Journée Interdisciplinaire Mathématiques Musique

Size: px

Start display at page:

Download "Journée Interdisciplinaire Mathématiques Musique"

Maximilian Newman
5 years ago
Views:

1 Journée Interdisciplinaire Mathématiques Musique Music Information Geometry Arnaud Dessein 1,2 and Arshia Cont 1 1 Institute for Research and Coordination of Acoustics and Music, Paris, France 2 Japanese-French Laboratory for Informatics, Tokyo, Japan IRMA, Strasbourg, April 7th 2011 arnaud.dessein@ircam.fr April 7th 2011 Journée Interdisciplinaire Mathématiques Musique 1/21

2 Outline Introduction 1 Introduction arnaud.dessein@ircam.fr April 7th 2011 Journée Interdisciplinaire Mathématiques Musique 2/21

3 Outline Introduction A bit of history about science and music Motivations towards information geometry 1 Introduction A bit of history about science and music Motivations towards information geometry arnaud.dessein@ircam.fr April 7th 2011 Journée Interdisciplinaire Mathématiques Musique 3/21

4 Where do we come from? Introduction A bit of history about science and music Motivations towards information geometry Pythagoras ( BC): relation between string length and produced sound, Pythagorean tuning. There is geometry in the humming of the strings, there is music in the spacing of the spheres. arnaud.dessein@ircam.fr April 7th 2011 Journée Interdisciplinaire Mathématiques Musique 4/21

5 Where do we come from? Introduction A bit of history about science and music Motivations towards information geometry Pythagoras ( BC): relation between string length and produced sound, Pythagorean tuning. There is geometry in the humming of the strings, there is music in the spacing of the spheres. Helmholtz ( ): Helmholtz resonator, harmonics and frequency spectrum of sounds. arnaud.dessein@ircam.fr April 7th 2011 Journée Interdisciplinaire Mathématiques Musique 4/21

6 Where do we come from? Introduction A bit of history about science and music Motivations towards information geometry Pythagoras ( BC): relation between string length and produced sound, Pythagorean tuning. There is geometry in the humming of the strings, there is music in the spacing of the spheres. Helmholtz ( ): Helmholtz resonator, harmonics and frequency spectrum of sounds. But also indirectly Fourier, Shannon, etc. arnaud.dessein@ircam.fr April 7th 2011 Journée Interdisciplinaire Mathématiques Musique 4/21

7 What do we need? Introduction A bit of history about science and music Motivations towards information geometry Figure: Levels of representation of audio, waveform and spectrogram representations. arnaud.dessein@ircam.fr April 7th 2011 Journée Interdisciplinaire Mathématiques Musique 5/21

8 What do we need? Introduction A bit of history about science and music Motivations towards information geometry Figure: Levels of representation of audio, waveform and spectrogram representations. Develop a comprehensive framework that allows to quantify, process and represent the information contained in audio signals. arnaud.dessein@ircam.fr April 7th 2011 Journée Interdisciplinaire Mathématiques Musique 5/21

9 What do we need? Introduction A bit of history about science and music Motivations towards information geometry Figure: Levels of representation of audio, waveform and spectrogram representations. Develop a comprehensive framework that allows to quantify, process and represent the information contained in audio signals. Fill in the gap between signal and symbolic representations. arnaud.dessein@ircam.fr April 7th 2011 Journée Interdisciplinaire Mathématiques Musique 5/21

10 Outline Introduction Background Exponential families 1 Introduction 2 Background Exponential families arnaud.dessein@ircam.fr April 7th 2011 Journée Interdisciplinaire Mathématiques Musique 6/21

11 What is information geometry? Background Exponential families Statistical differentiable manifold. Under certain assumptions, a parametric statistical model S = {p ξ : ξ Ξ} of probability distributions defined on X forms a differentiable manifold. arnaud.dessein@ircam.fr April 7th 2011 Journée Interdisciplinaire Mathématiques Musique 7/21

12 What is information geometry? Background Exponential families Statistical differentiable manifold. Under certain assumptions, a parametric statistical model S = {p ξ : ξ Ξ} of probability distributions defined on X forms a differentiable manifold. { } 1 Example: p ξ (x) = exp (x µ)2 for all x X = R, with 2πσ 2 2σ 2 ξ = [µ, σ 2 ] Ξ = R R ++. arnaud.dessein@ircam.fr April 7th 2011 Journée Interdisciplinaire Mathématiques Musique 7/21

