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- 1. Sparse and Low Rank Representations in Music Signal Analysis Constantine Kotropoulos, Yannis Panagakis Artiﬁcial Intelligence & Information Analysis Laboratory Department of Informatics Aristotle University of Thessaloniki Thessaloniki 54124, GREECE 2nd Greek Signal Processing Jam, Thessaloniki, May 17th, 2012Sparse and Low Rank Representations in Music Signal Analysis 1/54
- 2. Outline 1 Introduction 2 Auditory spectro-temporal modulations 3 Suitable data representations for classiﬁcation 4 Joint sparse low-rank representations in the ideal case 5 Joint sparse low-rank representations in the presence of noise 6 Joint sparse low-rank subspace-based classiﬁcation 7 Music signal analysis 8 ConclusionsSparse and Low Rank Representations in Music Signal Analysis 2/54
- 3. 1 Introduction 2 Auditory spectro-temporal modulations 3 Suitable data representations for classiﬁcation 4 Joint sparse low-rank representations in the ideal case 5 Joint sparse low-rank representations in the presence of noise 6 Joint sparse low-rank subspace-based classiﬁcation 7 Music signal analysis 8 ConclusionsSparse and Low Rank Representations in Music Signal Analysis 3/54
- 4. Introduction Music genre classiﬁcation Genre: the most popular description of music content despite the lack of a commonly agreed deﬁnition. To classify music recordings into distinguishable genres using information extracted from the audio signal. Musical structure analysis To derive the musical form, i.e., the structural description of a music piece at the time scale of segments, such as intro, verse, chorus, bridge, from the audio signal. Music tagging Tags: text-based labels encoding semantic information related to music (i.e., instrumentation, genres, emotions, etc.). Manual tagging (expensive, time consuming, applicable to popular music); Automatic tagging (fast, applied to new and unpopular music).Sparse and Low Rank Representations in Music Signal Analysis 4/54
- 5. Introduction Music genre classiﬁcation Genre: the most popular description of music content despite the lack of a commonly agreed deﬁnition. To classify music recordings into distinguishable genres using information extracted from the audio signal. Musical structure analysis To derive the musical form, i.e., the structural description of a music piece at the time scale of segments, such as intro, verse, chorus, bridge, from the audio signal. Music tagging Tags: text-based labels encoding semantic information related to music (i.e., instrumentation, genres, emotions, etc.). Manual tagging (expensive, time consuming, applicable to popular music); Automatic tagging (fast, applied to new and unpopular music).Sparse and Low Rank Representations in Music Signal Analysis 4/54
- 6. Introduction Music genre classiﬁcation Genre: the most popular description of music content despite the lack of a commonly agreed deﬁnition. To classify music recordings into distinguishable genres using information extracted from the audio signal. Musical structure analysis To derive the musical form, i.e., the structural description of a music piece at the time scale of segments, such as intro, verse, chorus, bridge, from the audio signal. Music tagging Tags: text-based labels encoding semantic information related to music (i.e., instrumentation, genres, emotions, etc.). Manual tagging (expensive, time consuming, applicable to popular music); Automatic tagging (fast, applied to new and unpopular music).Sparse and Low Rank Representations in Music Signal Analysis 4/54
- 7. Introduction Motivation The appealing properties of slow temporal and spectro-temporal modulations from the human perceptual point of viewa ; The strong theoretical foundations of sparse representationsbc and low-rank representationsd . a K. Wang and S. A. Shamma, “Spectral shape analysis in the central auditory system,” IEEE Trans. Speech and Audio Processing, vol. 3, no. 5, pp. 382–396, 1995. b ` E. J. Candes, J. Romberg, and T. Tao,“Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Information Theory, vol. 52, no. 2, pp. 489–509, February 2006. c D. L. Donoho, “Compressed sensing,” IEEE Trans. Information Theory, vol. 52, no. 4, pp. 1289–1306, April 2006. d G. Liu, Z. Lin, S. Yan, J. Sun, and Y. Ma, “Robust recovery of subspace structures by low-rank representation, IEEE Trans. Pattern Analysis and Machine Intelligence, 2011, arXiv:1010.2955v4 (preprint).Sparse and Low Rank Representations in Music Signal Analysis 5/54
- 8. Introduction Motivation The appealing properties of slow temporal and spectro-temporal modulations from the human perceptual point of viewa ; The strong theoretical foundations of sparse representationsbc and low-rank representationsd . a K. Wang and S. A. Shamma, “Spectral shape analysis in the central auditory system,” IEEE Trans. Speech and Audio Processing, vol. 3, no. 5, pp. 382–396, 1995. b ` E. J. Candes, J. Romberg, and T. Tao,“Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Information Theory, vol. 52, no. 2, pp. 489–509, February 2006. c D. L. Donoho, “Compressed sensing,” IEEE Trans. Information Theory, vol. 52, no. 4, pp. 1289–1306, April 2006. d G. Liu, Z. Lin, S. Yan, J. Sun, and Y. Ma, “Robust recovery of subspace structures by low-rank representation, IEEE Trans. Pattern Analysis and Machine Intelligence, 2011, arXiv:1010.2955v4 (preprint).Sparse and Low Rank Representations in Music Signal Analysis 5/54
- 9. Notations Span Let span(X) denote the linear space spanned by the columns of X. Then, Y ∈ span(X) denotes that all column vectors of Y belong to span(X).Sparse and Low Rank Representations in Music Signal Analysis 6/54
- 10. Notations Vector norms x 0 is 0 quasi-norm counting the number of nonzero entries in x. If |.| denotes the absolute value operator, x 1 = i |xi | and x 2 norm of x, respectively. 2 = i xi are the 1 and the 2Sparse and Low Rank Representations in Music Signal Analysis 6/54
- 11. Notations Matrix norms 1 q q p mixed p,q matrix norm: X p,q = i j |xij |p . For p = q = 0, the matrix 0 quasi-norm, X 0 , returns the number of nonzero entries in X. For p = q = 1, the matrix 1 norm is obtained X 1 = i j |xij |. Frobenius norm: X = 2 xij . F i j norm of X: X 2 2/ 1 2,1 = j i xij . nuclear norm of X, X ∗ , is the sum of singular values of X.Sparse and Low Rank Representations in Music Signal Analysis 6/54
- 12. Notations Support A vector x is said to be q-sparse if the size of the support of x (i.e., the set of indices associated to non-zero vector elements) is no larger than q. The support of a collection of vectors X = [x1 , x2 , . . . , xN ] is deﬁned as the union over all the individual supports. A matrix X is called q joint sparse, if |supp(X)| ≤ q. That is, there are at most q rows in X that contain nonzero elements, because X 0,q = |supp(X)| for any q a . a M. Davies and Y. Eldar, “Rank awareness in joint sparse recovery,” arXiv:1004.4529v1, 2010.Sparse and Low Rank Representations in Music Signal Analysis 6/54
- 13. 1 Introduction 2 Auditory spectro-temporal modulations 3 Suitable data representations for classiﬁcation 4 Joint sparse low-rank representations in the ideal case 5 Joint sparse low-rank representations in the presence of noise 6 Joint sparse low-rank subspace-based classiﬁcation 7 Music signal analysis 8 ConclusionsSparse and Low Rank Representations in Music Signal Analysis 7/54
- 14. Auditory spectro-temporal modulations Computational auditory model It is inspired by psychoacoustical and neurophysiological investigations in the early and central stages of the human auditory system. Auditory Spectro-Temporal Modulations (Cortical Representation) Auditory Spectrogram Central auditory model Early auditory model Auditory Temporal ModulationsSparse and Low Rank Representations in Music Signal Analysis 8/54
- 15. Auditory spectro-temporal modulations Computational auditory model It is inspired by psychoacoustical and neurophysiological investigations in the early and central stages of the human auditory system. Auditory Spectro-Temporal Modulations (Cortical Representation) Auditory Spectrogram Central auditory model Early auditory model Auditory Temporal ModulationsSparse and Low Rank Representations in Music Signal Analysis 8/54
