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Blind source separation based on independent low-rank matrix analysis and its extension to Student's t-distribution

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Daichi Kitamura, "Blind source separation based on independent low-rank matrix analysis and its extension to Student's t-distribution," Télécom ParisTech, Invited Lecture, September 4th, 2017.

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Blind source separation based on independent low-rank matrix analysis and its extension to Student's t-distribution

  1. 1. Blind source separation based on independent low-rank matrix analysis and its extension to Student's t-distribution Télécom ParisTech Visiting September 4th The University of Tokyo, Japan Project Research Associate Daichi Kitamura
  2. 2. • Name: Daichi Kitamura • Age: 27 (born in 1990) – Kagawa Pref. in Japan • Background: – NAIST, Japan • Master degree (received in 2014) – SOKENDAI, Japan • Ph.D. degree (received in 2017) – The University of Tokyo, Japan • Project Research Associate • Research topics – Acoustic signal processing, statistical signal processing, audio source separation, etc. Self introduction 2 Japan Kagawa Tokyo
  3. 3. Contents • Background – Blind source separation (BSS) for audio signals – Motivation • Related Methods – Frequency-domain independent component analysis (FDICA) – Independent vector analysis (IVA) – Itakura–Saito nonnegative matrix factorization (ISNMF) • Independent Low-Rank Matrix Analysis (ILRMA) – Employ low-rank TF structures of each source in BSS – Gaussian source model with TF-varying variance – Relationship between ILRMA and multichannel NMF – Student’s t source model with TF-varying scale parameters • Conclusion 3
  4. 4. Contents • Background – Blind source separation (BSS) for audio signals – Motivation • Related Methods – Frequency-domain independent component analysis (FDICA) – Independent vector analysis (IVA) – Itakura–Saito nonnegative matrix factorization (ISNMF) • Independent Low-Rank Matrix Analysis (ILRMA) – Employ low-rank TF structures of each source in BSS – Gaussian source model with TF-varying variance – Relationship between ILRMA and multichannel NMF – Student’s t source model with TF-varying scale parameters • Conclusion 4
  5. 5. • Blind source separation (BSS) for audio signals – separates original audio sources – does not require prior information of recording conditions • locations of mics and sources, room geometry, timbres, etc. – can be available for many audio app. • Consider only “determined” situation Background 5 Recording mixture Separated guitar BSS Sources Observed Estimated Mixing system Demixing system # of mics # of sources
  6. 6. • Basic theories and their evolution History of BSS for audio signals 6 1994 1998 2013 1999 2012 Age Many permutation solvers for FDICA Apply NMF to many tasks Generative models in NMF Many extensions of NMF Independent component analysis (ICA) Frequency-domain ICA (FDICA) Itakura–Saito NMF (ISNMF) Independent vector analysis (IVA) Multichannel NMF Independent low-rank matrix analysis (ILRMA) *Depicting only popular methods 2016 2009 2006 2011 Auxiliary-function-based IVA (AuxIVA) Time-varying Gaussian IVA Nonnegative matrix factorization (NMF)
  7. 7. Motivation of ILRMA • Conventional BSS techniques based on ICA –  Minimum distortion (linear demixing) –  Relatively fast and stable optimization • FastICA [A. Hyvarinen, 1999], natural gradient [S. Amari, 1996], and auxiliary function technique [N. Ono+, 2010], [N. Ono, 2011] –  Could not use “specific” assumption of sources • Only assumes non-Gaussian p.d.f. for sources –  Permutation problem is crucial and still difficult to solve • IVA often fails causing a “block permutation problem” [Y. Liang+, 2012] • Better to use a “specific source model” in TF domain – Independent low-rank matrix analysis (ILRMA) employs a low-rank property 7 : frequency bins Observed signal Source signalsFrequency-wise mixing matrix : time frames Estimated signal Frequency-wise demixing matrix