13 What is information geometry? Background Exponential families Statistical differentiable manifold. Under certain assumptions, a parametric statistical model S = {p ξ : ξ Ξ} of probability distributions defined on X forms a differentiable manifold. { } 1 Example: p ξ (x) = exp (x µ)2 for all x X = R, with 2πσ 2 2σ 2 ξ = [µ, σ 2 ] Ξ = R R ++. Fisher information metric [Rao, 1945, Chentsov, 1982]. Under certain assumptions, the Fisher information matrix defines the unique Riemannian metric g on S: g ij(ξ) = i log p ξ (x) j log p ξ (x) p ξ (x) dx. x X arnaud.dessein@ircam.fr April 7th 2011 Journée Interdisciplinaire Mathématiques Musique 7/21

14 What is information geometry? Background Exponential families Statistical differentiable manifold. Under certain assumptions, a parametric statistical model S = {p ξ : ξ Ξ} of probability distributions defined on X forms a differentiable manifold. { } 1 Example: p ξ (x) = exp (x µ)2 for all x X = R, with 2πσ 2 2σ 2 ξ = [µ, σ 2 ] Ξ = R R ++. Fisher information metric [Rao, 1945, Chentsov, 1982]. Under certain assumptions, the Fisher information matrix defines the unique Riemannian metric g on S: g ij(ξ) = i log p ξ (x) j log p ξ (x) p ξ (x) dx. x X Dual affine connections [Chentsov, 1982, Amari & Nagaoka, 2000]. Under certain assumptions, there is a unique family of dual affine connections { (α), ( α) } α R on (S, g) called α-connections. arnaud.dessein@ircam.fr April 7th 2011 Journée Interdisciplinaire Mathématiques Musique 7/21

15 Background Exponential families How to use information geometry from a computational viewpoint? Exponential family. p θ (x) = exp ( θ T T (x) F (θ) + C(x) ) for all x X. θ: natural parameters, vector belonging to a convex open set Θ. F : log-normalizer, real-valued, strictly convex smooth function on Θ. C: carrier measure, real-valued function on X. T : sufficient statistic, vector-valued function on X. arnaud.dessein@ircam.fr April 7th 2011 Journée Interdisciplinaire Mathématiques Musique 8/21

16 Background Exponential families How to use information geometry from a computational viewpoint? Exponential family. A taxonomy of probability measures p θ (x) = exp ( θ T T (x) F (θ) + C(x) ) for all x X. Probability measure Parametric Non-parametric Exponential families Non-exponential families Univariate Multivariate Uniform Cauchy Lévy skew α-stable uniparameter Bi-parameter multi-parameter Binomial Beta β Gamma Γ Multinomial Dirichlet Weibull Bernoulli Poisson Exponential Rayleigh Gaussian Figure: A taxonomy of exponential families. arnaud.dessein@ircam.fr April 7th 2011 Journée Interdisciplinaire Mathématiques Musique 8/21

17 Background Exponential families How to use information geometry from a computational viewpoint? Exponential family. p θ (x) = exp ( θ T T (x) F (θ) + C(x) ) for all x X. We consider a statistical manifold S = {p θ : θ Θ} equipped with g and the dual exponential and mixture connections (1) and ( 1). arnaud.dessein@ircam.fr April 7th 2011 Journée Interdisciplinaire Mathématiques Musique 8/21

18 Background Exponential families How to use information geometry from a computational viewpoint? Exponential family. p θ (x) = exp ( θ T T (x) F (θ) + C(x) ) for all x X. We consider a statistical manifold S = {p θ : θ Θ} equipped with g and the dual exponential and mixture connections (1) and ( 1). (S, g, (1), ( 1) ) possesses two dual affine coordinate systems, natural parameters θ and expectation parameters η = F (θ). arnaud.dessein@ircam.fr April 7th 2011 Journée Interdisciplinaire Mathématiques Musique 8/21