- 16. Auditory spectro-temporal modulations Early auditory system Auditory Spectrogram: time-frequency distribution of energy along a tonotopic (logarithmic frequency) axis. Auditory Spectrogram Early auditory modelSparse and Low Rank Representations in Music Signal Analysis 9/54
- 17. Auditory spectro-temporal modulations Central auditory system - Temporal modulations Auditory Spectrogram Auditory Temporal Modulations z) (H ω z) (H ωSparse and Low Rank Representations in Music Signal Analysis 10/54
- 18. Auditory spectro-temporal modulations Auditory temporal modulations across 10 music genres Blues Classical Country Disco Hiphop Jazz Metal Pop Reggae RockSparse and Low Rank Representations in Music Signal Analysis 11/54
- 19. Auditory spectro-temporal modulations Central auditory system - Spectro-temporal modulations Auditory Spectrogram Auditory Spectro-Temporal Modulations Ω(c/o) z) (H ωSparse and Low Rank Representations in Music Signal Analysis 12/54
- 20. Auditory spectro-temporal modulations Efﬁcient implementation through constant Q transform (CQT)Sparse and Low Rank Representations in Music Signal Analysis 13/54
- 21. Auditory spectro-temporal modulations Parameters and implementation (1) The audio signal is analyzed by employing 128 constant-Q ﬁlters covering 8 octaves from 44.9 Hz to 11 KHz (i.e., 16 ﬁlters per octave). The magnitude of the CQT is compressed by raising each element of the CQT matrix to the power of 0.1a . The 2D multiresolution wavelet analysis is implemented via a bank of 2D Gaussian ﬁlters with scales ∈ {0.25, 0.5, 1, 2, 4, 8} (Cycles/Octave) and rates ∈ {±2, ±4, ±8, ±16, ±32} (Hz). For each music recording, the extracted 4D cortical representation is time- averaged and the resulting rate-scale-frequency 3D cortical representation is thus obtained. a C. Schoerkhuber and A. Klapuri, “Constant-Q transform toolbox for music processing, ” in 7th Sound and Music Computing Conf., Barcelona, Spain, 2010.Sparse and Low Rank Representations in Music Signal Analysis 14/54
- 22. Auditory spectro-temporal modulations Parameters and implementation (1) The audio signal is analyzed by employing 128 constant-Q ﬁlters covering 8 octaves from 44.9 Hz to 11 KHz (i.e., 16 ﬁlters per octave). The magnitude of the CQT is compressed by raising each element of the CQT matrix to the power of 0.1a . The 2D multiresolution wavelet analysis is implemented via a bank of 2D Gaussian ﬁlters with scales ∈ {0.25, 0.5, 1, 2, 4, 8} (Cycles/Octave) and rates ∈ {±2, ±4, ±8, ±16, ±32} (Hz). For each music recording, the extracted 4D cortical representation is time- averaged and the resulting rate-scale-frequency 3D cortical representation is thus obtained. a C. Schoerkhuber and A. Klapuri, “Constant-Q transform toolbox for music processing, ” in 7th Sound and Music Computing Conf., Barcelona, Spain, 2010.Sparse and Low Rank Representations in Music Signal Analysis 14/54
- 23. Auditory spectro-temporal modulations Parameters and implementation (1) The audio signal is analyzed by employing 128 constant-Q ﬁlters covering 8 octaves from 44.9 Hz to 11 KHz (i.e., 16 ﬁlters per octave). The magnitude of the CQT is compressed by raising each element of the CQT matrix to the power of 0.1a . The 2D multiresolution wavelet analysis is implemented via a bank of 2D Gaussian ﬁlters with scales ∈ {0.25, 0.5, 1, 2, 4, 8} (Cycles/Octave) and rates ∈ {±2, ±4, ±8, ±16, ±32} (Hz). For each music recording, the extracted 4D cortical representation is time- averaged and the resulting rate-scale-frequency 3D cortical representation is thus obtained. a C. Schoerkhuber and A. Klapuri, “Constant-Q transform toolbox for music processing, ” in 7th Sound and Music Computing Conf., Barcelona, Spain, 2010.Sparse and Low Rank Representations in Music Signal Analysis 14/54
- 24. Auditory spectro-temporal modulations Parameters and implementation (2) To sum up, each music recording is represented by a vector x ∈ R7680 by stacking the elements of the 3D cortical + representation into a vector. An ensemble of music recordings is represented by the data matrix X ∈ R7680×S , where S is the number of the available + recordings. Each row of X is normalized to the range [0, 1] by subtracting from each entry the row minimum and then by dividing it with the difference between the row maximum and the row minimum.Sparse and Low Rank Representations in Music Signal Analysis 15/54
- 25. Auditory spectro-temporal modulations Parameters and implementation (2) To sum up, each music recording is represented by a vector x ∈ R7680 by stacking the elements of the 3D cortical + representation into a vector. An ensemble of music recordings is represented by the data matrix X ∈ R7680×S , where S is the number of the available + recordings. Each row of X is normalized to the range [0, 1] by subtracting from each entry the row minimum and then by dividing it with the difference between the row maximum and the row minimum.Sparse and Low Rank Representations in Music Signal Analysis 15/54
- 26. Auditory spectro-temporal modulations Parameters and implementation (2) To sum up, each music recording is represented by a vector x ∈ R7680 by stacking the elements of the 3D cortical + representation into a vector. An ensemble of music recordings is represented by the data matrix X ∈ R7680×S , where S is the number of the available + recordings. Each row of X is normalized to the range [0, 1] by subtracting from each entry the row minimum and then by dividing it with the difference between the row maximum and the row minimum.Sparse and Low Rank Representations in Music Signal Analysis 15/54
- 27. 1 Introduction 2 Auditory spectro-temporal modulations 3 Suitable data representations for classiﬁcation 4 Joint sparse low-rank representations in the ideal case 5 Joint sparse low-rank representations in the presence of noise 6 Joint sparse low-rank subspace-based classiﬁcation 7 Music signal analysis 8 ConclusionsSparse and Low Rank Representations in Music Signal Analysis 16/54
- 28. Learning Problem Statement Let X ∈ Rd×S be the data matrix that contains S vector samples in its columns of size d. That is, xs ∈ Rd , s = 1, 2, . . . , S. Without loss of generality, the data matrix can be partitioned as X = [A | Y], where A = [A1 |A2 | . . . |AK ] ∈ Rd×N represents a set of N training samples that belong to K classes Y = [Y1 |Y2 | . . . |YK ] ∈ Rd×M contains M = S − N test vector samples in its columns. If certain assumptions hold, learn a block diagonal matrix Z = diag[Z1 , Z2 , . . . , ZK ] ∈ RN×M such that Y = AZ.Sparse and Low Rank Representations in Music Signal Analysis 17/54
- 29. Learning Problem Statement Let X ∈ Rd×S be the data matrix that contains S vector samples in its columns of size d. That is, xs ∈ Rd , s = 1, 2, . . . , S. Without loss of generality, the data matrix can be partitioned as X = [A | Y], where A = [A1 |A2 | . . . |AK ] ∈ Rd×N represents a set of N training samples that belong to K classes Y = [Y1 |Y2 | . . . |YK ] ∈ Rd×M contains M = S − N test vector samples in its columns. If certain assumptions hold, learn a block diagonal matrix Z = diag[Z1 , Z2 , . . . , ZK ] ∈ RN×M such that Y = AZ.Sparse and Low Rank Representations in Music Signal Analysis 17/54
- 30. Learning Problem Statement Let X ∈ Rd×S be the data matrix that contains S vector samples in its columns of size d. That is, xs ∈ Rd , s = 1, 2, . . . , S. Without loss of generality, the data matrix can be partitioned as X = [A | Y], where A = [A1 |A2 | . . . |AK ] ∈ Rd×N represents a set of N training samples that belong to K classes Y = [Y1 |Y2 | . . . |YK ] ∈ Rd×M contains M = S − N test vector samples in its columns. If certain assumptions hold, learn a block diagonal matrix Z = diag[Z1 , Z2 , . . . , ZK ] ∈ RN×M such that Y = AZ.Sparse and Low Rank Representations in Music Signal Analysis 17/54
- 31. Learning Problem Statement Let X ∈ Rd×S be the data matrix that contains S vector samples in its columns of size d. That is, xs ∈ Rd , s = 1, 2, . . . , S. Without loss of generality, the data matrix can be partitioned as X = [A | Y], where A = [A1 |A2 | . . . |AK ] ∈ Rd×N represents a set of N training samples that belong to K classes Y = [Y1 |Y2 | . . . |YK ] ∈ Rd×M contains M = S − N test vector samples in its columns. If certain assumptions hold, learn a block diagonal matrix Z = diag[Z1 , Z2 , . . . , ZK ] ∈ RN×M such that Y = AZ.Sparse and Low Rank Representations in Music Signal Analysis 17/54
