In music similarity and in the related task of genre classification, a distance measure between Gaussian mixture models is frequently needed. We present a comparison of the Kullback-Leibler distance, the earth movers distance and the normalized L2 distance for this application. Although the normalized L2 distance was slightly inferior to the Kullback-Leibler distance with respect to classification performance, it has the advantage of obeying the triangle inequality, which allows for efficient searching

Christensen, Mads G.

Ellis, Daniel P. W.

Jensen, Jesper Hojvang

Jensen, Soren Holdt

English

Columbia University Academic Commons

EVALUATION OF DISTANCE MEASURES BETWEEN GAUSSIANMIXTURE MODELS OF MFCCSJesper Højvang Jensen Daniel P.W. Ellis Mads G. Christensen Søren Holdt JensenAalborg University Columbia University Aalborg University Aalborg UniversityDept. Electron. Syst. LabROSA Dept. Electron. Syst. Dept. Electron. Syst.ABSTRACTIn music similarity and in the related task of genre clas-sification, a distance measure between Gaussian mixturemodels is frequently needed. We present a comparison ofthe Kullback-Leibler distance, the earth movers distanceand the normalized L2 distance for this application. Al-though the normalized L2 distance was slightly inferiorto the Kullback-Leibler distance with respect to classifi-cation performance, it has the advantage of obeying thetriangle inequality, which allows for efficient searching.1 INTRODUCTIONA common approach in computational music similarityis to extract mel-frequency cepstral coefficients (MFCCs)from a song, model them by a Gaussian mixture model(GMM) and use a distance measure between the GMMsas a measure of the musical distance between the songs[2, 3, 5]. Through the years, a number of distance mea-sures between GMMs have been suggested, such as theKullback-Leibler (KL) distance [2], optionally combinedwith the earth movers distance (EMD) [3]. In this arti-cle, we evaluate the performance of these two distancemeasures between GMMs together with the normalizedL2 distance, which to our knowledge has not previouslybeen used for this application.2 MEASURING MUSICAL DISTANCEIn the following, we shortly describe the Gaussian mixturemodel and the three distance measures between GMMswe have tested. Note that if a distance measure satis-fies the triangle inequality, i.e., d(p1, p3) ≤ d(p1, p2) +d(p2, p3) for all values of p1, p2 and p3, then a nearestneighbor search can be speeded up by precomputing somedistances. Assume we are searching for the nearest neigh-bor to p, and that we have just computed the distance to p1.If we already know the distance between p1 and p2, thenthe distance to p2 is bounded by d(p, p2) ≥ d(p1, p2) −This research was supported by the Intelligent Sound project, Dan-ish Technical Research Council grant no. 26–04–0092, and the Paramet-ric Audio Processing project, Danish Research Council for Technologyand Production Sciences grant no. 274–06–0521.c© 2007 Austrian Computer Society (OCG).d(p1, p). If the distance to the currently best candidate issmaller than d(p1, p2)− d(p1, p), we can discard p2 with-out computing d(p, p2).2.1 Gaussian Mixture ModelsDue to intractability, theMFCCs extracted from a song aretypically not stored but are instead modelled by a GMM.A GMM is a weighted sum of multivariate Gaussians:p(x)=K∑k=1ck1√|2piΣk| exp(− 12 (x−µk)TΣ−1k (x−µk)),where K is the number of mixtures. For K = 1, a simpleclosed-form expression exists for the maximum-likelihoodestimate of the parameters. ForK > 1, the k-means algo-rithm and optionally the expectation-maximization algo-rithm are used to estimate the parameters.2.2 Kullback-Leibler DistanceThe KL distance is an information-theoretic distance mea-sure between probability density functions. It is given bydKL(p1, p2) =∫p1(x) logp1(x)p2(x)dx. As the KL distanceis not symmetric, a symmetrized version, dsKL(p1, p2) =dKL(p1, p2) + dKL(p2, p1), is usually used in music infor-mation retrieval. For Gaussian mixtures, a closed formexpression for dKL(p1, p2) only exists for K = 1. ForK > 1, dKL(p1, p2) is estimated using stochastic integra-tion or the approximation in [4]. The KL distance doesnot obey the triangle inequality.2.3 Earth Movers DistanceIn this context the EMD is the minimum cost of changingone mixture into another when the cost of moving proba-bility mass from componentm in the first mixture to com-ponent n in the second mixture is given [3]. A commonchoice of cost is the symmetrized KL distance betweenthe individual Gaussian components. With this cost, theEMD does not obey the triangle inequality.2.4 Normalized L2 DistanceLet p′i(x) = pi(x)/√∫pi(x)2dx, i.e., pi(x) scaled tounit L2-norm. We then define the normalized L2 distanceby dnL2(p1, p2) =∫(p′1(x)− p′2(x))2dx. Since the ordi-Figure 1. Instrument recognition results. Labels on thex-axis denotes the number of MFCCs retained, i.e. 0:10means retaining the first 11 coefficients including the 0th.