Informatics and Mathematical Modelling Intelligent Signal Processing Extensions

Informatics and Mathematical Modelling / Intelligent Signal Processing Extensions of Non-negative Matrix Factorization to Higher Order data Morten Mørup Informatics and Mathematical Modeling Intelligent Signal Processing Technical University of Denmark Morten Mørup 1

Informatics and Mathematical Modelling / Intelligent Signal Processing 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 Morten Mørup Sidse M. Arnfred, Dr. Med. Ph. D Mikkel N. Schmidt, Stud. Ph. D Cognitive Research Unit Hvidovre Hospital University Hospital of Copenhagen Department of Signal Processing Informatics and Mathematical Modeling, Technical University of Denmark 2

Informatics and Mathematical Modelling / Intelligent Signal Processing Outline n Non-negativity Matrix Factorization (NMF) n Sparse coding (SNMF) n Convolutive PARAFAC models (c. PARAFAC) n Higher Order Non-negative Matrix Factorization (an extension of NMF to the Tucker model) Morten Mørup 3

Informatics and Mathematical Modelling / Intelligent Signal Processing NMF is based on Gradient Descent NMF: V WH s. t. Wi, d, Hd, j 0 Let C be a given cost function, then update the parameters according to: Morten Mørup 4

Informatics and Mathematical Modelling / Intelligent Signal Processing The idea behind multiplicative updates Positive term Morten Mørup 5 Negative term

Informatics and Mathematical Modelling / Intelligent Signal Processing Non-negative matrix factorization (NMF) (Lee & Seung - 2001) NMF gives Part based representation (Lee & Seung – Nature 1999) Morten Mørup 6

Informatics and Mathematical Modelling / Intelligent Signal Processing The NMF decomposition is not unique Simplical Cone NMF only unique when data adequately spans the positive orthant (Donoho & Stodden - 2004) Morten Mørup 7

Informatics and Mathematical Modelling / Intelligent Signal Processing Sparse Coding NMF (SNMF) (Eggert & Körner, 2004) Morten Mørup 8

Informatics and Mathematical Modelling / Intelligent Signal Processing Swimmer Articulations Illustration (the swimmer problem) True Expressions Morten Mørup NMF Expressions 9 SNMF Expressions

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

Informatics and Mathematical Modelling / Intelligent Signal Processing PART I: Convolutive PARAFAC (c. PARAFAC) Morten Mørup 11

Informatics and Mathematical Modelling / Intelligent Signal Processing By c. PARAFAC means PARAFAC convolutive in at least one modality Convolution: The process of generating X by convolving (sending) the sources S through the filter A Deconvolution: The process of estimating the filter A from X and S Convolution can be in any combination of modalities -Single convolutive, double convolutive etc. Morten Mørup 12

Informatics and Mathematical Modelling / Intelligent Signal Processing Relation to other models n PARAFAC 2 (Harshman, Kiers, Bro) n Shifted PARAFAC (Hong and Harshman, 2003) 3 3 c. PARAFAC can account for echo effects Morten Mørup 13 c. PARAFAC becomes shifted PARAFAC when convolutive filter is sparse

Informatics and Mathematical Modelling / Intelligent Signal Processing Application example of c. PARAFAC Transcription and separation of music Morten Mørup 14

Informatics and Mathematical Modelling / Intelligent Signal Processing The ‘ideal’ Log-frequency Magnitude Spectrogram of an instrument n 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 15

Informatics and Mathematical Modelling / Intelligent Signal Processing NMF 2 D deconvolution (NMF 2 D 1): The Basic Idea n 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-logfrequency signature but varying in fundamental frequency (shift) model convolutive in the log-frequency axis. (1 Mørup & Scmidt, 2006) Morten Mørup 16

Informatics and Mathematical Modelling / Intelligent Signal Processing NMF 2 D Model n NMF 2 D Model – extension of NMFD 1: (1 Smaragdis, 2004, Eggert et al. 2004, Fitzgerald et al. 2005) Morten Mørup 17

Informatics and Mathematical Modelling / Intelligent Signal Processing Understanding the NMF 2 D Model Morten Mørup 18

Informatics and Mathematical Modelling / Intelligent Signal Processing The NMF 2 D 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 19

Informatics and Mathematical Modelling / Intelligent Signal Processing Real music example of how imposing sparseness resolves the ambiguity between W and H NMF 2 D Morten Mørup 20 SNMF 2 D

Informatics and Mathematical Modelling / Intelligent Signal Processing Extension to multi channel analysis by the PARAFAC model Not Un uni que iqu Factor analysis (Charles Spearman ~1900) Morten Mørup e!! PARAFAC (Harshman & Carrol and Chang 1970) 21

