Dictionary-based Tensor Canonical Polyadic Decomposition

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

A Tour of Constrained Tensor Canonical Polyadic Decomposition

This paper surveys the use of constraints in tensor decomposition models. Constrained tensor decompositions have been extensively applied to chemometrics and array processing, but there is a growing interest in understanding these methods independently of the application of interest. We suggest a formalism that unifies various instances of constrained tensor decomposition, while shedding light on some possible extensions of existing methods

Fast Decomposition of Large Nonnegative Tensors

International audienceIn Signal processing, tensor decompositions have gained in popularity this last decade. In the meantime, the volume of data to be processed has drastically increased. This calls for novel methods to handle Big Data tensors. Since most of these huge data are issued from physical measurements, which are intrinsically real nonnegative, being able to compress nonnegative tensors has become mandatory. Following recent works on HOSVD compression for Big Data, we detail solutions to decompose a nonnegative tensor into decomposable terms in a compressed domain

Joint Tensor Compression for Coupled Canonical Polyadic Decompositions

International audienceTo deal with large multimodal datasets, coupled canonical polyadic decompositions are used as an approximation model. In this paper, a joint compression scheme is introduced to reduce the dimensions of the dataset. Joint compression allows to solve the approximation problem in a compressed domain using standard coupled decomposition algorithms. Computational complexity required to obtain the coupled decomposition is therefore reduced. Also, we propose to approximate the update of the coupled factor by a simple weighted average of the independent updates of the coupled factors. The proposed approach and its simplified version are tested with synthetic data and we show that both do not incur substantial loss in approximation performance

Barwise Music Structure Analysis with the Correlation Block-Matching Segmentation Algorithm

Music Structure Analysis (MSA) is a Music Information Retrieval task consisting of representing a song in a simplified, organized manner by breaking it down into sections typically corresponding to ``chorus'', ``verse'', ``solo'', etc. In this work, we extend an MSA algorithm called the Correlation Block-Matching (CBM) algorithm introduced by (Marmoret et al., 2020, 2022b). The CBM algorithm is a dynamic programming algorithm that segments self-similarity matrices, which are a standard description used in MSA and in numerous other applications. In this work, self-similarity matrices are computed from the feature representation of an audio signal and time is sampled at the bar-scale. This study examines three different standard similarity functions for the computation of self-similarity matrices. Results show that, in optimal conditions, the proposed algorithm achieves a level of performance which is competitive with supervised state-of-the-art methods while only requiring knowledge of bar positions. In addition, the algorithm is made open-source and is highly customizable.Comment: 19 pages, 13 figures, 11 tables, 1 algorithm, published in Transactions of the International Society for Music Information Retrieva

Convolutive Block-Matching Segmentation Algorithm with Application to Music Structure Analysis

Music Structure Analysis (MSA) consists of representing a song in sections (such as ``chorus'', ``verse'', ``solo'' etc), and can be seen as the retrieval of a simplified organization of the song. This work presents a new algorithm, called Convolutive Block-Matching (CBM) algorithm, devoted to MSA. In particular, the CBM algorithm is a dynamic programming algorithm, applying on autosimilarity matrices, a standard tool in MSA. In this work, autosimilarity matrices are computed from the feature representation of an audio signal, and time is sampled on the barscale. We study three different similarity functions for the computation of autosimilarity matrices. We report that the proposed algorithm achieves a level of performance competitive to that of supervised state-of-the-art methods on 3 among 4 metrics, while being fully unsupervised.Comment: 4 pages, 5 figures, 1 table. Submitted at ICASSP 2023. The associated toolbox is available at https://gitlab.inria.fr/amarmore/autosimilarity_segmentatio

A Homotopy-based Algorithm for Sparse Multiple Right-hand Sides Nonnegative Least Squares

Nonnegative least squares (NNLS) problems arise in models that rely on additive linear combinations. In particular, they are at the core of nonnegative matrix factorization (NMF) algorithms. The nonnegativity constraint is known to naturally favor sparsity, that is, solutions with few non-zero entries. However, it is often useful to further enhance this sparsity, as it improves the interpretability of the results and helps reducing noise. While the $\ell_0$-"norm", equal to the number of non-zeros entries in a vector, is a natural sparsity measure, its combinatorial nature makes it difficult to use in practical optimization schemes. Most existing approaches thus rely either on its convex surrogate, the $\ell_1$-norm, or on heuristics such as greedy algorithms. In the case of multiple right-hand sides NNLS (MNNLS), which are used within NMF algorithms, sparsity is often enforced column- or row-wise, and the fact that the solution is a matrix is not exploited. In this paper, we first introduce a novel formulation for sparse MNNLS, with a matrix-wise $\ell_0$ sparsity constraint. Then, we present a two-step algorithm to tackle this problem. The first step uses a homotopy algorithm to produce the whole regularization path for all the $\ell_1$-penalized NNLS problems arising in MNNLS, that is, to produce a set of solutions representing different tradeoffs between reconstruction error and sparsity. The second step selects solutions among these paths in order to build a sparsity-constrained matrix that minimizes the reconstruction error. We illustrate the advantages of our proposed algorithm for the unmixing of facial and hyperspectral images.Comment: 20 pages + 7 pages supplementary materia

EXTRAPOLATED ALTERNATING ALGORITHMS FOR APPROXIMATE CANONICAL POLYADIC DECOMPOSITION

Tensor decompositions have become a central tool in machine learning to extract interpretable patterns from multiway arrays of data. However, computing the approximate Canonical Polyadic Decomposition (aCPD), one of the most important tensor decomposition model, remains a challenge. In this work, we propose several algorithms based on extrapolation that improve over existing alternating methods for aCPD. We show on several simulated and real data sets that carefully designed extrapolation can significantly improve the convergence speed hence reduce the computational time, especially in difficult scenarios

Nonnegative tensor CP decomposition of hyperspectral data

International audienceNew hyperspectral missions will collect huge amounts of hyperspectral data. Besides, it is possible now to acquire time series and multiangular hyperspectral images. The process and analysis of these big data collections will require common hyperspectral techniques to be adapted or reformulated. The tensor decomposition, \textit{a.k.a.} multiway analysis, is a technique to decompose multiway arrays, that is, hypermatrices with more than two dimensions (ways). Hyperspectral time series and multiangular acquisitions can be represented as a 3-way tensor. Here, we apply Canonical Polyadic tensor decomposition techniques to the blind analysis of hyperspectral big data. In order to do so, we use a novel compression-based nonnegative CP decomposition. We show that the proposed methodology can be interpreted as multilinear blind spectral unmixing, a higher order extension of the widely known spectral unmixing. In the proposed approach, the big hyperspectral tensor is decomposed in three sets of factors which can be interpreted as spectral signatures, their spatial distribution and temporal/angular changes. We provide experimental validation using a study case of the snow coverage of the French Alps during the snow season

A primer for resonant tunnelling

Resonant tunnelling is studied numerically and analytically with the help of a three-well quantum one-dimensional time-independent model. The simplest cases are considered where the three-well potential is polynomial or piecewise constant.Comment: accepted to EJP, 19 pages, 8 figure