- 32. Learning Problem Statement Let X ∈ Rd×S be the data matrix that contains S vector samples in its columns of size d. That is, xs ∈ Rd , s = 1, 2, . . . , S. Without loss of generality, the data matrix can be partitioned as X = [A | Y], where A = [A1 |A2 | . . . |AK ] ∈ Rd×N represents a set of N training samples that belong to K classes Y = [Y1 |Y2 | . . . |YK ] ∈ Rd×M contains M = S − N test vector samples in its columns. If certain assumptions hold, learn a block diagonal matrix Z = diag[Z1 , Z2 , . . . , ZK ] ∈ RN×M such that Y = AZ.Sparse and Low Rank Representations in Music Signal Analysis 17/54
- 33. Learning Problem Assumptions If 1 the data are exactly drawn from independent linear subspaces, i.e., span(Ak ) linearly spans the k th class data space, k = 1, 2, . . . , K , 2 Y ∈ span(A), 3 the data contain neither outliers nor noise, then each test vector sample that belongs to the k th class can be represented as a linear combination of the training samples in Ak .Sparse and Low Rank Representations in Music Signal Analysis 18/54
- 34. Learning Problem Assumptions If 1 the data are exactly drawn from independent linear subspaces, i.e., span(Ak ) linearly spans the k th class data space, k = 1, 2, . . . , K , 2 Y ∈ span(A), 3 the data contain neither outliers nor noise, then each test vector sample that belongs to the k th class can be represented as a linear combination of the training samples in Ak .Sparse and Low Rank Representations in Music Signal Analysis 18/54
- 35. Learning Problem Assumptions If 1 the data are exactly drawn from independent linear subspaces, i.e., span(Ak ) linearly spans the k th class data space, k = 1, 2, . . . , K , 2 Y ∈ span(A), 3 the data contain neither outliers nor noise, then each test vector sample that belongs to the k th class can be represented as a linear combination of the training samples in Ak .Sparse and Low Rank Representations in Music Signal Analysis 18/54
- 36. Learning Problem Assumptions If 1 the data are exactly drawn from independent linear subspaces, i.e., span(Ak ) linearly spans the k th class data space, k = 1, 2, . . . , K , 2 Y ∈ span(A), 3 the data contain neither outliers nor noise, then each test vector sample that belongs to the k th class can be represented as a linear combination of the training samples in Ak .Sparse and Low Rank Representations in Music Signal Analysis 18/54
- 37. Learning Problem Assumptions If 1 the data are exactly drawn from independent linear subspaces, i.e., span(Ak ) linearly spans the k th class data space, k = 1, 2, . . . , K , 2 Y ∈ span(A), 3 the data contain neither outliers nor noise, then each test vector sample that belongs to the k th class can be represented as a linear combination of the training samples in Ak .Sparse and Low Rank Representations in Music Signal Analysis 18/54
- 38. Solutions Sparsest Representation (SR) Z ∈ RN×M is the sparsest representation of the test data Y ∈ Rd×M with respect to the training data A ∈ Rd×N obtained by solving the optimization problema : SR: argmin zi 0 subject to yi = A zi , (1) zi a E. Elhamifar and R. Vidal, “Sparse subspace clustering,” in IEEE Int. Conf. Computer Vision and Pattern Recognition, Miami, FL, USA, 2009, pp. 2790-2797.Sparse and Low Rank Representations in Music Signal Analysis 19/54
- 39. Solutions Lowest-rank representation (LRR) or Z ∈ RN×M is the lowest-rank representation of the test data Y ∈ Rd×M with respect to the training data A ∈ Rd×N obtained by solving the optimization problema : LRR: argmin rank(Z) subject to Y = A Z. (2) Z a G. Liu, Z. Lin, S. Yan, J. Sun, and Y. Ma (2011)Sparse and Low Rank Representations in Music Signal Analysis 19/54
- 40. Solutions Convex relaxations The convex envelope of the 0 norm is the 1 norma , while the convex envelope of the rank function is the nuclear normb . Convex relaxations can be obtained by replacing the 0 norm and the rank function by their convex envelopes: SR: argmin zi 1 subject to yi = A zi , (3) zi LRR: argmin Z ∗ subject to Y = A Z. (4) Z a D. Donoho, “For most large underdetermined systems of equations, the minimal l1-norm near-solution approximates the sparsest near-solution,” Communications on Pure and Applied Mathematics, vol. 59, no. 7, pp. 907-934, 2006. b M. Fazel, Matrix Rank Minimization with Applications, Ph.D. thesis, Dept. Electrical Engineering, Stanford University, CA, USA, 2002.Sparse and Low Rank Representations in Music Signal Analysis 19/54
- 41. Solutions SR pros and cons The SR matrix Z ∈ RN×M is sparse block-diagonal and has good discriminative properties, as has been demonstrated for the SR based classiﬁersa . However, the SR 1 can not model generic subspace structures. Indeed the SR models accurately subregions on subspaces, the so-called bouquets, rather than generic subspacesb . 2 does not capture the global structure of the data, since it is computed for each data sample individually. Indeed, although the sparsity offers an efﬁcient representation, it damages the high within-class homogeneity, which is desirable for classiﬁcation, especially in the presence of noise. a J. Wright, A. Yang, A. Ganesh, S. Sastry, and Y. Ma, “Robust face recognition via sparse representation,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 31, no. 2, pp. 210-227, 2009. b J. Wright and Y. Ma, “Dense error correction via l1-minimization,” IEEE Trans. Information Theory, vol. 56, no. 7, pp. 3540-3560, 2010.Sparse and Low Rank Representations in Music Signal Analysis 20/54
- 42. Solutions SR pros and cons The SR matrix Z ∈ RN×M is sparse block-diagonal and has good discriminative properties, as has been demonstrated for the SR based classiﬁersa . However, the SR 1 can not model generic subspace structures. Indeed the SR models accurately subregions on subspaces, the so-called bouquets, rather than generic subspacesb . 2 does not capture the global structure of the data, since it is computed for each data sample individually. Indeed, although the sparsity offers an efﬁcient representation, it damages the high within-class homogeneity, which is desirable for classiﬁcation, especially in the presence of noise. a J. Wright, A. Yang, A. Ganesh, S. Sastry, and Y. Ma, “Robust face recognition via sparse representation,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 31, no. 2, pp. 210-227, 2009. b J. Wright and Y. Ma, “Dense error correction via l1-minimization,” IEEE Trans. Information Theory, vol. 56, no. 7, pp. 3540-3560, 2010.Sparse and Low Rank Representations in Music Signal Analysis 20/54
- 43. Solutions SR pros and cons The SR matrix Z ∈ RN×M is sparse block-diagonal and has good discriminative properties, as has been demonstrated for the SR based classiﬁersa . However, the SR 1 can not model generic subspace structures. Indeed the SR models accurately subregions on subspaces, the so-called bouquets, rather than generic subspacesb . 2 does not capture the global structure of the data, since it is computed for each data sample individually. Indeed, although the sparsity offers an efﬁcient representation, it damages the high within-class homogeneity, which is desirable for classiﬁcation, especially in the presence of noise. a J. Wright, A. Yang, A. Ganesh, S. Sastry, and Y. Ma, “Robust face recognition via sparse representation,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 31, no. 2, pp. 210-227, 2009. b J. Wright and Y. Ma, “Dense error correction via l1-minimization,” IEEE Trans. Information Theory, vol. 56, no. 7, pp. 3540-3560, 2010.Sparse and Low Rank Representations in Music Signal Analysis 20/54
- 44. Solutions SR pros and cons The SR matrix Z ∈ RN×M is sparse block-diagonal and has good discriminative properties, as has been demonstrated for the SR based classiﬁersa . However, the SR 1 can not model generic subspace structures. Indeed the SR models accurately subregions on subspaces, the so-called bouquets, rather than generic subspacesb . 2 does not capture the global structure of the data, since it is computed for each data sample individually. Indeed, although the sparsity offers an efﬁcient representation, it damages the high within-class homogeneity, which is desirable for classiﬁcation, especially in the presence of noise. a J. Wright, A. Yang, A. Ganesh, S. Sastry, and Y. Ma, “Robust face recognition via sparse representation,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 31, no. 2, pp. 210-227, 2009. b J. Wright and Y. Ma, “Dense error correction via l1-minimization,” IEEE Trans. Information Theory, vol. 56, no. 7, pp. 3540-3560, 2010.Sparse and Low Rank Representations in Music Signal Analysis 20/54
