong, 2005) Semantic Differential Data(Murakami and Kroonenberg, 2003) And many more…… Many of the data sets previously explored by the Tucker model are non-negative and could with good reason be decomposed under constraints of non-negativity on all modalities including the core. Hopefully, the devised algorithms for sparse non-negative TUCKER will prove useful Morten Mørup

  24. Conclusion • HONMF and NTF not in general unique, however when imposing sparseness uniqueness can be achieved. • Algorithms devised for LS and KL able to impose sparseness on any combination of modalities • The HONMF decompositions more part based hence easier to interpret than other Tucker decompositions such as the HOSVD. • Imposing sparseness can work as model selection turning of excess components Morten Mørup

  25. Released 14th September 2006 ERPWAVELAB Morten Mørup

  26. Sparse Non-negative Tensor Factor double deconvolution for music separation and transcription Morten Mørup

  27. The ‘ideal’ Log-frequency Magnitude Spectrogram of an instrument Tchaikovsky: Violin Concert in D Major • Different notes played by aninstrument corresponds on a logarithmic frequency scale to a translation of the same harmonicstructure of a fixed temporal pattern Mozart Sonate no,. 16 in C Major Morten Mørup

  28. NMF 2D deconvolution (NMF2D1): The Basic Idea • Model a log-spectrogram of polyphonic music by an extended type of non-negative matrix factorization: • The frequency signature of a specific note played by an instrument has a fixed temporal pattern (echo) model convolutive in time • Different notes of same instrument has same time-log-frequency signature but varying in fundamental frequency (shift)  model convolutive in the log-frequency axis. (1Mørup & Scmidt, 2006) Morten Mørup

  29. Understanding the NMF2D Model H W V Morten Mørup

  30. The NMF2D has inherent ambiguity between the structure in W and H To resolve this ambiguity sparsity is imposed on H to force ambiguous structure onto W Morten Mørup

  31. Real music example of how imposing sparseness resolves the ambiguity between W and H NMF2D SNMF2D Morten Mørup

  32. Tchaikovsky: Violin Concert in D Major Mozart Sonate no. 16 in C Major Morten Mørup

  33. Sparse Non-negative Tensor Factor 2D deconvolution (SNTF2D) (Extension of Fitzgerald et al. 2005, 2006 to form a sparse double deconvolution) Morten Mørup

  34. Stereo recording of ”Fog is Lifting” by Carl Nielsen Morten Mørup

  35. Applications • Applications • Source separation. • Music information retrieval. • Automatic music transcription (MIDI compression). • Source localization (beam forming) Morten Mørup

