Appearing in Proceedings of the 29th International Conference on Machine Learning, Edinburgh, Scotland, UK, 2012.The tree representation, using rhythm for defining the tree structure and pitch information for node labeling has proven to be effective in melodic similarity computation. In this paper we propose a solution representing melodies by tree grammars. For that, we infer a probabilistic context-free grammars for the melodies in a database, using their tree coding (with duration and pitch) and classify queries represented as a string of pitches. We aim to assess their ability to identify a noisy snippet query among a set of songs stored in symbolic format

Bernabeu Briones, José Francisco

Calera Rubio, Jorge

Iñesta, José M.

Rizo, David

Query parsing using probabilistic tree grammars

Repositorio Institucional de la Universidad de Alicante

Query Parsing Using Probabilistic Tree GrammarsJose´ F. Bernabeu jfbernabeu@dlsi.ua.esJorge Calera-Rubio calera@dlsi.ua.esJose´ M. In˜esta inesta@dlsi.ua.esDavid Rizo drizo@dlsi.ua.esDept. Lenguajes y Sistemas Informa´ticos, Universidad de Alicante, SpainAbstractThe tree representation, using rhythm fordefining the tree structure and pitch infor-mation for node labeling has proven to be ef-fective in melodic similarity computation. Inthis paper we propose a solution representingmelodies by tree grammars. For that, we in-fer a probabilistic context-free grammars forthe melodies in a database, using their treecoding (with duration and pitch) and classifyqueries represented as a string of pitches. Weaim to assess their ability to identify a noisysnippet query among a set of songs stored insymbolic format.1. IntroductionSymbolic music retrieval from queries has been ex-tensively studied in the MIR literature. In order tosearch a query in a dataset of songs, we can applyany of the well-known pattern matching algorithmslike local string editing to each of the songs in thedataset, and retrieve a ranking of most similar items.The main problem here is the efficiency when usinglarge datasets, but with the advantage that it can finda partial or approximate occurrence of the query inany part of the dataset. On the other hand, a previ-ous indexation of the dataset, e.g by means of motiveextraction, solves the scalability issue, but motive ex-traction usually needs to work with exact repetitions,making very inaccurate to build robust indices fromdifferent interpretations of the same song. In addi-tion, when searching only in motives, the music notpresent in the motives is hidden.Our proposal is able to solve intrinsically these prob-Appearing in Proceedings of the 29 th International Confer-ence on Machine Learning, Edinburgh, Scotland, UK, 2012.Copyright 2012 by the author(s)/owner(s).lems. The probabilistic grammar structure itself en-codes both motives and melody variations, giving moreweight to the most repeated themes, without the needto be exact, and enabling the possibility to learn fromdifferent renderings of the same song, leaving asidethe need to encode motives. Besides, there is no dif-ference between looking for a whole song or searchinga small query in the dataset. For it, we use probabilis-tic context-free grammars obtained from probabilistick-testable tree models (Knuutila, 1993). The durationinformation implicit in the tree representation is cap-tured by the grammar and this is used for classifyingsnippet melodies represented by strings that only havethe pitch information. This is a solution when dura-tion information is not available or unreliable for theinput data. This work is an extension of that by Bern-abeu et al. (Bernabeu et al., 2011) where the inputswere whole melodies.2. MethodologyFor representing the note pitches in a monophonicmelody s as a string, symbols σ from a pitch represen-tation alphabet Σp are used: s ∈ Σ∗p, s = σ1σ2...σ|s|.In this paper, the interval from the tonic of the songmodulo 12 is utilized as pitch descriptor and the sym-bol ‘−’ to represent rests. In the tree approach, eachmelody bar is represented by a tree, t ∈ TΣp . Barsare coded by separated trees and then they are linkedto a common root. The level of a node in the treedetermines the duration it represents. Once the treehas been built, a bottom-up propagation of the pitchlabels is performed to label all the internal nodes.Stochastic models based on k-grams predict the prob-ability of the next symbol in a sequence dependingon the k − 1 previous symbols (Downie, 1999). k-gram models can be regarded as a probabilistic exten-sion of locally testable languages. These models areeasy to learn and can be efficiently processed. Locallytestable languages, in the case of trees, were describedQuery Parsing Using Probabilistic Tree GrammarsN4N4N4T4 