  8. 8. Contents • Background – Blind source separation (BSS) for audio signals – Motivation • Related Methods – Frequency-domain independent component analysis (FDICA) – Independent vector analysis (IVA) – Itakura–Saito nonnegative matrix factorization (ISNMF) • Independent Low-Rank Matrix Analysis (ILRMA) – Employ low-rank TF structures of each source in BSS – Gaussian source model with TF-varying variance – Relationship between ILRMA and multichannel NMF – Student’s t source model with TF-varying scale parameters • Conclusion 8
  9. 9. • Independent component analysis (ICA)[P. Comon, 1994] – estimates without knowing – Source model (scalar) • is non-Gaussian and mutually independent – Spatial model • Mixing system is a time-invariant matrix • Mixing system in audio signals – Convolutive mixture with room reverberation Related methods: ICA 9 Mixing matrix Demixing matrix Source model Sources Observed Estimated Spatial model
  10. 10. • Frequency-domain ICA (FDICA) [P. Smaragdis, 1998] – estimates frequency-wise demixing matrix – Source model (scalar) • is complex-valued, non-Gaussian, and mutually independent – Spatial model • Frequency-wise mixing matrix is time-invariant – Instantaneous mixture in each frequency band – A.k.a. rank-1 spatial model [N.Q.K. Duong, 2010] • Permutation problem? – Order of estimated signals cannot be determined by ICA – Alignment of frequency-wise estimated signals is required • Many permutation solvers were proposed Related methods: FDICA 10 Spectrograms ICA1 … Frequencybin Time frame … ICA2 ICA I
  11. 11. • FDICA requires signal alignment for all frequency – Order of estimated signals cannot be determined by ICA* Permutation problem 11 ICA All frequency components Source 1 Source 2 Observed 1 Observed 2 Permutation Solver Estimated signal 1 Estimated signal 2 Time *Signal scale should also be restored by a back-projection technique
  12. 12. Related methods: IVA • Independent vector analysis (IVA)[A. Hiroe, 2006], [T. Kim, 2006] – extends ICA to multivariate probabilistic model to consider sourcewise frequency vector as a variable – Source model (vector) • is multivariate, spherical, complex-valued, non-Gaussian, and mutually independent – Spatial model • Mixing system is a time-invariant matrix (rank-1 spatial model) 12 … … Mixing matrix … … … Observed vector Demixing matrix Estimated vector Multivariate non- Gaussian dist. Have higher-order correlations Permutation-free estimation of is achieved! Source vector
  13. 13. • Spherical multivariate distribution[T. Kim, 2007] • Why spherical distribution? – Frequency bands that have similar activations will be merged together as one source avoid permutation problem Higher-order correlation assumed in IVA 13 x1 and x2 are mutually independent Spherical Laplace dist. Mutually independent two Laplace dist.s x1 and x2 have higher-order correlation Probability depends on only the norm
  14. 14. • Frequency-domain ICA (FDICA) [P. Smaragdis, 1998] • Independent vector analysis (IVA)[A. Hiroe, 2006], [T. Kim, 2006] Comparison of source models 14 Observed Update separation filter so that the estimated signals obey non-Gaussian distribution we assumed Estimated Demixing matrix Current empirical dist. Non-Gaussian source dist. STFT Frequency Time Frequency Time Observed Estimated Current empirical dist. STFT Frequency Time Frequency Time Non-Gaussian spherical source dist. Scalar r.v.s Vector (multivariate) r.v.s Update separation filter so that the estimated signals obey non-Gaussian distribution we assumed Mixture is close to Gaussian signal because of CLT Source obeys non- Gaussian dist. Mutually independent Demixing matrix Mutually independent
  15. 15. Related method: NMF • Nonnegative matrix factorization (NMF) [D. D. Lee, 1999] – Low-rank decomposition with nonnegative constraint • Limited number of nonnegative bases and their coefficients – Spectrogram is decomposed in acoustic signal processing • Frequently appearing spectral patterns and their activations 15 Amplitude Amplitude Nonnegative matrix (power spectrogram) Basis matrix (spectral patterns) Activation matrix (time-varying gains) Time : # of freq. bins : # of time frames : # of bases Time Frequency Frequency
  16. 16. • ISNMF[C. Févotte, 2009] – can be decomposed using “stable property” of • If we define , Related method: ISNMF 16 Equivalent Circularly symmetric complex Gaussian dist. Complex-valued observed signal Nonnegative variance Variance is also decomposed!
