ORIGINAL ARTICLE Probabilistic approaches for music similarity using restricted Boltzmann machines Son N. Tran 1 • Son Ngo 2 • Artur d’Avila Garcez 3 Received: 6 March 2018 / Accepted: 16 February 2019 Ó Springer-Verlag London Ltd., part of Springer Nature 2019 Abstract In music informatics, there has been increasing attention to relative similarity as it plays a central role in music retrieval, recommendation, and musicology. Most approaches for relative similarity are based on distance metric learning, in which similarity relationship is modelled by a parameterised distance function. Normally, these parameters can be learned by solving a constrained optimisation problem using kernel-based methods. In this paper, we study the use of restricted Boltzmann machines (RBMs) in similarity modelling. We take advantage of RBM as a probabilistic neural network to assign a true hypothesis ‘‘x is more similar to y than to z’’ with a higher probability. Such model can be trained by maximising the true hypotheses while, at the same time, minimising the false hypotheses using a stochastic method. Alternatively, we show that learning similarity relations can be done deterministically by minimising the free energy function of a bipolar RBM or using a classification approach. In the experiments, we evaluate our proposed approaches on music scripts extracted from MagnaTagATune dataset. The results show that an energy-based optimisation approach with bipolar RBM can achieve better performance than other methods, including support vector machine and machine learning rank which are the state-of-the-art for this dataset. Keywords Music similarity  Restricted Boltzmann machines  Machine learning 1 Introduction Relative similarity has been emerging as an effective tool in music information retrieval, especially for personalisa- tion and recommendation [15, 22, 25]. Also, in computa- tional musicology, relative similarity has been studied intensively for music recognition and perception [20, 27]. Different from other similarity learning approaches where absolute ranking scores of instances, e.g. ranking of music scripts, are needed, the idea of modelling similarity metrics by using relative relations is more practical. The issue of absolute ranking is that given a music script it is difficult for users to rank a list of other scripts in a similarity order by listening to them. However, if there are only two scripts to compare, it is very easy for users to identify which script is more similar to the given one. Formally, a relative similarity model is learned from a set of triplets x, y, z in which instance x is more similar to instance y than to instance z. Normally, most related approaches in music similarity learn a parameterised function using the triplets in order to quantify the similarity between two pairs of instances, such as d(x, y) and d(x, z). The similarity rela- tion of a newly given triplet then will be decided by comparing if dðx; yÞ [ dðx; zÞ or not. The most common measurement, i.e. function d, is based on distance metric whose parameters can be learned by solving a constrained optimisation problem using kernel-based methods [5, 15]. Such methods are very popular in music similarity [25, 27]. Besides kernel-based methods, neural networks also have & Son N. Tran sn.tran@utas.edu.au Son Ngo sonnt69@fe.edu.vn Artur d’Avila Garcez a.garcez@city.ac.uk 1 ICT Discipline, University of Tasmania, Launceston, Australia 2 Department of Computer Science, FPT University, Hanoi, Vietnam 3 Department of Computer Science, City, University of London, London, UK 123 Neural Computing and Applications https://doi.org/10.1007/s00521-019-04106-y