## 1 Introduction  
Automatic music transcription (AMT) is one of the fundamental problems of music information retrieval and is defined as the process of converting an acoustic music signal into some form of music notation [1]. A core problem of AMT is multi-pitch detection, the detection of multiple concurrent pitches from an audio recording. While much work has gone into the field of multi-pitch detection in recent years, it has frequently been constrained to instrumental music, most often piano recordings due to a wealth of available data. Vocal music has been less often studied, likely due to the complexity and variety of sounds that can be produced by a singer. Spectrogram factorization methods have been used extensively in the last decade for multi-pitch detection [1]. These approaches decompose an input time-frequency representation (such as a spectrogram) into a linear combination of non-negative factors, often consisting of spectral atoms and note activations. The most successful of these spectrogram factorization methods have been based on non-negative matrix factorisation (NMF) [2] or probabilistic latent component analysis (PLCA) [3]. While these spectrogram factorisation methods have shown promise for AMT, their parameter estimation can suffer from local optima, a problem that has motivated a variety of approachesthat incorporate additional knowledge in an attempt to achieve more meaningful decompositions. [PERSON] et al. [PERSON] et al. (2015) used an adaptive spectral decomposition for multi-pitch detection assuming that the input signal can be decomposed as a sum of narrowband spectra. [PERSON] et al. [PERSON] et al. (2015) exploited structural regularities in the spectrograms during the NMF process, adding constraints and regularization to reduce the degrees of freedom of their model. These constraints are based on time-varying basis spectra (e.g., using sound states: \"attack\", \"decay\", \"sustain\" and \"release\") and have since been included in other probabilistic models [PERSON] et al. (2015); [PERSON] et al. (2015). [PERSON] et al. [PERSON] et al. (2015) introduced the concept of brakes, slowing the convergence rate of any model parameter known to be properly initialized. Other approaches [PERSON] et al. (2015); [PERSON] et al. (2015); [PERSON] et al. (2015) avoid undesirable parameter convergence using pre-learning steps, where spectral atoms of specific instruments are extracted in a supervised manner. Using the constant-Q transform (CQT) [PERSON] et al. (2015) as the input time-frequency representation, some approaches developed techniques using shift-invariant models over log-frequency [PERSON] et al. (2015); [PERSON] et al. (2015); [PERSON] et al. (2015), allowing for the creation of a compact set of dictionary templates that can support tuning deviations and frequency modulations. Shift-invariant models are also used in several recent approaches for automatic music transcription [PERSON] et al. (2015); [PERSON] et al. (2015); [PERSON] et al. (2015). [PERSON] et al. [PERSON] et al. (2015) propose stepwise and gradient-based methods for non-negative group sparse decompositions, exploring the use of subspace modelling of note spectra. This group sparse NMF approach is used to tune a generic harmonic subspace dictionary, improving automatic music transcription results based on NMF. However, despite promising results of template-based techniques [PERSON] et al. (2015); [PERSON] et al. (2015); [PERSON] et al. (2015), the considerable variation in the spectral shape of pitches produced by different sources can still affect generalization performance. Recent research on multi-pitch detection has also focused on deep learning approaches: in [PERSON] et al. (2015); [PERSON] et al. (2015), feedforward, recurrent and convolutional neural networks were evaluated towards the problem of automatic piano transcription.