- 45. Solutions LRR pros and cons The LRR matrix Z ∈ RN×M 1 models data steming from generic subspace structures 2 preserves accurately the global data structure. 3 For clean data, the LRR also exhibits dense within-class homogeneity and zero between-class afﬁnities, making it an appealing representation for classiﬁcation purposes, e.g., in music mood classiﬁcationa . 4 For data contaminated with noise and outliers, the low-rank constraint seems to enforce noise correctionb . But LRR looses sparsity within the classes. a Y. Panagakis and C. Kotropoulos, “Automatic music mood classiﬁcation via low-rank representation,” in Proc. 19th European Signal Processing Conf., Barcelona, Spain, 2011, pp. 689–693. b E. J. Candes, X. Li, Y. Ma, and J. Wright, “Robust principal component analysis?,” Journal of ACM, vol. 58, no. 3, pp. 1-37, 2011.Sparse and Low Rank Representations in Music Signal Analysis 21/54
- 46. Solutions LRR pros and cons The LRR matrix Z ∈ RN×M 1 models data steming from generic subspace structures 2 preserves accurately the global data structure. 3 For clean data, the LRR also exhibits dense within-class homogeneity and zero between-class afﬁnities, making it an appealing representation for classiﬁcation purposes, e.g., in music mood classiﬁcationa . 4 For data contaminated with noise and outliers, the low-rank constraint seems to enforce noise correctionb . But LRR looses sparsity within the classes. a Y. Panagakis and C. Kotropoulos, “Automatic music mood classiﬁcation via low-rank representation,” in Proc. 19th European Signal Processing Conf., Barcelona, Spain, 2011, pp. 689–693. b E. J. Candes, X. Li, Y. Ma, and J. Wright, “Robust principal component analysis?,” Journal of ACM, vol. 58, no. 3, pp. 1-37, 2011.Sparse and Low Rank Representations in Music Signal Analysis 21/54
- 47. Solutions LRR pros and cons The LRR matrix Z ∈ RN×M 1 models data steming from generic subspace structures 2 preserves accurately the global data structure. 3 For clean data, the LRR also exhibits dense within-class homogeneity and zero between-class afﬁnities, making it an appealing representation for classiﬁcation purposes, e.g., in music mood classiﬁcationa . 4 For data contaminated with noise and outliers, the low-rank constraint seems to enforce noise correctionb . But LRR looses sparsity within the classes. a Y. Panagakis and C. Kotropoulos, “Automatic music mood classiﬁcation via low-rank representation,” in Proc. 19th European Signal Processing Conf., Barcelona, Spain, 2011, pp. 689–693. b E. J. Candes, X. Li, Y. Ma, and J. Wright, “Robust principal component analysis?,” Journal of ACM, vol. 58, no. 3, pp. 1-37, 2011.Sparse and Low Rank Representations in Music Signal Analysis 21/54
- 48. Solutions LRR pros and cons The LRR matrix Z ∈ RN×M 1 models data steming from generic subspace structures 2 preserves accurately the global data structure. 3 For clean data, the LRR also exhibits dense within-class homogeneity and zero between-class afﬁnities, making it an appealing representation for classiﬁcation purposes, e.g., in music mood classiﬁcationa . 4 For data contaminated with noise and outliers, the low-rank constraint seems to enforce noise correctionb . But LRR looses sparsity within the classes. a Y. Panagakis and C. Kotropoulos, “Automatic music mood classiﬁcation via low-rank representation,” in Proc. 19th European Signal Processing Conf., Barcelona, Spain, 2011, pp. 689–693. b E. J. Candes, X. Li, Y. Ma, and J. Wright, “Robust principal component analysis?,” Journal of ACM, vol. 58, no. 3, pp. 1-37, 2011.Sparse and Low Rank Representations in Music Signal Analysis 21/54
- 49. Solutions LRR pros and cons The LRR matrix Z ∈ RN×M 1 models data steming from generic subspace structures 2 preserves accurately the global data structure. 3 For clean data, the LRR also exhibits dense within-class homogeneity and zero between-class afﬁnities, making it an appealing representation for classiﬁcation purposes, e.g., in music mood classiﬁcationa . 4 For data contaminated with noise and outliers, the low-rank constraint seems to enforce noise correctionb . But LRR looses sparsity within the classes. a Y. Panagakis and C. Kotropoulos, “Automatic music mood classiﬁcation via low-rank representation,” in Proc. 19th European Signal Processing Conf., Barcelona, Spain, 2011, pp. 689–693. b E. J. Candes, X. Li, Y. Ma, and J. Wright, “Robust principal component analysis?,” Journal of ACM, vol. 58, no. 3, pp. 1-37, 2011.Sparse and Low Rank Representations in Music Signal Analysis 21/54
- 50. Solutions LRR pros and cons The LRR matrix Z ∈ RN×M 1 models data steming from generic subspace structures 2 preserves accurately the global data structure. 3 For clean data, the LRR also exhibits dense within-class homogeneity and zero between-class afﬁnities, making it an appealing representation for classiﬁcation purposes, e.g., in music mood classiﬁcationa . 4 For data contaminated with noise and outliers, the low-rank constraint seems to enforce noise correctionb . But LRR looses sparsity within the classes. a Y. Panagakis and C. Kotropoulos, “Automatic music mood classiﬁcation via low-rank representation,” in Proc. 19th European Signal Processing Conf., Barcelona, Spain, 2011, pp. 689–693. b E. J. Candes, X. Li, Y. Ma, and J. Wright, “Robust principal component analysis?,” Journal of ACM, vol. 58, no. 3, pp. 1-37, 2011.Sparse and Low Rank Representations in Music Signal Analysis 21/54
- 51. 1 Introduction 2 Auditory spectro-temporal modulations 3 Suitable data representations for classiﬁcation 4 Joint sparse low-rank representations in the ideal case 5 Joint sparse low-rank representations in the presence of noise 6 Joint sparse low-rank subspace-based classiﬁcation 7 Music signal analysis 8 ConclusionsSparse and Low Rank Representations in Music Signal Analysis 22/54
- 52. Joint sparse low-rank representations (JSLRR) Motivation Intuitively, a representation matrix that is able to reveal the most characteristic subregions of the subspaces must be simultaneously row sparse and low-rank. The row sparsity ensures that only a small fraction of the training samples is involved in the representation. The low-rank constraint ensures that the representation vectors (i.e., the columns of the representation matrix) are correlated in the sense that the data lying onto a single subspace are represented as a linear combination of the same few training samples.Sparse and Low Rank Representations in Music Signal Analysis 23/54
- 53. Joint sparse low-rank representations (JSLRR) Motivation Intuitively, a representation matrix that is able to reveal the most characteristic subregions of the subspaces must be simultaneously row sparse and low-rank. The row sparsity ensures that only a small fraction of the training samples is involved in the representation. The low-rank constraint ensures that the representation vectors (i.e., the columns of the representation matrix) are correlated in the sense that the data lying onto a single subspace are represented as a linear combination of the same few training samples.Sparse and Low Rank Representations in Music Signal Analysis 23/54
- 54. Joint sparse low-rank representations (JSLRR) Motivation Intuitively, a representation matrix that is able to reveal the most characteristic subregions of the subspaces must be simultaneously row sparse and low-rank. The row sparsity ensures that only a small fraction of the training samples is involved in the representation. The low-rank constraint ensures that the representation vectors (i.e., the columns of the representation matrix) are correlated in the sense that the data lying onto a single subspace are represented as a linear combination of the same few training samples.Sparse and Low Rank Representations in Music Signal Analysis 23/54