19 Background Exponential families How to use information geometry from a computational viewpoint? Exponential family. p θ (x) = exp ( θ T T (x) F (θ) + C(x) ) for all x X. We consider a statistical manifold S = {p θ : θ Θ} equipped with g and the dual exponential and mixture connections (1) and ( 1). (S, g, (1), ( 1) ) possesses two dual affine coordinate systems, natural parameters θ and expectation parameters η = F (θ). Dually flat geometry, Hessian structure (g = 2 F ), generated by the potential F together with its conjugate potential F defined by the Legendre-Fenchel transform: F (η) = sup θ Θ θ T η F (θ), which verifies F = ( F ) 1 so that θ = F (η). arnaud.dessein@ircam.fr April 7th 2011 Journée Interdisciplinaire Mathématiques Musique 8/21

20 Background Exponential families How to use information geometry from a computational viewpoint? Exponential family. p θ (x) = exp ( θ T T (x) F (θ) + C(x) ) for all x X. We consider a statistical manifold S = {p θ : θ Θ} equipped with g and the dual exponential and mixture connections (1) and ( 1). (S, g, (1), ( 1) ) possesses two dual affine coordinate systems, natural parameters θ and expectation parameters η = F (θ). Dually flat geometry, Hessian structure (g = 2 F ), generated by the potential F together with its conjugate potential F defined by the Legendre-Fenchel transform: F (η) = sup θ Θ θ T η F (θ), which verifies F = ( F ) 1 so that θ = F (η). Generalizes the self-dual Euclidean geometry, with notably two canonically associated Bregman divergences B F and B F instead of the self-dual Euclidean distance, but also dual geodesics, a generalized Pythagorean theorem and dual projections. arnaud.dessein@ircam.fr April 7th 2011 Journée Interdisciplinaire Mathématiques Musique 8/21

21 Background Exponential families How to use information geometry from a computational viewpoint? Exponential family. p θ (x) = exp ( θ T T (x) F (θ) + C(x) ) for all x X. Bregman divergence. B F (θ, θ ) = F (θ) F (θ ) (θ θ ) T F (θ ). arnaud.dessein@ircam.fr April 7th 2011 Journée Interdisciplinaire Mathématiques Musique 8/21

22 Background Exponential families How to use information geometry from a computational viewpoint? Exponential family. p θ (x) = exp ( θ T T (x) F (θ) + C(x) ) for all x X. Bregman divergence. B F (θ, θ ) = F (θ) F (θ ) (θ θ ) T F (θ ). Canonical divergences of dually flat spaces, bijection with exponential families [Amari & Nagaoka, 2000, Banerjee et al., 2005]: D KL (p ξ p ξ ) = B F (θ θ) = B F (η η ). arnaud.dessein@ircam.fr April 7th 2011 Journée Interdisciplinaire Mathématiques Musique 8/21

23 Background Exponential families How to use information geometry from a computational viewpoint? Exponential family. p θ (x) = exp ( θ T T (x) F (θ) + C(x) ) for all x X. Bregman divergence. B F (θ, θ ) = F (θ) F (θ ) (θ θ ) T F (θ ). Canonical divergences of dually flat spaces, bijection with exponential families [Amari & Nagaoka, 2000, Banerjee et al., 2005]: D KL (p ξ p ξ ) = B F (θ θ) = B F (η η ). No symmetry nor triangular inequality in general, but an information-theoretic interpretation. arnaud.dessein@ircam.fr April 7th 2011 Journée Interdisciplinaire Mathématiques Musique 8/21

24 Background Exponential families How to use information geometry from a computational viewpoint? Exponential family. p θ (x) = exp ( θ T T (x) F (θ) + C(x) ) for all x X. Bregman divergence. B F (θ, θ ) = F (θ) F (θ ) (θ θ ) T F (θ ). Canonical divergences of dually flat spaces, bijection with exponential families [Amari & Nagaoka, 2000, Banerjee et al., 2005]: D KL (p ξ p ξ ) = B F (θ θ) = B F (η η ). No symmetry nor triangular inequality in general, but an information-theoretic interpretation. Generic algorithms that handle many generalized distances [Banerjee et al., 2005, Cayton, 2008, Cayton, 2009, Nielsen & Nock, 2009, Nielsen et al., 2009, Garcia et al., 2009]: Centroid computation and hard clustering (k-means). Parameter estimation and soft clustering (expectation-maximization). Proximity queries in ball trees (nearest-neighbors and range search). arnaud.dessein@ircam.fr April 7th 2011 Journée Interdisciplinaire Mathématiques Musique 8/21

