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Slide 2 http://www.cmpe.boun.edu.tr/pilab smail Ar, A. Taylan Cemgil, Lale Akarun Bogazici University, stanbul Sep 2012, IEEE MLSP Slide 3 Ar, Cemgil, Akarun - Probabilistic Interpolative Decomposition 2 Slide 4 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 Ar, Cemgil, Akarun - Probabilistic Interpolative Decomposition 3 Slide 5 Interpolative Decomposition (ID) in brief Probabilistic ID Analytical Solution Numerical Solution Probabilistic CUR Experiments Synthetic Experiment Polyphonic Music Transcription Results & Discussions Conclusions Ar, Cemgil, Akarun - Probabilistic Interpolative Decomposition 4 Slide 6 ID uses actual columns Better interpretability SVD uses linear combinations If the data are sparse ID maintains sparsity SVD does not Ar, Cemgil, Akarun - Probabilistic Interpolative Decomposition 5 Slide 7 6 Slide 8 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] Ar, Cemgil, Akarun - Probabilistic Interpolative Decomposition 7 Slide 9 8 Slide 10 9 Slide 11 10 Slide 12 Ar, Cemgil, Akarun - Probabilistic Interpolative Decomposition 11 Slide 13 Ar, Cemgil, Akarun - Probabilistic Interpolative Decomposition 12 Slide 14 Ar, Cemgil, Akarun - Probabilistic Interpolative Decomposition 13 Slide 15 Ar, Cemgil, Akarun - Probabilistic Interpolative Decomposition 14 Slide 16 Ar, Cemgil, Akarun - Probabilistic Interpolative Decomposition 15 Slide 17 Ar, Cemgil, Akarun - Probabilistic Interpolative Decomposition 16 Slide 18 Ar, Cemgil, Akarun - Probabilistic Interpolative Decomposition 17 A 2 E 3 C 5 Frequency (Hz) Piano keys time (sec) Slide 19 Ar, 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 Slide 20 Ar, Cemgil, Akarun - Probabilistic Interpolative Decomposition 19... Slide 21 Ar, 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. Slide 22 F-measure seems to be more than 60% for each polyphony order Ar, Cemgil, Akarun - Probabilistic Interpolative Decomposition 21 Polyphony order: Number of notes played simultaneously Slide 23 Ar, Cemgil, Akarun - Probabilistic Interpolative Decomposition 22 Slide 24 Ar, Cemgil, Akarun - Probabilistic Interpolative Decomposition 23