- 55. JSLRR Problem statement and solution The JSLRR of Y ∈ Rd×M with respect to A ∈ Rd×N is the matrix Z ∈ RN×M with rank r min(q, M), where q N is the size of the support of Z. It can be found by minimizing the rank function regularized by the 0,q quasi-norm. The 0,q regularization term ensures that the low-rank matrix is also row sparse, since Z 0,q = |supp(Z)| for any q. A convex relaxation of the just mentioned problem is solved: JSLRR: argmin Z ∗ + θ1 Z 1 subject to Y = A Z, (5) Z where the term Z 1 promotes sparsity to the LRR matrix and θ1 > 0 balances the two norms in (5).Sparse and Low Rank Representations in Music Signal Analysis 24/54
- 56. JSLRR Problem statement and solution The JSLRR of Y ∈ Rd×M with respect to A ∈ Rd×N is the matrix Z ∈ RN×M with rank r min(q, M), where q N is the size of the support of Z. It can be found by minimizing the rank function regularized by the 0,q quasi-norm. The 0,q regularization term ensures that the low-rank matrix is also row sparse, since Z 0,q = |supp(Z)| for any q. A convex relaxation of the just mentioned problem is solved: JSLRR: argmin Z ∗ + θ1 Z 1 subject to Y = A Z, (5) Z where the term Z 1 promotes sparsity to the LRR matrix and θ1 > 0 balances the two norms in (5).Sparse and Low Rank Representations in Music Signal Analysis 24/54
- 57. JSLRR Problem statement and solution The JSLRR of Y ∈ Rd×M with respect to A ∈ Rd×N is the matrix Z ∈ RN×M with rank r min(q, M), where q N is the size of the support of Z. It can be found by minimizing the rank function regularized by the 0,q quasi-norm. The 0,q regularization term ensures that the low-rank matrix is also row sparse, since Z 0,q = |supp(Z)| for any q. A convex relaxation of the just mentioned problem is solved: JSLRR: argmin Z ∗ + θ1 Z 1 subject to Y = A Z, (5) Z where the term Z 1 promotes sparsity to the LRR matrix and θ1 > 0 balances the two norms in (5).Sparse and Low Rank Representations in Music Signal Analysis 24/54
- 58. JSLRR Problem statement and solution The JSLRR of Y ∈ Rd×M with respect to A ∈ Rd×N is the matrix Z ∈ RN×M with rank r min(q, M), where q N is the size of the support of Z. It can be found by minimizing the rank function regularized by the 0,q quasi-norm. The 0,q regularization term ensures that the low-rank matrix is also row sparse, since Z 0,q = |supp(Z)| for any q. A convex relaxation of the just mentioned problem is solved: JSLRR: argmin Z ∗ + θ1 Z 1 subject to Y = A Z, (5) Z where the term Z 1 promotes sparsity to the LRR matrix and θ1 > 0 balances the two norms in (5).Sparse and Low Rank Representations in Music Signal Analysis 24/54
- 59. JSLRR Any theoretical quarantee? The JSLRR has a block-diagonal structure, a property that makes it appealing for classiﬁcation. This fact is proved in Theorem 1, which is a consequence of Lemma 1.Sparse and Low Rank Representations in Music Signal Analysis 25/54
- 60. JSLRR Lemma 1 Let . θ = . ∗ + θ . 1 , with θ > 0. For any four matrices B, C, D, and F of compatible dimensions, B C B 0 ≥ = B θ + F θ. (6) D F θ 0 F θSparse and Low Rank Representations in Music Signal Analysis 25/54
- 61. JSLRR Theorem 1 Assume that the data are exactly drawn from independent linear subspaces. That is, span(Ak ) linearly spans the training vectors of the k th class, k = 1, 2, . . . , K , and Y ∈ span(A). Then, the minimizer of (5) is block-diagonal.Sparse and Low Rank Representations in Music Signal Analysis 25/54
- 62. Example 1 Ideal case 4 linear pairwise independent subspaces are constructed whose bases {Ui }4 are computed by Ui+1 = RUi , i = 1, 2, 3. i=1 U1 ∈ R600×110 is a column orthonormal random matrix and R ∈ R600×600 is a random rotation matrix. The data matrix X = [X1 , X2 , X3 , X4 ] ∈ R600×400 is obtained by picking 100 samples from each subspace. That is, Xi ∈ R600×100 , i = 1, 2, 3, 4. Next, the data matrix is partitioned into the training matrix A ∈ R600×360 and the test matrix Y ∈ R600×40 by employing a 10-fold cross validation.Sparse and Low Rank Representations in Music Signal Analysis 26/54
- 63. Example 1 Ideal case 4 linear pairwise independent subspaces are constructed whose bases {Ui }4 are computed by Ui+1 = RUi , i = 1, 2, 3. i=1 U1 ∈ R600×110 is a column orthonormal random matrix and R ∈ R600×600 is a random rotation matrix. The data matrix X = [X1 , X2 , X3 , X4 ] ∈ R600×400 is obtained by picking 100 samples from each subspace. That is, Xi ∈ R600×100 , i = 1, 2, 3, 4. Next, the data matrix is partitioned into the training matrix A ∈ R600×360 and the test matrix Y ∈ R600×40 by employing a 10-fold cross validation.Sparse and Low Rank Representations in Music Signal Analysis 26/54
- 64. Example 1 Ideal case 4 linear pairwise independent subspaces are constructed whose bases {Ui }4 are computed by Ui+1 = RUi , i = 1, 2, 3. i=1 U1 ∈ R600×110 is a column orthonormal random matrix and R ∈ R600×600 is a random rotation matrix. The data matrix X = [X1 , X2 , X3 , X4 ] ∈ R600×400 is obtained by picking 100 samples from each subspace. That is, Xi ∈ R600×100 , i = 1, 2, 3, 4. Next, the data matrix is partitioned into the training matrix A ∈ R600×360 and the test matrix Y ∈ R600×40 by employing a 10-fold cross validation.Sparse and Low Rank Representations in Music Signal Analysis 26/54
- 65. Example 1 JSLRR, LRR, SR matrices Z ∈ R360×40Sparse and Low Rank Representations in Music Signal Analysis 27/54
- 66. 1 Introduction 2 Auditory spectro-temporal modulations 3 Suitable data representations for classiﬁcation 4 Joint sparse low-rank representations in the ideal case 5 Joint sparse low-rank representations in the presence of noise 6 Joint sparse low-rank subspace-based classiﬁcation 7 Music signal analysis 8 ConclusionsSparse and Low Rank Representations in Music Signal Analysis 28/54
- 67. JSLRR Revisiting The data are approximately drawn from a union of subspaces. The deviations from the ideal assumptions can be treated collectively as additive noise contaminating the ideal model, i.e., Y = AZ + E. The noise term E models both small (but densely supported) deviations and grossly (but sparse) corrupted observations (i.e., outliers or missing data). In the presence of noise, both the rank and the density of the representation matrix Z increases, since the columns in Z contain non-zero elements associated to more than one class. If one requests to reduce the rank of Z or to increase the sparsity of Z, the noise in the test set can be smoothed and Z simultaneously admits a close to block-diagonal structure.Sparse and Low Rank Representations in Music Signal Analysis 29/54
- 68. JSLRR Revisiting The data are approximately drawn from a union of subspaces. The deviations from the ideal assumptions can be treated collectively as additive noise contaminating the ideal model, i.e., Y = AZ + E. The noise term E models both small (but densely supported) deviations and grossly (but sparse) corrupted observations (i.e., outliers or missing data). In the presence of noise, both the rank and the density of the representation matrix Z increases, since the columns in Z contain non-zero elements associated to more than one class. If one requests to reduce the rank of Z or to increase the sparsity of Z, the noise in the test set can be smoothed and Z simultaneously admits a close to block-diagonal structure.Sparse and Low Rank Representations in Music Signal Analysis 29/54
- 69. JSLRR Revisiting The data are approximately drawn from a union of subspaces. The deviations from the ideal assumptions can be treated collectively as additive noise contaminating the ideal model, i.e., Y = AZ + E. The noise term E models both small (but densely supported) deviations and grossly (but sparse) corrupted observations (i.e., outliers or missing data). In the presence of noise, both the rank and the density of the representation matrix Z increases, since the columns in Z contain non-zero elements associated to more than one class. If one requests to reduce the rank of Z or to increase the sparsity of Z, the noise in the test set can be smoothed and Z simultaneously admits a close to block-diagonal structure.Sparse and Low Rank Representations in Music Signal Analysis 29/54
- 70. JSLRR Revisiting The data are approximately drawn from a union of subspaces. The deviations from the ideal assumptions can be treated collectively as additive noise contaminating the ideal model, i.e., Y = AZ + E. The noise term E models both small (but densely supported) deviations and grossly (but sparse) corrupted observations (i.e., outliers or missing data). In the presence of noise, both the rank and the density of the representation matrix Z increases, since the columns in Z contain non-zero elements associated to more than one class. If one requests to reduce the rank of Z or to increase the sparsity of Z, the noise in the test set can be smoothed and Z simultaneously admits a close to block-diagonal structure.Sparse and Low Rank Representations in Music Signal Analysis 29/54