25 Outline Introduction General architecture Sound descriptors modeling Temporal modeling 1 Introduction 2 3 General architecture Sound descriptors modeling Temporal modeling 4 5 arnaud.dessein@ircam.fr April 7th 2011 Journée Interdisciplinaire Mathématiques Musique 9/21

26 General architecture Sound descriptors modeling Temporal modeling How to design an audio system based on information geometry? Audio stream decomposition (on-line) Scheme: 1 Represent the incoming audio stream with short-time sound descriptors d j. 2 Model these descriptors as probability distributions p θj from a given exponential family. 3 Use the framework of computational information geometry on these distributions. Auditory scene Short-time sound representation dj Sound descriptors modeling pθj Temporal modeling Figure: Schema of the general architecture of the system. arnaud.dessein@ircam.fr April 7th 2011 Journée Interdisciplinaire Mathématiques Musique 10/21

27 General architecture Sound descriptors modeling Temporal modeling How to design an audio system based on information geometry? Audio stream decomposition (on-line) Scheme: 1 Represent the incoming audio stream with short-time sound descriptors d j. 2 Model these descriptors as probability distributions p θj from a given exponential family. 3 Use the framework of computational information geometry on these distributions. In particular, it allows to define the notion of similarity in an information setup through divergences. Auditory scene Short-time sound representation dj Sound descriptors modeling pθj Temporal modeling Figure: Schema of the general architecture of the system. arnaud.dessein@ircam.fr April 7th 2011 Journée Interdisciplinaire Mathématiques Musique 10/21

28 General architecture Sound descriptors modeling Temporal modeling How to design an audio system based on information geometry? Audio stream decomposition (on-line) Scheme: 1 Represent the incoming audio stream with short-time sound descriptors d j. 2 Model these descriptors as probability distributions p θj from a given exponential family. 3 Use the framework of computational information geometry on these distributions. In particular, it allows to define the notion of similarity in an information setup through divergences. Important need for temporal modeling. Auditory scene Short-time sound representation dj Sound descriptors modeling pθj Temporal modeling Figure: Schema of the general architecture of the system. arnaud.dessein@ircam.fr April 7th 2011 Journée Interdisciplinaire Mathématiques Musique 10/21

29 General architecture Sound descriptors modeling Temporal modeling How to design an audio system based on information geometry? Audio stream decomposition (on-line) Scheme: 1 Represent the incoming audio stream with short-time sound descriptors d j. 2 Model these descriptors as probability distributions p θj from a given exponential family. 3 Use the framework of computational information geometry on these distributions. In particular, it allows to define the notion of similarity in an information setup through divergences. Important need for temporal modeling. Potential applications [Cont et al., 2011]: Audio content analysis. Segmentation of audio streams. Automatic structure discovery of audio signals. Sound processing and synthesis. Auditory scene Short-time sound representation dj Sound descriptors modeling pθj Temporal modeling Figure: Schema of the general architecture of the system. arnaud.dessein@ircam.fr April 7th 2011 Journée Interdisciplinaire Mathématiques Musique 10/21

30 How to model sounds? Introduction General architecture Sound descriptors modeling Temporal modeling Computation of a sound descriptor d j: Fourier or constant-q transforms for information on the spectral content. Mel-frequency cepstral coefficients for information on the timbre. Many other possibilities. Figure: Sound descriptors modeling. arnaud.dessein@ircam.fr April 7th 2011 Journée Interdisciplinaire Mathématiques Musique 11/21

31 How to model sounds? Introduction General architecture Sound descriptors modeling Temporal modeling Computation of a sound descriptor d j: Fourier or constant-q transforms for information on the spectral content. Mel-frequency cepstral coefficients for information on the timbre. Many other possibilities. Modeling with a probability distribution p θj from an exponential family: Categorical distributions. Many other possibilities. Figure: Sound descriptors modeling. arnaud.dessein@ircam.fr April 7th 2011 Journée Interdisciplinaire Mathématiques Musique 11/21

32 How to take time into account? General architecture Sound descriptors modeling Temporal modeling Model formation: from signal to symbol. Assumption of quasi-stationary audio chunks. Change detection adapted from CuSum [Basseville & Nikiforov, 1993]. Figure: Model formation at time t. April 7th 2011 Journée Interdisciplinaire Mathématiques Musique 12/21

