İsmail Arı, A. Taylan Cemgil, Lale Akarun Bogazici University, İstanbul Sep 2012, IEEE MLSP

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 Express ID as a statistical model within a Bayesian framework  Derivation of an analytical solution for an example case  Numerical solution for general case using importance sampling  Application to polyphonic music transcription Arı, Cemgil, Akarun - Probabilistic Interpolative Decomposition 3

 Interpolative Decomposition (ID) in brief  Probabilistic ID  Analytical Solution  Numerical Solution  Probabilistic CUR  Experiments  Synthetic Experiment  Polyphonic Music Transcription  Results & Discussions  Conclusions Arı, Cemgil, Akarun - Probabilistic Interpolative Decomposition 4

 ID uses actual columns  Better interpretability  SVD uses linear combinations  If the data are sparse  ID maintains sparsity  SVD does not Arı, Cemgil, Akarun - Probabilistic Interpolative Decomposition 5

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Question: Which columns should we select?  Plain column-pivoted Gram-Schmidt method based on vector norms [Halko et al. 2011]  Randomized methods based on  Euclidean norm [Drineas et al. 2007, Frieze et al. 2004]  Norm of right singular vectors [Mahoney and Drineas 2009]  Vector sparseness value [Lee and Choi 2008] Arı, Cemgil, Akarun - Probabilistic Interpolative Decomposition 7

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Arı, Cemgil, Akarun - Probabilistic Interpolative Decomposition 17 A 2 E 3 C ♯ 5 Frequency (Hz) Piano keys time (sec)

Arı, Cemgil, Akarun - Probabilistic Interpolative Decomposition ‡ P. Smaragdis, «Polyphonic pitch tracking by example,» IEEE WASPAA, apply thresholding compute weights via NMF A 2 E 3 C ♯ 5 Dictionary is very big

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Arı, Cemgil, Akarun - Probabilistic Interpolative Decomposition 20 Column-pivoted QR: Halko et al., «Finding Structure with Randomness: Prob. Alg. for Constructing Approx. Matrix Dec.,» SIAM Review, Randomized CUR: Mahoney and Drineas, “CUR Matrix Decompositions for Improved Data Analysis,” Proc. of the National Acad. of Sci., 2009.

 F-measure seems to be more than 60% for each polyphony order ‡ Arı, Cemgil, Akarun - Probabilistic Interpolative Decomposition 21 ‡ Polyphony order: Number of notes played simultaneously

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