- 71. Robust JSLRR Optimization Problem A solution is sought for the convex optimization problem: Robust JSLRR: argmin Z ∗ + θ1 Z 1 + θ2 E 2,1 Z,E subject to Y = A Z + E, (7) where θ2 > 0 is a regularization parameter and . 2,1 denotes the 2 / 1 norm. Problem (7) can be solved iteratively by employing the Linearized Alternating Direction Augmented Lagrange Multiplier (LADALM) methoda , a variant of the Alternating Direction Augmented Lagrange Multiplier methodb . a J. Yang and X. M. Yuan,“Linearized augmented Lagrangian and alternating direction methods for nuclear norm minimization,” Math. Comput., (to appear) 2011. b D. P. Bertsekas, Constrained Optimization and Lagrange Multiplier Methods, Athena Scientiﬁc, Belmont, MA, 2/e, 1996.Sparse and Low Rank Representations in Music Signal Analysis 30/54
- 72. Robust JSLRR Optimization Problem A solution is sought for the convex optimization problem: Robust JSLRR: argmin Z ∗ + θ1 Z 1 + θ2 E 2,1 Z,E subject to Y = A Z + E, (7) where θ2 > 0 is a regularization parameter and . 2,1 denotes the 2 / 1 norm. Problem (7) can be solved iteratively by employing the Linearized Alternating Direction Augmented Lagrange Multiplier (LADALM) methoda , a variant of the Alternating Direction Augmented Lagrange Multiplier methodb . a J. Yang and X. M. Yuan,“Linearized augmented Lagrangian and alternating direction methods for nuclear norm minimization,” Math. Comput., (to appear) 2011. b D. P. Bertsekas, Constrained Optimization and Lagrange Multiplier Methods, Athena Scientiﬁc, Belmont, MA, 2/e, 1996.Sparse and Low Rank Representations in Music Signal Analysis 30/54
- 73. Robust JSLRR LADALM That is, one solves argmin J ∗ + θ1 W 1 + θ2 E 2,1 J,Z,W,E subject to Y = A Z + E, Z = J, J = W, (8) by minimizing the augmented Lagrangian function: L(J, Z, W, E, Λ1 , Λ2 , Λ3 ) = J ∗ + θ1 W 1 + θ2 E 2,1 +tr ΛT (Y 1 − AZ − E) + tr ΛT (Z 2 − J) + tr ΛT (J 3 − W) µ 2 2 2 + Y − AZ − E F + Z−J F + J−W F , (9) 2 where Λ1 , Λ2 , and Λ3 are the Lagrange multipliers and µ > 0 is a penalty parameter.Sparse and Low Rank Representations in Music Signal Analysis 31/54
- 74. Robust JSLRR LADALM That is, one solves argmin J ∗ + θ1 W 1 + θ2 E 2,1 J,Z,W,E subject to Y = A Z + E, Z = J, J = W, (8) by minimizing the augmented Lagrangian function: L(J, Z, W, E, Λ1 , Λ2 , Λ3 ) = J ∗ + θ1 W 1 + θ2 E 2,1 +tr ΛT (Y 1 − AZ − E) + tr ΛT (Z 2 − J) + tr ΛT (J 3 − W) µ 2 2 2 + Y − AZ − E F + Z−J F + J−W F , (9) 2 where Λ1 , Λ2 , and Λ3 are the Lagrange multipliers and µ > 0 is a penalty parameter.Sparse and Low Rank Representations in Music Signal Analysis 31/54
- 75. Robust JSLRR Optimization with respect to J[t] J[t+1] = argmin L(J[t] , Z[t] , W[t] , E[t] , Λ1[t] , Λ2[t] , Λ3[t] ) J[t] 1 ≈ argmin J ∗ J[t] µ [t] 1 2 + J − (Z[t] − J[t] − Λ3[t] /µ + W[t] + Λ2[t] /µ) 2 [t] F J[t+1] ← Dµ−1 Z[t] − J[t] − Λ3[t] /µ + W[t] + Λ2[t] /µ . (10) The solution is obtained via the singular value thresholding operator deﬁned for any matrix Q as Dτ [Q] = USτ VT with Q = UΣVT being the singular value decomposition and Sτ [q] = sgn(q)max(|q| − τ, 0) being the shrinkage operator.Sparse and Low Rank Representations in Music Signal Analysis 32/54
- 76. Robust JSLRR Optimization with respect to Z[t] Z[t+1] = argmin L(J[t+1] , Z[t] , W[t] , E[t] , Λ1[t] , Λ2[t] , Λ3[t] ) Z[t] −1 Z[t+1] = I + AT A AT (Y − E[t] ) + J[t+1] + (AT Λ1[t] − Λ2[t] )/µ . (11) i.e., an unconstrained least squares problem.Sparse and Low Rank Representations in Music Signal Analysis 32/54
- 77. Robust JSLRR Optimization with respect to W[t] W[t+1] = argmin L(J[t+1] , Z[t+1] , W[t] , E[t] , Λ1[t] , Λ2[t] , Λ3[t] ) W[t] θ1 1 2 = argmin W[t] 1 + W[t] − (J[t+1] + Λ3[t] /µ) F W[t] µ 2 W[t+1] ← Sθ1 µ−1 J[t+1] + Λ3[t] /µ . (12)Sparse and Low Rank Representations in Music Signal Analysis 32/54
- 78. Robust JSLRR Optimization with respect to E[t] E[t+1] = argmin L(J[t+1] , Z[t+1] , W[t+1] , E[t] , Λ1[t] , Λ2[t] , Λ3[t] ) E[t] θ2 = argmin E 2,1 + E[t] µ [t] 1 2 + E − (Y − AZ[t+1] + Λ1[t] /µ) F. (13) 2 [t] Let M[t] = Y − AZ[t+1] + Λ1[t] /µ. Update E[t+1] column-wise as follows: mj[t] ej[t+1] ← Sθ2 µ−1 mj[t] 2 . (14) mj[t] 2Sparse and Low Rank Representations in Music Signal Analysis 32/54
- 79. Robust JSLRR Updating of Lagrange multiplier matrices Λ1[t+1] = Λ1[t] + µ(Y − AZ[t+1] − E[t+1] ), Λ2[t+1] = Λ2[t] + µ(Z[t+1] − J[t+1] ), Λ3[t+1] = Λ3[t] + µ(J[t+1] − W[t+1] ). (15)Sparse and Low Rank Representations in Music Signal Analysis 32/54
- 80. Special cases Robust joint SR (JSR) The solution of the convex optimization problem is sought: Robust JSR: argmin Z 1 + θ2 E 2,1 subject to Y = A Z + E. Z,E (16) (16) takes into account the correlations between the test samples, while seeking to jointly represent the test samples from a speciﬁc class by a few columns of the training matrix. L1 (Z, J, E, Λ1 , Λ2 ) = J 1 + θ2 E 2,1 + tr ΛT (Y − AZ − E) 1 µ +tr ΛT (Z − J) + 2 Y − AZ − E 2 + Z − J 2 , F F (17) 2 where Λ1 , Λ2 are Lagrange multipliers and µ > 0 is a penalty parameter.Sparse and Low Rank Representations in Music Signal Analysis 33/54
- 81. Special cases Robust joint SR (JSR) The solution of the convex optimization problem is sought: Robust JSR: argmin Z 1 + θ2 E 2,1 subject to Y = A Z + E. Z,E (16) (16) takes into account the correlations between the test samples, while seeking to jointly represent the test samples from a speciﬁc class by a few columns of the training matrix. L1 (Z, J, E, Λ1 , Λ2 ) = J 1 + θ2 E 2,1 + tr ΛT (Y − AZ − E) 1 µ +tr ΛT (Z − J) + 2 Y − AZ − E 2 + Z − J 2 , F F (17) 2 where Λ1 , Λ2 are Lagrange multipliers and µ > 0 is a penalty parameter.Sparse and Low Rank Representations in Music Signal Analysis 33/54
- 82. Special cases Robust joint SR (JSR) The solution of the convex optimization problem is sought: Robust JSR: argmin Z 1 + θ2 E 2,1 subject to Y = A Z + E. Z,E (16) (16) takes into account the correlations between the test samples, while seeking to jointly represent the test samples from a speciﬁc class by a few columns of the training matrix. L1 (Z, J, E, Λ1 , Λ2 ) = J 1 + θ2 E 2,1 + tr ΛT (Y − AZ − E) 1 µ +tr ΛT (Z − J) + 2 Y − AZ − E 2 + Z − J 2 , F F (17) 2 where Λ1 , Λ2 are Lagrange multipliers and µ > 0 is a penalty parameter.Sparse and Low Rank Representations in Music Signal Analysis 33/54
- 83. Special cases Robust LRR The solution of the convex optimization problem is sought: Robust LRR: argmin Z ∗ + θ2 E 2,1 subject to Y = A Z + E. Z,E (18) by minimizing an augmented Lagrangian function similar to (17) where the ﬁrst term J 1 is replaced by J ∗ .Sparse and Low Rank Representations in Music Signal Analysis 34/54
- 84. Special cases Robust LRR The solution of the convex optimization problem is sought: Robust LRR: argmin Z ∗ + θ2 E 2,1 subject to Y = A Z + E. Z,E (18) by minimizing an augmented Lagrangian function similar to (17) where the ﬁrst term J 1 is replaced by J ∗ .Sparse and Low Rank Representations in Music Signal Analysis 34/54
- 85. Example 2 Noisy case 4 linear pairwise independent subspaces are constructed as in the Example 1 and the matrices A ∈ R600×360 and Y ∈ R600×40 are obtained. We pick randomly 50 columns of A and we replace them by a linear combination of randomly chosen vectors from two subspaces with random weights. Thus, the training set is now contaminated by outliers. The 5th column of the test matrix Y is replaced by a linear combination of vectors not drawn from any of the 4 subspaces and the 15th column of Y is replaced by a vector drawn from the 1st and the 4th subspace, as previously said.Sparse and Low Rank Representations in Music Signal Analysis 35/54
- 86. Example 2 Noisy case 4 linear pairwise independent subspaces are constructed as in the Example 1 and the matrices A ∈ R600×360 and Y ∈ R600×40 are obtained. We pick randomly 50 columns of A and we replace them by a linear combination of randomly chosen vectors from two subspaces with random weights. Thus, the training set is now contaminated by outliers. The 5th column of the test matrix Y is replaced by a linear combination of vectors not drawn from any of the 4 subspaces and the 15th column of Y is replaced by a vector drawn from the 1st and the 4th subspace, as previously said.Sparse and Low Rank Representations in Music Signal Analysis 35/54