  17. 17. • Power spectrogram corresponds to variances in TF plane Related method: ISNMF 17 Frequencybin Time frame : Power spectrogram Small value of power Large value of power Complex Gaussian distribution with TF-varying variance If we marginalize in terms of time or frequency, the distribution becomes non-Gaussian even though each TF grid is defined in Gaussian distribution Grayscale shows the value of variance
  18. 18. Contents • Background – Blind source separation (BSS) for audio signals – Motivation • Related Methods – Frequency-domain independent component analysis (FDICA) – Independent vector analysis (IVA) – Itakura–Saito nonnegative matrix factorization (ISNMF) • Independent Low-Rank Matrix Analysis (ILRMA) – Employ low-rank TF structures of each source in BSS – Gaussian source model with TF-varying variance – Relationship between ILRMA and multichannel NMF – Student’s t source model with TF-varying scale parameters • Conclusion 18
  19. 19. Extension of source model in IVA • Source model in IVA – has a frequency-uniform scale • Multivariate Laplace with fixed scale • Since scale cannot be determined, it is not equivalent to the flat spectral basis – Almost an NMF with only one basis • Extend to ISNMF-based source model – NMF with arbitrary number of bases • can represent complicated TF structures – can learn “co-occurrence” of each source in TF domain • Co-occurrence is captured as the variance – The structure can easily be estimated by NMF 19 Frequency Time Frequency Time
  20. 20. • Spherical Laplace distribution in IVA • Gaussian distribution with TF-varying variance in ISNMF[C. Févotte+, 2009] 20 Frequency-uniform scale Extension of source model in IVA Complex-valued Gaussian in each TF bin Low-rank decomposition with NMF Spherical Laplace (bivariate) Frequency vector (I-dimensional) Time-frequency-varying variance Time-frequency matrix (IJ-dimensional)
  21. 21. • Negative log-likelihood in ILRMA Cost function in ILRMA and partitioning function 21 All the variables can easily be optimized by an alternative update Update rules in ICA Update rules in ISNMF Estimated signal: Cost function in ICA (estimates demixing matrix) Cost function in ISNMF (estimates low-rank source model)
  22. 22. Update rules of ILRMA • ML-based iterative update rules – Update rule for is based on iterative projection [N. Ono, 2011] – Update rules for NMF variables is based on MM algorithm – Pseudo code is available at • http://d-kitamura.net/pdf/misc/AlgorithmsForIndependentLowRankMatrixAnalysis.pdf 22 Spatial model (demixing matrix) Source model (NMF source model) where and is a one-hot vector that has 1 at th element
  23. 23. • ILRMA with partitioning function – Appropriate number of bases for each source can automatically be determined – Useful when various types of sources are mixed • Ex. drums are very low-rank but vocals are not so low-rank Cost function in ILRMA and partitioning function 23 andwhere
  24. 24. Update rules of ILRMA • ML-based iterative update rules – Update rule for is based on iterative projection [N. Ono, 2011] – Update rules for NMF variables is based on MM algorithm 24 Spatial model (demixing matrix) Source model (NMF source model) where and is a one-hot vector that has 1 at th element
  25. 25. Optimization process in ILRMA • Demixing matrix and source model are alternatively updated – The precise modeling of low-rank TF structures will improve the estimation accuracy of demixing matrix 25 Estimating demixing matrix Mixture Separated Source model Update NMF NMF Estimating NMF variables
  26. 26. Comparison of source models 26 FDICA source model Non-Gaussian scalar variable IVA source model Non-Gaussian vector variable with higher-order correlation ILRMA source model Non-Gaussian matrix variable with low-rank time-frequency structure Rank of TF matrix of mixture Rank of TF matrix of each source
