Non-negative Tensor Decompositions 16-04-2017 Non-negative Tensor Decompositions Morten Mørup Informatics and Mathematical Modeling Intelligent Signal Processing Technical University of Denmark Morten Mørup Jan Larsen

Parts of the work done in collaboration with 16-04-2017 Sæby, May 22-2006 Parts of the work done in collaboration with Lars Kai Hansen, Professor Department of Signal Processing Informatics and Mathematical Modeling, Technical University of Denmark Sidse M. Arnfred, Dr. Med. PhD Cognitive Research Unit Hvidovre Hospital University Hospital of Copenhagen Mikkel N. Schmidt, Stud. PhD Department of Signal Processing Informatics and Mathematical Modeling, Technical University of Denmark Morten Mørup Jan Larsen

Overview Non-negativity Matrix Factorization (NMF) 16-04-2017 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 Jan Larsen

Non-negative Matrix Factorization (NMF): 16-04-2017 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 Jan Larsen

The idea behind multiplicative updates 16-04-2017 The idea behind multiplicative updates Positive term Negative term Morten Mørup Jan Larsen

Non-negative matrix factorization (NMF) 16-04-2017 Non-negative matrix factorization (NMF) (Lee & Seung - 2001) NMF gives Part based representation (Lee & Seung – Nature 1999) Morten Mørup Jan Larsen

The NMF decomposition is not unique 16-04-2017 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 Jan Larsen

Sparse Coding NMF (SNMF) 16-04-2017 Sparse Coding NMF (SNMF) (Eggert & Körner, 2004) (Mørup & Schmidt, 2006) Morten Mørup Jan Larsen

Illustration (the swimmer problem) 16-04-2017 Illustration (the swimmer problem) Swimmer Articulations True Expressions NMF Expressions SNMF Expressions Morten Mørup Jan Larsen

Why sparseness? Ensures uniqueness 16-04-2017 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 Jan Larsen

d d = Extensions to tensors Factor Analysis PARAFAC TUCKER TUCKER 16-04-2017 Extensions to tensors Factor Analysis PARAFAC TUCKER TUCKER d d = Morten Mørup Jan Larsen

16-04-2017 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 Jan Larsen

Example why Non-negative PARAFAC isn’t unique 16-04-2017 Example why Non-negative PARAFAC isn’t unique Morten Mørup Jan Larsen

PARAFAC model estimation 16-04-2017 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 Jan Larsen

TUCKER model estimation 16-04-2017 TUCKER model estimation TUCKER Morten Mørup Jan Larsen

Algorithms for Non-negative TUCKER (PARAFAC follows by setting C=I) 16-04-2017 Algorithms for Non-negative TUCKER (PARAFAC follows by setting C=I) (Mørup et al. 2006) Morten Mørup Jan Larsen

16-04-2017 Application of Non-negative TUCKER and PARAFAC Non-negative TUCKER in the following called HONMF (Higher order non-negative matrix factorization) Non-negative PARAFAC called NTF (Non-negative tensor factorization) Morten Mørup Jan Larsen

Continuous Wavelet transform 16-04-2017 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 Jan Larsen

Channel x Time-Frequency x Subjects 16-04-2017 Channel x Time-Frequency x Subjects Subjects channel time-frequency Morten Mørup Jan Larsen

16-04-2017 Results HONMF with sparseness, above imposed on the core can be 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 Jan Larsen

Evaluation of uniqueness 16-04-2017 Evaluation of uniqueness Morten Mørup Jan Larsen

Data of a Flow Injection Analysis (Nørrgaard, 1994) 16-04-2017 Data of a Flow Injection Analysis (Nørrgaard, 1994) HONMF with sparse core and mixing captures unsupervised the true mixing and model order! Morten Mørup Jan Larsen

Semantic Differential Data (Murakami and Kroonenberg, 2003) 16-04-2017 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. 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…… Hopefully, the devised algorithms for sparse non-negative TUCKER will prove useful Morten Mørup Jan Larsen

16-04-2017 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 Jan Larsen

Released 14th September 2006 ERPWAVELAB Morten Mørup 16-04-2017 Jan Larsen

16-04-2017 Sparse Non-negative Tensor Factor double deconvolution for music separation and transcription Morten Mørup Jan Larsen

The ‘ideal’ Log-frequency Magnitude Spectrogram of an instrument 16-04-2017 The ‘ideal’ Log-frequency Magnitude Spectrogram of an instrument Different notes played by an instrument corresponds on a logarithmic frequency scale to a translation of the same harmonic structure of a fixed temporal pattern Tchaikovsky: Violin Concert in D Major Mozart Sonate no,. 16 in C Major Morten Mørup Jan Larsen

NMF 2D deconvolution (NMF2D1): The Basic Idea 16-04-2017 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 Jan Larsen

Understanding the NMF2D Model 16-04-2017 Understanding the NMF2D Model V W H Morten Mørup Jan Larsen

The NMF2D has inherent ambiguity between the structure in W and H 16-04-2017 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 Jan Larsen

16-04-2017 Real music example of how imposing sparseness resolves the ambiguity between W and H NMF2D SNMF2D Morten Mørup Jan Larsen

Morten Mørup 16-04-2017 Tchaikovsky: Violin Concert in D Major Mozart Sonate no. 16 in C Major Morten Mørup Jan Larsen

Sparse Non-negative Tensor Factor 2D deconvolution (SNTF2D) 16-04-2017 Sparse Non-negative Tensor Factor 2D deconvolution (SNTF2D) (Extension of Fitzgerald et al. 2005, 2006 to form a sparse double deconvolution) Morten Mørup Jan Larsen

Stereo recording of ”Fog is Lifting” by Carl Nielsen 16-04-2017 Stereo recording of ”Fog is Lifting” by Carl Nielsen Morten Mørup Jan Larsen

Applications Applications Source separation. 16-04-2017 Applications Applications Source separation. Music information retrieval. Automatic music transcription (MIDI compression). Source localization (beam forming) Morten Mørup Jan Larsen

References Morten Mørup 16-04-2017 Jan Larsen 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. 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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. 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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 Jan Larsen