158 research outputs found
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 -"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 -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
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 -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
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