- 87. Example 2 Noisy case 4 linear pairwise independent subspaces are constructed as in the Example 1 and the matrices A ∈ R600×360 and Y ∈ R600×40 are obtained. We pick randomly 50 columns of A and we replace them by a linear combination of randomly chosen vectors from two subspaces with random weights. Thus, the training set is now contaminated by outliers. The 5th column of the test matrix Y is replaced by a linear combination of vectors not drawn from any of the 4 subspaces and the 15th column of Y is replaced by a vector drawn from the 1st and the 4th subspace, as previously said.Sparse and Low Rank Representations in Music Signal Analysis 35/54
- 88. Example 2 Representation matrices (zoom in the 5th and 15th test samples)Sparse and Low Rank Representations in Music Signal Analysis 36/54
- 89. 1 Introduction 2 Auditory spectro-temporal modulations 3 Suitable data representations for classiﬁcation 4 Joint sparse low-rank representations in the ideal case 5 Joint sparse low-rank representations in the presence of noise 6 Joint sparse low-rank subspace-based classiﬁcation 7 Music signal analysis 8 ConclusionsSparse and Low Rank Representations in Music Signal Analysis 37/54
- 90. Joint sparse low-rank subspace-based classiﬁcation Algorithm Input: Training matrix A ∈ Rd×N and test matrix Y ∈ Rd×M . Output: A class label for each column of Y. 1 Solve (8) to obtain Z ∈ RN×M and E ∈ Rd×M . 2 for m = 1 to M 3 ¯ ym = ym − em . 4 for k = 1 to K 5 ¯ ¯ Compute the residuals rk (ym ) = ym − A δk (zm ) 2 . 6 end for 7 ¯ ¯ class(ym ) = argmink rk (ym ). 8 end forSparse and Low Rank Representations in Music Signal Analysis 38/54
- 91. Joint sparse low-rank subspace-based classiﬁcation Algorithm Input: Training matrix A ∈ Rd×N and test matrix Y ∈ Rd×M . Output: A class label for each column of Y. 1 Solve (8) to obtain Z ∈ RN×M and E ∈ Rd×M . 2 for m = 1 to M 3 ¯ ym = ym − em . 4 for k = 1 to K 5 ¯ ¯ Compute the residuals rk (ym ) = ym − A δk (zm ) 2 . 6 end for 7 ¯ ¯ class(ym ) = argmink rk (ym ). 8 end forSparse and Low Rank Representations in Music Signal Analysis 38/54
- 92. Joint sparse low-rank subspace-based classiﬁcation Algorithm Input: Training matrix A ∈ Rd×N and test matrix Y ∈ Rd×M . Output: A class label for each column of Y. 1 Solve (8) to obtain Z ∈ RN×M and E ∈ Rd×M . 2 for m = 1 to M 3 ¯ ym = ym − em . 4 for k = 1 to K 5 ¯ ¯ Compute the residuals rk (ym ) = ym − A δk (zm ) 2 . 6 end for 7 ¯ ¯ class(ym ) = argmink rk (ym ). 8 end forSparse and Low Rank Representations in Music Signal Analysis 38/54
- 93. Joint sparse low-rank subspace-based classiﬁcation Algorithm Input: Training matrix A ∈ Rd×N and test matrix Y ∈ Rd×M . Output: A class label for each column of Y. 1 Solve (8) to obtain Z ∈ RN×M and E ∈ Rd×M . 2 for m = 1 to M 3 ¯ ym = ym − em . 4 for k = 1 to K 5 ¯ ¯ Compute the residuals rk (ym ) = ym − A δk (zm ) 2 . 6 end for 7 ¯ ¯ class(ym ) = argmink rk (ym ). 8 end forSparse and Low Rank Representations in Music Signal Analysis 38/54
- 94. Joint sparse low-rank subspace-based classiﬁcation Algorithm Input: Training matrix A ∈ Rd×N and test matrix Y ∈ Rd×M . Output: A class label for each column of Y. 1 Solve (8) to obtain Z ∈ RN×M and E ∈ Rd×M . 2 for m = 1 to M 3 ¯ ym = ym − em . 4 for k = 1 to K 5 ¯ ¯ Compute the residuals rk (ym ) = ym − A δk (zm ) 2 . 6 end for 7 ¯ ¯ class(ym ) = argmink rk (ym ). 8 end forSparse and Low Rank Representations in Music Signal Analysis 38/54
- 95. Joint sparse low-rank subspace-based classiﬁcation Algorithm Input: Training matrix A ∈ Rd×N and test matrix Y ∈ Rd×M . Output: A class label for each column of Y. 1 Solve (8) to obtain Z ∈ RN×M and E ∈ Rd×M . 2 for m = 1 to M 3 ¯ ym = ym − em . 4 for k = 1 to K 5 ¯ ¯ Compute the residuals rk (ym ) = ym − A δk (zm ) 2 . 6 end for 7 ¯ ¯ class(ym ) = argmink rk (ym ). 8 end forSparse and Low Rank Representations in Music Signal Analysis 38/54
- 96. Joint sparse low-rank subspace-based classiﬁcation Algorithm Input: Training matrix A ∈ Rd×N and test matrix Y ∈ Rd×M . Output: A class label for each column of Y. 1 Solve (8) to obtain Z ∈ RN×M and E ∈ Rd×M . 2 for m = 1 to M 3 ¯ ym = ym − em . 4 for k = 1 to K 5 ¯ ¯ Compute the residuals rk (ym ) = ym − A δk (zm ) 2 . 6 end for 7 ¯ ¯ class(ym ) = argmink rk (ym ). 8 end forSparse and Low Rank Representations in Music Signal Analysis 38/54
- 97. Joint sparse low-rank subspace-based classiﬁcation Algorithm Input: Training matrix A ∈ Rd×N and test matrix Y ∈ Rd×M . Output: A class label for each column of Y. 1 Solve (8) to obtain Z ∈ RN×M and E ∈ Rd×M . 2 for m = 1 to M 3 ¯ ym = ym − em . 4 for k = 1 to K 5 ¯ ¯ Compute the residuals rk (ym ) = ym − A δk (zm ) 2 . 6 end for 7 ¯ ¯ class(ym ) = argmink rk (ym ). 8 end forSparse and Low Rank Representations in Music Signal Analysis 38/54
- 98. Joint sparse low-rank subspace-based classiﬁcation Linearity concentration index The LCI of a coefﬁcient vector zm ∈ RN associated to the mth test sample is deﬁned as K · maxk δk (zm ) 2 / zm 2 −1 LCI(zm ) = ∈ [0, 1]. (19) K −1 If LCI(zm ) = 1, the test sample is drawn from a single subspace. If LCI(zm ) = 0 the test sample is drawn evenly from all subspaces. By choosing a threshold c ∈ (0, 1), the mth test sample is claimed to be valid if LCI(zm ) > c. Otherwise, the test sample can be either rejected as totally invalid (for very small values of LCI(zm )) or be classiﬁed into multiple classes by assigning to it the labels associated to the larger values δk (zm ) 2 .Sparse and Low Rank Representations in Music Signal Analysis 39/54
- 99. Joint sparse low-rank subspace-based classiﬁcation Linearity concentration index The LCI of a coefﬁcient vector zm ∈ RN associated to the mth test sample is deﬁned as K · maxk δk (zm ) 2 / zm 2 −1 LCI(zm ) = ∈ [0, 1]. (19) K −1 If LCI(zm ) = 1, the test sample is drawn from a single subspace. If LCI(zm ) = 0 the test sample is drawn evenly from all subspaces. By choosing a threshold c ∈ (0, 1), the mth test sample is claimed to be valid if LCI(zm ) > c. Otherwise, the test sample can be either rejected as totally invalid (for very small values of LCI(zm )) or be classiﬁed into multiple classes by assigning to it the labels associated to the larger values δk (zm ) 2 .Sparse and Low Rank Representations in Music Signal Analysis 39/54
- 100. 1 Introduction 2 Auditory spectro-temporal modulations 3 Suitable data representations for classiﬁcation 4 Joint sparse low-rank representations in the ideal case 5 Joint sparse low-rank representations in the presence of noise 6 Joint sparse low-rank subspace-based classiﬁcation 7 Music signal analysis 8 ConclusionsSparse and Low Rank Representations in Music Signal Analysis 40/54
- 101. Music genre classiﬁcation: Datasets and evaluationprocedure GTZAN dataset 1000 audio recordings 30 seconds longa ; 10 genre classes: Blues, Classical, Country, Disco, HipHop, Jazz, Metal, Pop, Reggae, and Rock; Each genre class contains 100 audio recordings. The recordings are converted to monaural wave format at 16 kHz sampling rate with 16 bits and normalized, so that they have zero mean amplitude with unit variance. a G. Tzanetakis and P. Cook, “Musical genre classiﬁcation of audio signals,” IEEE Trans. Speech and Audio Processing, vol. 10, no. 5, pp. 293–302, July 2002.Sparse and Low Rank Representations in Music Signal Analysis 41/54
- 102. Music genre classiﬁcation: Datasets and evaluationprocedure ISMIR 2004 Genre dataset 1458 full audio recordings; 6 genre classes: Classical (640), Electronic (229), Jazz Blues(52), MetalPunk(90), RockPop(203), World (244).Sparse and Low Rank Representations in Music Signal Analysis 41/54
- 103. Music genre classiﬁcation: Datasets and evaluationprocedure Protocols GTZAN dataset: stratiﬁed 10-fold cross-validation: Each training set consists of 900 audio recordings yielding a training matrix AGTZAN . ISMIR 2004 Genre dataset: The ISMIR2004 Audio Description Contest protocol deﬁnes training and evaluation sets, which consist of 729 audio ﬁles each.Sparse and Low Rank Representations in Music Signal Analysis 41/54
