1/16 Dynamic Programming Carmine Casciato MUMT 611 Thursday March 31 st 2005

Overview Problem Space Origins Dynamic Time Warping Overview of Usage in MIR

Problem Space Multi-stage decision processes system S characterized as evolution of vector p N stages * M decisions/stage multi-dimensional maximization problems

Origins Dynamic Programming (DP) “…the optimal decision to be made at any state of the system. ” Bellman (1957) “Dynamic” refers to temporal nature of S Each decision is determined by max/min cost of previous state Allocation problem, x = y + x-y f N (x) = Max/Min [g(y) + h(x-y) + f N-1 (ay + b(x-y)) 0 <= y <= x

Rabiner and Huang 1993 Dynamic Time Warping (DTW) as solution for time-alignment and normalization of two utterances (Dis)similarity measurement of two vectors of short-time spectral features is equal to “best” path through feature grid

DTW Path Constraints endpoint monotonicity local path constraints global path constraints slope weighting locally and globally Dissimilarity metric, constraints, weightings, are all heuristically determined

Paulus and Klapuri 2002 Adopts Rabiner and Huang (1993) DTW to rhythmic similarity Depends on correct segmentation of rhythms from audio signal Finds optimal path between feature vectors of loudness and spectral centroid

Paulus and Klapuri 2002

Usage in MIR Query by humming Heo et al Adams et al Nishimura et. al 2001 Tempo tracking Raphael 2002 Feature selection Chang 1972

References Adams, N., M. Bartsch, J. Shifrin, and G.Wakefield Time series alignment for music information retrieval. In Proceedings of the International Conference on Music Information Retrieval: 303  10 Bellman, R Dynamic Programming. Princeton: Princeton University Press. Chang, C Dynamic programming as applied to feature subset selection in a pattern recognition system. In Proceedings of the ACM annual conference 1: 94  103. Guo, A., and H. Siegelman Time-warped longest common subsequence algorithm for music retrieval. In Proceedings of the International Conference on Music Information Retrieval: 258  61. Heo, S., M. Suzuki, A. Ito, and S. Makino Three dimensional continuous DP algorithm for multiple pitch candidates in music information retrieval system. In Proceedings of the International Conference on Music Information Retrieval. Nishimura, T., H. Hashiguchi, J. Takita, J. Zhang, M. Goto, and R. Oka Music signal spotting retrieval by a humming query using start frame feature dependent continuous dynamic programming. In Proceedings of the International Conference on Music Information Retrieval. Paulus, J., and A. Klapuri Measuring the similarity of rhythmic patterns. In Proceedings of the International Conference on Music Information Retrieval. Raphael, C A hybrid graphical model for rhythmic parsing. Artificial Intelligence 137: 217–38.