  27. 27. • Multichannel NMF[A. Ozerov+, 2010], [H. Sawada+, 2013] Multichannel extension of NMF 27 Spatial covariances in each time-frequency slot Observed multichannel signal Spatial covariances of each source Basis matrix Activation matrix Spatial model Source model Partitioning function Spectral patterns Gains Spatial property of each source Timber patterns of all sources Multichannel vector Simultaneous spatial covariance
  28. 28. Relationship b/w ILRMA and multichannel NMF • Difference b/w ILRMA and multichannel NMF? – Source distribution: complex Gaussian distribution (same) – ILRMA assumes – Multichannel NMF assumes full-rank spatial covariance • Assumption: rank-1 spatial model – Spatial covariance of each source is rank-1 matrix – Equivalent to simultaneous mixing assumption 28 Sourcewise steering vector ,
  29. 29. Relationship b/w ILRMA and multichannel NMF • Multichannel NMF with rank-1 spatial model 30 Substitute into the cost function Transform the variables as
  30. 30. Relationship b/w MNMF, IVA, and ILRMA • From multichannel NMF side, – Rank-1 spatial model is introduced, transform the problem from the estimation of mixing system to that of demixing matrix • From IVA side, – Increase the number of spectral bases in source model 31 Source model Spatialmodel FlexibleLimited FlexibleLimited IVA Multichannel NMF ILRMA NMF source model Rank-1 spatial model
  31. 31. Experimental evaluation • Conditions 32 Source signals Music signals obtained from SiSEC Convolve impulse response, two microphones and two sources Window length 512 ms of Hamming window Shift length 128 ms (1/4 shift) Number of bases 30 per each source (ILRMA w/o partitioning function) 60 for all source (ILRMA with partitioning function) Evaluation score Improvement ot signal-to-distortion ratio (SDR) 2 m Source 1 5.66cm 50 50 Source 2 2 m Source 1 5.66cm 60 60 Source 2 Impulse response E2A (reverberation time: 300 ms) Impulse response JR2 (reverberation time: 470 ms)
  32. 32. Results: fort_minor-remember_the_name 33 16 12 8 4 0 -4 -8 SDRimprovement[dB] Sawada’s MNMF IVA Ozerov’s MNMF Ozerov’s MNMF with random initialization Sawada’s MNMF initialized by ILRMA ILRMA w/o partitioning function ILRMA with partitioning function Directional clustering Sawada’s MNMF IVA Ozerov’s MNMF Ozerov’s MNMF with random initialization Sawada’s MNMF initialized by ILRMA ILRMA w/o partitioning function ILRMA with partitioning function Directional clustering 16 12 8 4 0 -4 -8 SDRimprovement[dB] Violin synth. Vocals Violin synth. Vocals E2A (300 ms) JR2 (470 ms) Poor Good Poor Good
  33. 33. Results: ultimate_nz_tour 34 Sawada’s MNMF IVA Ozerov’s MNMF Ozerov’s MNMF with random initialization Sawada’s MNMF initialized by ILRMA ILRMA w/o partitioning function ILRMA with partitioning function Directional clustering 20 15 10 5 0 -5 SDRimprovement[dB] Sawada’s MNMF IVA Ozerov’s MNMF Ozerov’s MNMF with random initialization Sawada’s MNMF initialized by ILRMA ILRMA w/o partitioning function ILRMA with partitioning function Directional clustering 20 15 10 5 0 -5 SDRimprovement[dB] Guitar Synth. Guitar Synth. Poor Good Poor Good E2A (300 ms) JR2 (470 ms)
  34. 34. • Signal length: 14 s 12 10 8 6 4 2 0 -2 SDRimprovement[dB] 4003002001000 Iteration steps IVA MNMF ILRMA ILRMA Results: bearlin-roads 35 without Z with Z 11.5 s 15.1 s 60.7 s 7647.3 s Poor Good
  35. 35. Demonstration: music source separation • Music source separation 36 Guitar Vocal Keyboard Guitar Vocal Keyboard Source separation Pay attention to listen three parts in the mixture Another demo is available at http://d-kitamura.net/en/index_en.html