- 104. Music genre classiﬁcation: Datasets and evaluationprocedure Classiﬁers JSLRSC, the JSSC, and the LRSC; SRCa with the coefﬁcients estimated by the LASSOb ; linear regression classiﬁer (LRC)c ; the SVM with a linear kernel, and the NN classiﬁer with the cosine similarity. a J. Wright, A. Yang, A. Ganesh, S. Sastry, and Y. Ma, “Robust face recognition via sparse representation,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 31, no. 2, pp. 210-227, 2009. b R. Tibshirani, “Regression shrinkage and selection via the LASSO,” J. Royal. Statist. Soc B., vol. 58, no. 1, pp. 267-288, 1996. c I. Naseem, R. Togneri, and M. Bennamoun, “Linear regression for face recognition,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 32, no. 11, pp. 2106-2112, 2010.Sparse and Low Rank Representations in Music Signal Analysis 41/54
- 105. Music genre classiﬁcation Parameters θ1 > 0 and θ2 > 0Sparse and Low Rank Representations in Music Signal Analysis 42/54
- 106. Music genre classiﬁcation Classiﬁcation accuracy Dataset: GTZAN ISMIR 2004 Genre Classiﬁer Parameters Accuracy (%) Parameters Accuracy (%) JSLRSC tuning 90.40 (3.06) tuning 88.75 JSSC tuning 88.80 (3.22) tuning 87.51 LRSC tuning 88.70 (2.79) tuning 85.18 JSLRSC θ1 = 0.2, 88.30 (2.21) θ1 = 0.5, 86.18 θ2 = 0.5, θ2 = 0.2, ρ = 1.1 ρ = 1.4 JSSC θ2 = 0.5, 88.00 (3.26) θ2 = 0.5, 87.51 ρ = 1.2 ρ = 1.2 LRSC θ2 = 0.2, 87.70 (2.54) θ2 = 0.2, 84.63 ρ = 1.4 ρ = 1.4 SRC - 86.50 (2.46) - 83.95 LRC - 87.30 (3.05) - 62.41 SVM - 86.20 (2.52) - 83.25 NN - 81.30 (2.79) - 79.42Sparse and Low Rank Representations in Music Signal Analysis 43/54
- 107. Music genre classiﬁcation Comparison with the state-of-the-art Dataset: GTZAN ISMIR 2004 Genre Rank Reference Accuracy (%) Reference Accuracy (%) 1) Chang et al.a 92.70 Lee et al.b 86.83 2) Lee et al. 90.60 Holzapfel et al.c 83.50 3) Panagakis et al.d 84.30 Panagakis et al. 83.15 4) Bergstra et al.e 82.50 Pampalk et al. 82.30 5) Tsunoo et al.f 77.20 a K. Chang, J. S. R. Jang, and C. S. Iliopoulos, “Music genre classiﬁcation via compressive sampling,” in Proc. 11th Int. Symp. Music Information Retrieval, pp. 387-392, 2010. b C. H. Lee, J. L. Shih, K. M. Yu, and H. S. Lin, ”Automatic music genre classiﬁcation based on modulation spectral analysis of spectral and cepstral features,” IEEE Trans. Multimedia, vol. 11, no. 4, pp. 670-682, 2009. c A. Holzapfel and Y. Stylianou, “Musical genre classiﬁcation using nonnegative matrix factorization-based features,” IEEE Trans. Audio, Speech, and Language Processing, vol. 16, no. 2, pp. 424-434, February 2008. d Y. Panagakis, C. Kotropoulos, and G. R. Arce, “Non-negative multilinear principal component analysis of auditory temporal modulations for music genre classiﬁcation,” IEEE Trans. Audio, Speech, and Language Technology, vol. 18, no. 3, pp. 576-588, 2010. e J. Bergstra, N. Casagrande, D. Erhan, D. Eck, and B. Kegl, “Aggregate features and AdaBoost for music classiﬁcation,” Machine Learning, vol. 65, no. 2-3, pp. 473–484, 2006. f E. Tsunoo, G. Tzanetakis, N. Ono, and S. Sagayama, “Beyond timbral statistics: Improving music classiﬁcation using percussive patterns and bass lines,” IEEE Trans. Audio, Speech, and Language Processing, vol. 19, no. 4, pp. 1003-1014, 2011.Sparse and Low Rank Representations in Music Signal Analysis 44/54
- 108. Music genre classiﬁcation Confusion matricesSparse and Low Rank Representations in Music Signal Analysis 45/54
- 109. Music genre classiﬁcation Dimensionality reduction via random projections Let the true low dimensionality of the data be denoted by r . A random projection matrix, drawn from a normal zero-mean distribution, provides with high probability a stable embeddinga with the dimensionality of the projection d selected as the minimum value such that d > 2r log(7680/d ). r is estimated by the robust principal component analysis on a training set for each dataset. d = 1581 for the GTZAN dataset and d = 1398 for the ISMIR 2004 Genre dataset is found. a R.G. Baraniuk, V. Cevher, and M.B. Wakin, “Low-dimensional models for dimensionality reduction and signal recovery: A geometric perspective,” Proceedings of the IEEE, vol. 98, no. 6, pp. 959–971, 2010.Sparse and Low Rank Representations in Music Signal Analysis 46/54
- 110. Music genre classiﬁcation Dimensionality reduction via random projections Let the true low dimensionality of the data be denoted by r . A random projection matrix, drawn from a normal zero-mean distribution, provides with high probability a stable embeddinga with the dimensionality of the projection d selected as the minimum value such that d > 2r log(7680/d ). r is estimated by the robust principal component analysis on a training set for each dataset. d = 1581 for the GTZAN dataset and d = 1398 for the ISMIR 2004 Genre dataset is found. a R.G. Baraniuk, V. Cevher, and M.B. Wakin, “Low-dimensional models for dimensionality reduction and signal recovery: A geometric perspective,” Proceedings of the IEEE, vol. 98, no. 6, pp. 959–971, 2010.Sparse and Low Rank Representations in Music Signal Analysis 46/54
- 111. Music genre classiﬁcation Dimensionality reduction via random projections Let the true low dimensionality of the data be denoted by r . A random projection matrix, drawn from a normal zero-mean distribution, provides with high probability a stable embeddinga with the dimensionality of the projection d selected as the minimum value such that d > 2r log(7680/d ). r is estimated by the robust principal component analysis on a training set for each dataset. d = 1581 for the GTZAN dataset and d = 1398 for the ISMIR 2004 Genre dataset is found. a R.G. Baraniuk, V. Cevher, and M.B. Wakin, “Low-dimensional models for dimensionality reduction and signal recovery: A geometric perspective,” Proceedings of the IEEE, vol. 98, no. 6, pp. 959–971, 2010.Sparse and Low Rank Representations in Music Signal Analysis 46/54
- 112. Music genre classiﬁcation Accuracy after dimensionality reduction Dataset: GTZAN ISMIR 2004 Genre Classiﬁer Parameters Accuracy (%) Parameters Accuracy (%) JSLRSC θ1 = 1.8, 87.5 (2.41) θ1 = 1.5, 85.87 θ2 = 0.7 , θ2 = 0.2, ρ = 1.4 ρ = 1.4 JSSC θ2 = 1.5 , 86.9 (3.28) θ2 = 0.6, 87.30 ρ = 1.2 ρ = 1.1 LRSC θ2 = 0.7 , 86.6 (2.75) θ2 = 0.8 , 84.08 ρ = 1.4 ρ = 2.4 SRC - 86.90 (3.21) - 83.67 LRC - 85.30 (3.16) - 54.18 SVM - 86.00 (2.53) - 83.26 NN - 80.80 (3.01) - 78.87Sparse and Low Rank Representations in Music Signal Analysis 47/54
- 113. Music genre classiﬁcation Accuracy after rejecting 1 out of 5 test samples JLRSC achieves classiﬁcation accuracy 95.51% in GTZAN dataset. For the ISMIR 2004 Genre dataset, the accuracy of the JSSC is 92.63%, while that of the JLRSC is 91.55%. 96 92 94 Classification accuracy (%) Classification accuracy (%) 92 90 90 88 88 86 86 JSLRSC JSLRSC JSSC JSSC 84 LRSC LRSC 84 LRC LRC SRC 82 SRC 82 SVM SVM NN NN 80 80 0.29 0.3 0.31 0.32 0.33 0.34 0.35 0.41 0.42 0.43 0.44 0.45 0.46 0.47 0.48 Threshold c Threshold cSparse and Low Rank Representations in Music Signal Analysis 48/54
- 114. Music structure analysis Optimization problem Given a music recording of K music segments be represented by a sequence of beat-synchronous feature vectors X = [x1 |x2 | . . . |xN ] ∈ Rd×N learn Z ∈ RN×N by minimizing 1 ˜ ˜˜ Let Z = U Σ VT . Deﬁne U = U Σ 2 . Set M = UUT . Build a nonnegative symmetric afﬁnity matrix W ∈ RN×N with elements + wij = mij and apply the normalized cutsa . 2 a J. Shi and J. Malik, “Normalized cuts and image segmentation,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 22, no. 8, pp. 888-905, 2000.Sparse and Low Rank Representations in Music Signal Analysis 49/54
- 115. Music structure analysis Optimization problem Given a music recording of K music segments be represented by a sequence of beat-synchronous feature vectors X = [x1 |x2 | . . . |xN ] ∈ Rd×N learn Z ∈ RN×N by minimizing λ2 2 argmin λ1 Z 1 + Z F subject to X = X Z, zii = 0 Z 2 1 ˜ ˜˜ Let Z = U Σ VT . Deﬁne U = U Σ 2 . Set M = UUT . Build a N×N nonnegative symmetric afﬁnity matrix W ∈ R+ with elements wij = mij and apply the normalized cutsa . 2 a J. Shi and J. Malik, “Normalized cuts and image segmentation,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 22, no. 8, pp. 888-905, 2000.Sparse and Low Rank Representations in Music Signal Analysis 49/54

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