  36. References Carroll, J. D. and Chang, J. J. Analysis of individual differences in multidimensional scaling via an N-way generalization of "Eckart-Young" decomposition, Psychometrika 35 1970 283—319 Donoho, D. and Stodden, V. When does non-negative matrix factorization give a correct decomposition into parts? NIPS2003 Eggert, J. and Korner, E. Sparse coding and NMF. In Neural Networks volume 4, pages 2529-2533, 2004 Eggert, J et al Transformation-invariant representation and nmf. In Neural Networks, volume 4 , pages 535-2539, 2004 Fiitzgerald, D. et al. Non-negative tensor factorization for sound source separation. In proceedings of Irish Signals and Systems Conference, 2005 FitzGerald, D. and Coyle, E. C Sound source separation using shifted non.-negative tensor factorization. In ICASSP2006, 2006 Fitzgerald, D et al. Shifted non-negative matrix factorization for sound source separation. In Proceedings of the IEEE conference on Statistics in Signal Processing. 2005 Kruskal, J.B. Three-way analysis: rank and uniqueness of trilinear decompostions, with application to arithmetic complexity and statistics. Linear Algebra Appl., 18: 95-138, 1977 Harshman, R. A. Foundations of the PARAFAC procedure: Models and conditions for an "explanatory" multi-modal factor analysis},UCLA Working Papers in Phonetics 16 1970 1—84 Harshman, Richard A.Harshman and Hong, Sungjin Lundy, Margaret E. Shifted factor analysis—Part I: Models and properties J. Chemometrics (17) pages 379–388, 2003 Lathauwer, Lieven De and Moor, Bart De and Vandewalle, Joos MULTILINEAR SINGULAR VALUE DECOMPOSITION.SIAM J. MATRIX ANAL. APPL.2000 (21)1253–1278 Lee, D.D. and Seung, H.S. Algorithms for non-negative matrix factorization. In NIPS, pages 556-462, 2000 Lee, D.D and Seung, H.S. Learning the parts of objects by non-negative matrix factorization, NATURE 1999 Lim, Lek-Heng - http://www.etis.ensea.fr/~wtda/Articles/wtda-nnparafac-slides.pdf Lim, L.-H. and Golub, G.H., "Nonnegative decomposition and approximation of nonnegative matrices and tensors," SCCM Technical Report, 06-01, forthcoming, 2006. Murakami, Takashi and Kroonenberg, Pieter M. Three-Mode Models and Individual Differences in Semantic Differential Data, Multivariate Behavioral Research(38) no. 2 pages 247-283, 2003 Mørup, M. and Hansen, L.K.and Arnfred, S.M.Decomposing the time-frequency representation of EEG using nonnegative matrix and multi-way factorization Technical report, Institute for Mathematical Modeling, Technical University of Denmark, 2006b Mørup, M., Hansen, L. K., Arnfred, S. M., ERPWAVELAB A toolbox for multi-channel analysis of time-frequency transformed event related potentials, Journal of Neuroscience Methods, vol. 161, pp. 361-368, 2007a Mørup, M., Hansen, L. K., Parnes, Josef, Hermann, C, Arnfred, S. M., Parallel Factor Analysis as an exploratory tool for wavelet transformed event-related EEG Neuroimage NeuroImage 29 938 – 947, 2006a Mørup, M., Schmidt, M. N., Hansen, L. K., Shift Invariant Sparse Coding of Image and Music Data, submitted, JMLR, 2007b Mørup, M., Hansen, L. K., Arnfred, S. M., Algorithms for Sparse Non-negative TUCKER, Submitted Neural Computation, 2006e Mørup, M. and Hansen, L.K.and Arnfred, S.M.Decomposing the time-frequency representation of EEG using nonnegative matrix and multi-way factorization Technical report, Institute for Mathematical Modeling, Technical University of Denmark, 2006a Schmidt, M.N. and Mørup, M. Non-negative matrix factor 2D deconvolution for blind single channel source separation. In ICA2006, pages 700-707, 2006d Nørgaard, L and Ridder, C.Rank annihilation factor analysis applied to flow injection analysis with photodiode-array detection Chemometrics and Intelligent Laboratory Systems 1994 (23) 107-114 Schmidt, M.N. and Mørup, M. Sparse Non-negative Matrix Factor 2-D Deconvolution for Automatic Transcription of Polyphonic Music, Technical report, Institute for Mathematical Modelling, Tehcnical University of Denmark, 2005 Smaragdis, P. Non-negative Matrix Factor deconvolution; Extraction of multiple sound sources from monophonic inputs. International Symposium on independent Component Analysis and Blind Source Separation (ICA)W Smilde, Age K. Smilde and Tauller, Roma and Saurina, Javier and Bro, Rasmus, Calibration methods for complex second-order data Analytica Chimica Acta 1999 237-251 Sun, Jian-Tao and Zeng, Hua-Jun and Liu, Huanand Lu Yuchang and Chen Zheng CubeSVD: a novel approach to personalized Web search WWW '05: Proceedings of the 14th international conference on World Wide Web pages 382—390, 2005 Tamara G. Kolda Multilinear operators for higher-order decompositions technical report Sandia national laboratory 2006 SAND2006-2081. Tucker, L. R. Some mathematical notes on three-mode factor analysis Psychometrika 31 1966 279—311 Welling, M. and Weber, M. Positive tensor factorization. Pattern Recogn. Lett. 2001 Vasilescu , M. A. O. and Terzopoulos , Demetri Multilinear Analysis of Image Ensembles: TensorFaces, ECCV '02: Proceedings of the 7th European Conference on Computer Vision-Part I, 2002 Morten Mørup

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