“Fluid” and “SGM” denotes the Fluid R3 and SGM 180sound fonts, respectively.nary L2 distance obeys the triangle inequality, and sincewe can simply prescale all GMMs to have unit L2-normand then consider the ordinary L2 distance between thescaled GMMs, the normalized L2 distance will also obeythe triangle inequality. Also note that dnL2(p1, p2) is noth-ing but a continuous version of the cosine distance [6],since dnL2(p1, p2) = 2(1−∫p′1(x)p′2(x)dx). For GMMs,closed form expressions for the normalized L2 distancecan be derived for anyK from [1, Eq. (5.1) and (5.2)].3 EVALUATIONWe have evaluated the symmetrized KL distance com-puted by stochastic integration using 100 samples, EMDwith the exact, symmetrized KL distance as cost, and thenormalized L2 distance. We extract the MFCCs with theISP toolbox R1 using default options 1 . To model theMFCCs we have both used a single Gaussian with fullcovariance matrix and a mixture of ten Gaussians with di-agonal covariance matrices. With a single Gaussian, theEMD reduces to the exact, symmetrized KL distance. Fur-thermore, we have used different numbers of MFCCs. AstheMFCCs are timbral features and therefore are expectedto model instrumentation rather than melody or rhythm,we have evaluated the distance measures in a syntheticnearest neighbor instrument classification task using 900synthesized MIDI songs with 30 different melodies and30 different instruments. In Figure 1, results for usinga single sound font and results where the query song issynthesized by a different sound font than the songs it iscompared to are shown. The former test can be considereda sanity test, and the latter test reflects generalization be-haviour. Moreover, we have evaluated the distance mea-sures using 30 s excerpts of the training songs from theMIREX 2004 genre classification contest, which consistsof 729 songs from 6 genres. Results for genre classifica-tion, artist identification and genre classification with anartist filter (see [5]) are shown in Figure 2.1 http://isound.kom.auc.dk/Figure 2. Genre and artist classification results for theMIREX 2004 database.4 DISCUSSIONAs the results show, all three distance measures performapproximately equal when using a single Gaussian withfull covariance matrix, except that the normalized L2 dis-tance performs a little worse when mixing instrumentsfrom different sound fonts. Using a mixture of ten diago-nal Gaussians generally decrease recognition rates slightly,although it should be noted that [2] recommends usingmore than ten mixtures. For ten mixtures, the recognitionrate for the Kullback-Leibler distance seems to decreaseless than for the EMD and the normalized L2 distance.From these results we conclude that the cosine distanceperforms slightly worse than the Kullback-Leibler distancein terms of accuracy. However, with a single Gaussianhaving full covariance matrix this difference is negligible,and since the cosine distance obeys the triangle inequality,it might be preferable in applications with large datasets.5 REFERENCES[1] P. Ahrendt, “The multivariate gaussian probability dis-tribution,” Technical University of Denmark, Tech.Rep., 2005.[2] J.-J. Aucouturier, “Ten experiments on the modellingof polyphonic timbre,” Ph.D. dissertation, Universityof Paris 6, France, 2006.[3] B. Logan and A. Salomon, “A music similarity func-tion based on signal analysis,” in Proc. IEEE Int. Conf.Multimedia Expo, 2001, pp. 745 – 748.[4] E. Pampalk, “Speeding up music similarity,” in 2ndAnnual Music Information Retrieval eXchange, Lon-don, 2005.[5] ——, “Computational models of music similarity andtheir application to music information retrieval,” Ph.D.dissertation, Vienna University of Technology, Aus-tria, 2006.[6] J. R. Smith, “Integrated spatial and feature image sys-tems: Retrieval, analysis and compression,” Ph.D. dis-sertation, Columbia University, New York, 1997.

Evaluation Distance Measures Between Gaussian Mixture Models of MFCCs

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Evaluation Distance Measures Between Gaussian Mixture Models of MFCCs

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