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http://www.cmpe.boun.edu.tr/pilab İsmail Arı, A. Taylan Cemgil, Lale Akarun Bogazici University, İstanbul Sep 2012, IEEE MLSP

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Arı, Cemgil, Akarun - Probabilistic Interpolative Decomposition 2

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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

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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

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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)

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Arı, Cemgil, Akarun - Probabilistic Interpolative Decomposition 18... ‡ P. Smaragdis, «Polyphonic pitch tracking by example,» IEEE WASPAA, 2011. 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 19...

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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, 2011. Randomized CUR: Mahoney and Drineas, “CUR Matrix Decompositions for Improved Data Analysis,” Proc. of the National Acad. of Sci., 2009.

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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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