Jérémy E. Cohen

Christos Chatzis, Carla Schenker, Jérémy E. Cohen et al.

Multiway datasets are commonly analyzed using unsupervised matrix and tensor factorization methods to reveal underlying patterns. Frequently, such datasets include timestamps and could correspond to, for example, health-related measurements of subjects collected over time. The temporal dimension is inherently different from the other dimensions, requiring methods that account for its intrinsic properties. Linear Dynamical Systems (LDS) are specifically designed to capture sequential dependencies in the observed data. In this work, we bridge the gap between tensor factorizations and dynamical modeling by exploring the relationship between LDS, Coupled Matrix Factorizations (CMF) and the PARAFAC2 model. We propose a time-aware coupled factorization model called d(ynamical)CMF that constrains the temporal evolution of the latent factors to adhere to a specific LDS structure. Using synthetic datasets, we compare the performance of dCMF with PARAFAC2 and t(emporal)PARAFAC2 which incorporates temporal smoothness. Our results show that dCMF and PARAFAC2-based approaches perform similarly when capturing smoothly evolving patterns that adhere to the PARAFAC2 structure. However, dCMF outperforms alternatives when the patterns evolve smoothly but deviate from the PARAFAC2 structure. Furthermore, we demonstrate that the proposed dCMF method enables to capture more complex dynamics when additional prior information about the temporal evolution is incorporated.

Haoran Wu, Axel Marmoret, Jérémy E. Cohen

Automatic Music Transcription, which consists in transforming an audio recording of a musical performance into symbolic format, remains a difficult Music Information Retrieval task. In this work, which focuses on piano transcription, we propose a semi-supervised approach using low-rank matrix factorization techniques, in particular Convolutive Nonnegative Matrix Factorization. In the semi-supervised setting, only a single recording of each individual notes is required. We show on the MAPS dataset that the proposed semi-supervised CNMF method performs better than state-of-the-art low-rank factorization techniques and a little worse than supervised deep learning state-of-the-art methods, while however suffering from generalization issues.

Axel Marmoret, Jérémy E. Cohen, Frédéric Bimbot

Music Structure Analysis (MSA) consists in segmenting a music piece in several distinct sections. We approach MSA within a compression framework, under the hypothesis that the structure is more easily revealed by a simplified representation of the original content of the song. More specifically, under the hypothesis that MSA is correlated with similarities occurring at the bar scale, this article introduces the use of linear and non-linear compression schemes on barwise audio signals. Compressed representations capture the most salient components of the different bars in the song and are then used to infer the song structure using a dynamic programming algorithm. This work explores both low-rank approximation models such as Principal Component Analysis or Nonnegative Matrix Factorization and "piece-specific" Auto-Encoding Neural Networks, with the objective to learn latent representations specific to a given song. Such approaches do not rely on supervision nor annotations, which are well-known to be tedious to collect and possibly ambiguous in MSA description. In our experiments, several unsupervised compression schemes achieve a level of performance comparable to that of state-of-the-art supervised methods (for 3s tolerance) on the RWC-Pop dataset, showcasing the importance of the barwise compression processing for MSA.

Axel Marmoret, Jérémy E. Cohen, Frédéric Bimbot

The ability of deep neural networks to learn complex data relations and representations is established nowadays, but it generally relies on large sets of training data. This work explores a "piece-specific" autoencoding scheme, in which a low-dimensional autoencoder is trained to learn a latent/compressed representation specific to a given song, which can then be used to infer the song structure. Such a model does not rely on supervision nor annotations, which are well-known to be tedious to collect and often ambiguous in Music Structure Analysis. We report that the proposed unsupervised auto-encoding scheme achieves the level of performance of supervised state-of-the-art methods with 3 seconds tolerance when using a Log Mel spectrogram representation on the RWC-Pop dataset.

Axel Marmoret, Florian Voorwinden, Valentin Leplat et al.

Nonnegative Tucker decomposition (NTD), a tensor decomposition model, has received increased interest in the recent years because of its ability to blindly extract meaningful patterns, in particular in Music Information Retrieval. Nevertheless, existing algorithms to compute NTD are mostly designed for the Euclidean loss. This work proposes a multiplicative updates algorithm to compute NTD with the beta-divergence loss, often considered a better loss for audio processing. We notably show how to implement efficiently the multiplicative rules using tensor algebra. Finally, we show on a music structure analysis task that unsupervised NTD fitted with beta-divergence loss outperforms earlier results obtained with the Euclidean loss.

Axel Marmoret, Jérémy E. Cohen, Nancy Bertin et al.

Recent work has proposed the use of tensor decomposition to model repetitions and to separate tracks in loop-based electronic music. The present work investigates further on the ability of Nonnegative Tucker Decompositon (NTD) to uncover musical patterns and structure in pop songs in their audio form. Exploiting the fact that NTD tends to express the content of bars as linear combinations of a few patterns, we illustrate the ability of the decomposition to capture and single out repeated motifs in the corresponding compressed space, which can be interpreted from a musical viewpoint. The resulting features also turn out to be efficient for structural segmentation, leading to experimental results on the RWC Pop data set which are potentially challenging state-of-the-art approaches that rely on extensive example-based learning schemes.

Jérémy E. Cohen, Nicolas Gillis

Sparse component analysis (SCA), also known as complete dictionary learning, is the following problem: Given an input matrix $M$ and an integer $r$, find a dictionary $D$ with $r$ columns and a matrix $B$ with $k$-sparse columns (that is, each column of $B$ has at most $k$ non-zero entries) such that $M \approx DB$. A key issue in SCA is identifiability, that is, characterizing the conditions under which $D$ and $B$ are essentially unique (that is, they are unique up to permutation and scaling of the columns of $D$ and rows of $B$). Although SCA has been vastly investigated in the last two decades, only a few works have tackled this issue in the deterministic scenario, and no work provides reasonable bounds in the minimum number of samples (that is, columns of $M$) that leads to identifiability. In this work, we provide new results in the deterministic scenario when the data has a low-rank structure, that is, when $D$ is (under)complete. While previous bounds feature a combinatorial term $r \choose k$, we exhibit a sufficient condition involving $\mathcal{O}(r^3/(r-k)^2)$ samples that yields an essentially unique decomposition, as long as these data points are well spread among the subspaces spanned by $r-1$ columns of $D$. We also exhibit a necessary lower bound on the number of samples that contradicts previous results in the literature when $k$ equals $r-1$. Our bounds provide a drastic improvement compared to the state of the art, and imply for example that for a fixed proportion of zeros (constant and independent of $r$, e.g., 10\% of zero entries in $B$), one only requires $\mathcal{O}(r)$ data points to guarantee identifiability.

Jérémy E. Cohen, Nicolas Gillis

To ensure interpretability of extracted sources in tensor decomposition, we introduce in this paper a dictionary-based tensor canonical polyadic decomposition which enforces one factor to belong exactly to a known dictionary. A new formulation of sparse coding is proposed which enables high dimensional tensors dictionary-based canonical polyadic decomposition. The benefits of using a dictionary in tensor decomposition models are explored both in terms of parameter identifiability and estimation accuracy. Performances of the proposed algorithms are evaluated on the decomposition of simulated data and the unmixing of hyperspectral images.