33 How to take time into account? General architecture Sound descriptors modeling Temporal modeling Model formation: from signal to symbol. Assumption of quasi-stationary audio chunks. Change detection adapted from CuSum [Basseville & Nikiforov, 1993]. Figure: Model formation at time t. Factor oracle: from symbol to syntax (and from genetics to music!). Forward transitions: original sequence factors. Backward links: suffix relations, common context. Figure: Factor oracle of the word abbbaab. April 7th 2011 Journée Interdisciplinaire Mathématiques Musique 12/21

34 Outline Introduction Audio segmentation Music similarity analysis Musical structure discovery Query by similarity Audio recombination by concatenative synthesis Computer-assisted improvisation 1 Introduction Audio segmentation Music similarity analysis Musical structure discovery Query by similarity Audio recombination by concatenative synthesis Computer-assisted improvisation 5 arnaud.dessein@ircam.fr April 7th 2011 Journée Interdisciplinaire Mathématiques Musique 13/21

Audio segmentation Introduction Audio segmentation Music similarity analysis Musical structure discovery Query by similarity Audio recombination by concatenative synthesis Computer-assisted

35 Audio segmentation Introduction Audio segmentation Music similarity analysis Musical structure discovery Query by similarity Audio recombination by concatenative synthesis Computer-assisted improvisation Figure: Segmentation of the 1st Piano Sonate, 1st Movement, 1st Theme, Beethoven. April 7th 2011 Journée Interdisciplinaire Mathématiques Musique 14/21

Music similarity analysis Introduction Audio segmentation Music similarity analysis Musical structure discovery Query by similarity Audio recombination by concatenative synthesis

36 Music similarity analysis Introduction Audio segmentation Music similarity analysis Musical structure discovery Query by similarity Audio recombination by concatenative synthesis Computer-assisted improvisation Figure: Similarity analysis of the 1st Piano Sonate, 3rd Movement, Beethoven. April 7th 2011 Journée Interdisciplinaire Mathématiques Musique 15/21

Musical structure discovery Introduction Audio segmentation Music similarity analysis Musical structure discovery Query by similarity Audio recombination by concatenative synthesis

37 Musical structure discovery Introduction Audio segmentation Music similarity analysis Musical structure discovery Query by similarity Audio recombination by concatenative synthesis Computer-assisted improvisation Figure: Structure discovery of the 1st Piano Sonate, 3rd Movement, Beethoven. April 7th 2011 Journée Interdisciplinaire Mathématiques Musique 16/21

38 Query by similarity Introduction Audio segmentation Music similarity analysis Musical structure discovery Query by similarity Audio recombination by concatenative synthesis Computer-assisted improvisation Figure: Query by similarity of the 1st Theme over the entire 1st Piano Sonate, 1st Movement, Beethoven. April 7th 2011 Journée Interdisciplinaire Mathématiques Musique 17/21

Audio recombination by concatenative synthesis Audio segmentation Music similarity analysis Musical structure discovery Query by similarity Audio recombination by concatenative synthesis

39 Audio recombination by concatenative synthesis Audio segmentation Music similarity analysis Musical structure discovery Query by similarity Audio recombination by concatenative synthesis Computer-assisted improvisation Figure: Audio recombination of African drums by concatenative synthesis of congas. April 7th 2011 Journée Interdisciplinaire Mathématiques Musique 18/21

Computer-assisted improvisation Audio segmentation Music similarity analysis Musical structure discovery Query by similarity Audio recombination by concatenative synthesis

40 Computer-assisted improvisation Audio segmentation Music similarity analysis Musical structure discovery Query by similarity Audio recombination by concatenative synthesis Computer-assisted improvisation Figure: Computer-assisted improvisation, Fabrizio Cassol and Philippe Leclerc. April 7th 2011 Journée Interdisciplinaire Mathématiques Musique 19/21

41 Outline Introduction 1 Introduction arnaud.dessein@ircam.fr April 7th 2011 Journée Interdisciplinaire Mathématiques Musique 20/21

42 What we (don t) have Introduction Summary and perspectives. Representations. Descriptors modeling. Temporal modeling. Temporality of events. April 7th 2011 Journée Interdisciplinaire Mathématiques Musique 21/21

43 What we (don t) have Introduction Summary and perspectives. Representations. Descriptors modeling. Temporal modeling. Temporality of events. Many possibilities. Combinations of descriptors. Complex representations. April 7th 2011 Journée Interdisciplinaire Mathématiques Musique 21/21