T4N2T2 T2T0(1/1) I → N4(1/3) N4→ N4 T0(1/3) N4→ N4 N2(1/3) N4→ T4 T4(1/1) N2→ T2 T2Figure 1. Example of a probabilistic tree grammar (k = 2).by Knuutila (Knuutila, 1993). These kind of tree lan-guages can be defined using the formalism of determin-istic tree automata (DTA) and can be extended to thecase where the sample Ω is stochastically generated,incorporating probabilities to the DTA (PDTA).We can classify a new melody in a particular class. Forthat, we need to infer a PDTA for each class, Cj , fromwell classified melodies. Once the PDTAs for the dif-ferent classes have been inferred and the probabilitiesestimated, a melody M can be classified in the classCˆ that maximizes the likelihood. In order to do this,both training and new melodies must be representedby trees. However, in a practical situation, the user ofa music retrieval system may make a query using justa sequence of pitches or with non reliable durations.Therefore, we need to transform the k-testable tree au-tomata in probabilistic context-free grammars to usethem for parsing the input melody strings. Thus, wecould use these grammars to obtain the correct struc-ture for a given melody represented by a pitch string.A probabilistic context-free grammar (PCFG) isshown in Figure 1 (right) corresponding to the treein the left. Therefore, we can obtain k-testable gram-mars with different values for k (depth of subtrees,see (Bernabeu et al., 2011) for details).In general, k-testable grammars with larger values ofk contain more specialized rules and, therefore, theyare less ambiguous and allow for faster parsing. Butthe larger they are, the more likely they can find un-seen strings, that will be assigned a zero probabilityin recognition. Therefore, the use of smoothing tech-niques becomes necessary if one wants to use thesemodels for parsing using the proper context size. Here,we use a rule-based backing-off approach. This re-quires the implementation of specific parsers, sincebuilding the whole grammar is unfeasible due to thelarge number of implicit rules. An alternative schemethat requires only minor modifications is to use aquasi-equivalent grammar G′ where the different k-grammars are combined to build a unique grammarfor each class Cj .After a grammar Gj is inferred for each class Cj weneed an algorithm for obtaining the probability thata given string s is generated by a grammar Gj . Forthis purpose, we have used the Stolcke algorithm andthe CYK+ algorithm for string parsing. These parsingalgorithms are able to give the probability p(s|G) thata string s is generated by a probabilistic Context-freegrammar G without requiring conversions to Chom-sky Normal Form (CNF). Then, a melody query M isclassified in the class Cˆ that maximizes the likelihoodCˆ = arg maxjl(M |Cj) (1)3. Results and conclusionsIn this work, we want to assess the ability of the treegrammars to identify a noisy snippet query among aset of songs stored in symbolic format. For that, weuse a corpus consisting of a set of 420 monophonic8-12 bar incipits of 20 worldwide well known tunes ofdifferent musical genres. Then, different (3−22) lengthqueries have been built from tunes using only the pitchdata. Then, a tree grammar for each tune has beentrained from the tunes represented by trees. Finally,the queries have been classified in the grammar whoseprobability to generate the query is higher.The obtained results show the system obtain a successrate of about 73% for 18 query length. Moreover, thesystem reaches a success rate of about 95% when aranking is formed with the best four classes.Thus, it seems that tree grammars are able to clas-sify noisy snippet queries. Although the success rateseems to increase with the query length we need find abalance between success and computation time. More-over, the ranking could be used as a pruning for dis-carding the parse of a whole melody for the worsegrammars given the snippet query. In other words,if we have a long melody, we could classify it usingonly its beginning. After that, we compute the prob-ability of the whole melody only for the best classes.This way, we can improve the computing time whenthe melody length is large.ReferencesBernabeu, J. F., Calera-Rubio, J., and In˜esta, J. M.Classifying melodies using tree grammars. volume6669, pp. 572–579, 2011.Downie, J. Stephen. Evaluating a Simple Approach toMusic Information Retrieval: Conceiving Melodic n-grams as Text. PhD thesis, University of WesternOntario, 1999.Knuutila, T. Inference of k-testable tree languages.In Bunke (Ed.), Advances in Structural and Syntac-tic Pattern Recognition (Proc. of the S+SSPR’92),World Scientific, Singapore, 1993.

Query parsing using probabilistic tree grammars

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