  36. 36. • Source model based on Symmetric a-stable (SaS) distribution[A. Liutkus+, 2015], [U. Şimşekli+, 2015], [S. Leglaive+, 2017], [M. Fontaine+, 2017] – which can validate the decomposition of complex-valued r.v.s as the decomposition of their parameters – Heavy tail (sparse) when a approaches to 0 • Student’s t-distribution is also used as a source model[C. Févotte+, 2006], [K. Yoshii+, 2016], [K. Kitamura+, 2016], [S. Leglaive+, 2017] – that includes Cauchy distribution ( ) and Gaussian distribution ( ) Stable and Student’s t-distributions 37 SaS (stable family) Student’s t (partially stable) Cauchy Gauss
  37. 37. Source model of Student’s t-distribution • Degree-of-freedom parameter – Heavy tail when approaches to 0 • Complex Student’s t-dist. – Circularly symmetric – Student’s t NMF (t-NMF) [K. Yoshii+ 2016] 38 Defined in each TF slot Scale corresponds to NMF model Phase is assumed to be uniform
  38. 38. Motivation for using Student’s t-dist. • Better separation with t-NMF was reported[K. Yoshii+, 2016] – in a very simple experiment using only C4, E4, and G4 piano tones • NMF with heavy tail distribution – tends to provide excessive low-rank approximation • Sparse components (which may increase the rank of model data) are considered as outliers • ILRMA based on Student’s t source model (t-ILRMA) – may improve the separation accuracy by forcing NMF source model to be excessively low-rank – will be presented at MLSP2017! (preprint is available on arXiv) • https://arxiv.org/abs/1708.04795 39
  39. 39. • th power spectrogram corresponds to scales in TF plane Source model based on Student’s t-distribution 40 Frequencybin Time frame : th power spectrogram Small value of power Large value of power Complex Student’s t-distribution with TF-varying scale Grayscale shows the value of scale
  40. 40. • Negative log-likelihood in ILRMA Cost function in ILRMA based on Student’s t-dist. 41 Gaussian ILRMA modeling power spectrogram by variance Student’s t ILRMA modeling pth power spectrogram by scale Generalization of p.d.f. and model domain
  41. 41. Experimental results: randomized t-ILRMA • Examples – Improved when – Stable when but score is not sufficient – Root spectrogram ( ) is preferable for speech signals • In the case of – Source model is over-fitted to mixture 42 Music signals Speech signals
  42. 42. Tempering parameter • Random initialization (previous result) • Initialization based on Gaussian ILRMA – (Tempering approach of parameter) 43 t-ILRMA (iteration: 200) Identity matrix Uniform random values Gauss ILRMA (iteration: 100) Identity matrix Uniform random values t-ILRMA (iteration: 100) t-NMF (iteration: 100)Uniform random values arbitrary val.
  43. 43. Experimental results: initialized t-ILRMA • Examples – Improved for all value of – Could avoid over- fitting problem in the case • Best parameter? – Completely depending on data 44 Music signals Speech signals
  44. 44. Average results: music signals 45
  45. 45. Average results: speech signals 46
  46. 46. Contents • Background – Blind source separation (BSS) for audio signals – Motivation • Related Methods – Frequency-domain independent component analysis (FDICA) – Independent vector analysis (IVA) – Itakura–Saito nonnegative matrix factorization (ISNMF) • Independent Low-Rank Matrix Analysis (ILRMA) – Employ low-rank TF structures of each source in BSS – Gaussian source model with TF-varying variance – Relationship between ILRMA and multichannel NMF – Student’s t source model with TF-varying scale parameters • Conclusion 47
  47. 47. Conclusion • Independent low-rank matrix analysis (ILRMA) – Assumption • Statistical independence between sources • Low-rank time-frequency structure of each source – Equivalent to multichannel NMF • when the mixing assumption is valid • Student’s t-distribution is newly introduced – including two symmetric a-stable distributions • Complex Cauchy distribution ( ) • Complex Gaussian distribution ( ) • Further extensions – Relaxation of rank-1 spatial model? – Employ another distribution? – Supervised ILRMA? User-guided ILRMA? 48

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