44 What we (don t) have Introduction Summary and perspectives. Representations. Descriptors modeling. Temporal modeling. Temporality of events. Exponential families and Bregman divergences. Mixture models of a given exponential family. Other geometries, divergences, metrics. arnaud.dessein@ircam.fr April 7th 2011 Journée Interdisciplinaire Mathématiques Musique 21/21

45 What we (don t) have Introduction Summary and perspectives. Representations. Descriptors modeling. Temporal modeling. Temporality of events. On-line segmentation and factor oracle. On-line clustering and equivalence between symbols. Overlap between symbols and other temporal models. April 7th 2011 Journée Interdisciplinaire Mathématiques Musique 21/21

46 What we (don t) have Introduction Summary and perspectives. Representations. Descriptors modeling. Temporal modeling. Temporality of events. Assumption of quasi-stationarity. Non-stationarity modeling. Time series. April 7th 2011 Journée Interdisciplinaire Mathématiques Musique 21/21

47 What we (don t) have Summary and perspectives. Representations. Descriptors modeling. Temporal modeling. Temporality of events. Resources on IG: National research group: IRCAM, Ecole Polytechnique, Thales, etc. Brillouin seminar: IGAIA April 7th 2011 Journée Interdisciplinaire Mathématiques Musique 21/21

48 What we (don t) have Summary and perspectives. Representations. Descriptors modeling. Temporal modeling. Temporality of events. Resources on IG: National research group: IRCAM, Ecole Polytechnique, Thales, etc. Brillouin seminar: IGAIA Thank you very much for your attention! Questions? This work was supported by a doctoral fellowship from the UPMC (EDITE) and by a grant from the JST-CNRS ICT (Improving the VR Experience). arnaud.dessein@ircam.fr April 7th 2011 Journée Interdisciplinaire Mathématiques Musique 21/21

49 Bibliography I Amari, S.-i. & Nagaoka, H. (2000). Methods of information geometry, volume 191 of Translations of Mathematical Monographs. American Mathematical Society. Banerjee, A., Merugu, S., Dhillon, I. S., & Ghosh, J. (2005). Clustering with Bregman divergences. Journal of Machine Learning Research, 6, Basseville, M. & Nikiforov, V. (1993). Detection of abrupt changes: Theory and application. Englewood Cliffs, NJ, USA: Prentice-Hall, Inc. Cayton, L. (2008). Fast nearest neighbor retrieval for Bregman divergences. In Proceedings of the 25th International Conference on Machine Learning, volume 307 Helsinki, Finland. Cayton, L. (2009). Efficient Bregman range search. In Y. Bengio, D. Schuurmans, J. Lafferty, C. K. I. Williams, & A. Culotta (Eds.), Advances in Neural Information Processing Systems, volume 22 (pp ). Curran Associates, Inc. Chentsov, N. N. (1982). Statistical decision rules and optimal inference, volume 53 of Translations of Mathematical Monographs. American Mathematical Society. arnaud.dessein@ircam.fr April 7th 2011 Journée Interdisciplinaire Mathématiques Musique 22/21

50 Bibliography II Cont, A., Dubnov, S., & Assayag, G. (2011). On the information geometry of audio streams with applications to similarity computing. IEEE Transactions on Audio, Speech and Language Processing, 19. To appear. Garcia, V., Nielsen, F., & Nock, R. (2009). Levels of details for Gaussian mixture models. In Proceedings of the 9th Asian Conference on Computer Vision, ACCV 2009 (pp ). Xi an, China. Nielsen, F. & Nock, R. (2009). Sided and symmetrized Bregman centroids. IEEE Transactions on Information Theory, 55(6), Nielsen, F., Piro, P., & Barlaud, M. (2009). Tailored Bregman ball trees for effective nearest neighbors. In Proceedings of the 25th European Workshop on Computational Geometry (EuroCG) (pp ). Brussels, Belgium. Rao, C. R. (1945). Information and accuracy attainable in the estimation of statistical parameters. Bulletin of the Calcutta Mathematical Society, 37, April 7th 2011 Journée Interdisciplinaire Mathématiques Musique 23/21

An information-geometric approach to real-time audio segmentation

An information-geometric approach to real-time audio segmentation Arnaud Dessein, Arshia Cont To cite this version: Arnaud Dessein, Arshia Cont. An information-geometric approach to real-time audio segmentation.