Informatics and Mathematical Modelling / Intelligent Signal Processing c. PARAFAC: Sparse Non-negative Tensor Factor 2 D deconvolution (SNTF 2 D) (Extension of Fitzgerald et al. 2005, 2006 to form a sparse double deconvolution) Morten Mørup 22

Informatics and Mathematical Modelling / Intelligent Signal Processing SNTF 2 D algorithms Morten Mørup 23

Informatics and Mathematical Modelling / Intelligent Signal Processing Tchaikovsky: Violin Concert in D Major Morten Mørup Mozart Sonate no. 16 in C Major 24

Informatics and Mathematical Modelling / Intelligent Signal Processing Stereo recording of ”Fog is Lifting” by Carl Nielsen Morten Mørup 25

Informatics and Mathematical Modelling / Intelligent Signal Processing Applications n Applications – – Source separation. Music information retrieval. Automatic music transcription (MIDI compression). Source localization (beam forming) Morten Mørup 26

Informatics and Mathematical Modelling / Intelligent Signal Processing PART II: Higher Order NMF (HONMF) Morten Mørup 27

Informatics and Mathematical Modelling / Intelligent Signal Processing Higher Order Non-negative Matrix Factorization (HONMF) Motivation: 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. n Spectroscopy data (Smilde et al. 1999, 2004, Andersson & Bro 1998, Nørgard & Ridder 1994) n Web mining (Sun et al. , 2004) n Image Analysis (Vasilescu and Terzopoulos, 2002, Wang and Ahuja, 2003, Jian and Gong, 2005) n Semantic Differential Data (Murakami and Kroonenberg, 2003) n And many more…… Morten Mørup 28

Informatics and Mathematical Modelling / Intelligent Signal Processing However, non-negative Tucker decompositions are not in general unique! But - Imposing sparseness overcomes this problem! Morten Mørup 29

Informatics and Mathematical Modelling / Intelligent Signal Processing The Tucker Model Morten Mørup 30

Informatics and Mathematical Modelling / Intelligent Signal Processing Algorithms for HONMF Morten Mørup 31

Informatics and Mathematical Modelling / Intelligent Signal Processing 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 32

Informatics and Mathematical Modelling / Intelligent Signal Processing Evaluation of uniqueness Morten Mørup 33

Informatics and Mathematical Modelling / Intelligent Signal Processing Data of a Flow Injection Analysis (Nørrgaard, 1994) HONMF with sparse core and mixing captures unsupervised the true mixing and model order! Morten Mørup 34

Informatics and Mathematical Modelling / Intelligent Signal Processing Conclusion n HONMF not in general unique, however when imposing sparseness uniqueness can be achieved. n Algorithms devised for LS and KL able to impose sparseness on any combination of modalities n The HONMF decompositions more part based hence easier to interpret than other Tucker decompositions such as the HOSVD. n Imposing sparseness can work as model selection turning of excess components Morten Mørup 35

Informatics and Mathematical Modelling / Intelligent Signal Processing Coming soon in a MATLAB implementation near You Morten Mørup 36

Informatics and Mathematical Modelling / Intelligent Signal Processing 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 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 Fitz. Gerald, D. and Coyle, E. C Sound source separation using shifted non. -negative tensor factorization. In ICASSP 2006, 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 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 Kiers, Henk A. L. and Berge, Jos M. F. ten and Bro, Rasmus PARAFAC 2 - Part I. A direct fitting algorithm for the PARAFAC 2 model, Journal of Chemometrics (13) nr. 3 -4 pages 275 -294, 1999 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 Murakami, Takashi and Kroonenberg, Pieter M. Three-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, 2006 a Mørup, M. and Schmidt, M. N. Sparse non-negative matrix factor 2 -D deconvolution. Technical report, Institute for Mathematical Modeling, Tehcnical University of Denmark, 2006 b Mørup, M and Schmidt, M. N. Non-negative Tensor Factor 2 D Deconvolution for multi-channel time-frequency analysis. Technical report, Institute for Mathematical Modeling, Technical University of Denmark, 2006 c Schmidt, M. N. and Mørup, M. Non-negative matrix factor 2 D deconvolution for blind single channel source separation. In ICA 2006, pages 700 -707, 2006 d Mørup, M. and Hansen, L. K. and Arnfred, S. M. Algorithms for Sparse Higher Order Non-negative Matrix Factorization (HONMF), Technical report, Institute for Mathematical Modeling, Technical University of Denmark, 2006 e 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 Cube. SVD: a novel approach to personalized Web search WWW '05: Proceedings of the 14 th 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 SAND 2006 -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: Tensor. Faces, ECCV '02: Proceedings of the 7 th European Conference on Computer Vision-Part I, 2002 Morten Mørup 37