Abstract-Automatic Music Transcription (AMT) is concerned with the problem of producing the pitch content of a piece of music given a recorded signal. Many methods rely on sparse or low rank models, where the observed magnitude spectra are represented as a linear combination of dictionary atoms corresponding to individual pitches. Some of the most successful approaches use Non-negative Matrix Decomposition (NMD) or Factorization (NMF), which can be used to learn a dictionary and pitch activation matrix from a given signal. Here we introduce a further refinement of NMD in which we assume the transcription itself is approximately low rank. The intuition behind this approach is that the total number of distinct activation patterns should be relatively small since the pitch content between adjacent frames should be similar. A rank penalty is introduced into the NMD objective function and solved using an iterative algorithm based on Singular Value thresholding. We find that the low rank assumption leads to a significant increase in performance compared to NMD using β-divergence on a standard AMT dataset.

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Proximal methods are an important tool in signal processing applications, where many problems can be characterized by the minimization of an expression involving a smooth fitting term and a convex regularization term -for example the classic 1 -Lasso. Such problems can be solved using the relevant proximal operator. Here we consider the use of proximal operators for the p -quasinorm where 0 ≤ p ≤ 1. Rather than seek a closed form solution, we develop an iterative algorithm using a Majorization-Minimization procedure which results in an inexact operator. Experiments on image denoising show that for p ≤ 1 the algorithm is effective in the high-noise scenario, outperforming the Lasso despite the inexactness of the proximal step.

Polyphonic music transcription is a challenging problem, requiring the identification of a collection of latent pitches which can explain an observed music signal. Many state-of-the-art methods are based on the Non-negative Matrix Factorization (NMF) framework, which itself can be cast as a latent variable model. However, the basic NMF algorithm fails to consider many important aspects of music signals such as lowrank or hierarchical structure and temporal continuity. In this work we propose a probabilistic model to address some of the shortcomings of NMF. Probabilistic Latent Component Analysis (PLCA) provides a probabilistic interpretation of NMF and has been widely applied to problems in audio signal processing. Based on PLCA, we propose an algorithm which represents signals using a collection of low-rank dictionaries built from a base pitch dictionary. This allows each dictionary to specialize to a given chord or interval template which will be used to represent collections of similar frames. Experiments on a standard music transcription data set show that our method can successfully decompose signals into a hierarchical and smooth structure, improving the quality of the transcription.

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