108 research outputs found
Tensor Decompositions for Signal Processing Applications From Two-way to Multiway Component Analysis
The widespread use of multi-sensor technology and the emergence of big
datasets has highlighted the limitations of standard flat-view matrix models
and the necessity to move towards more versatile data analysis tools. We show
that higher-order tensors (i.e., multiway arrays) enable such a fundamental
paradigm shift towards models that are essentially polynomial and whose
uniqueness, unlike the matrix methods, is guaranteed under verymild and natural
conditions. Benefiting fromthe power ofmultilinear algebra as theirmathematical
backbone, data analysis techniques using tensor decompositions are shown to
have great flexibility in the choice of constraints that match data properties,
and to find more general latent components in the data than matrix-based
methods. A comprehensive introduction to tensor decompositions is provided from
a signal processing perspective, starting from the algebraic foundations, via
basic Canonical Polyadic and Tucker models, through to advanced cause-effect
and multi-view data analysis schemes. We show that tensor decompositions enable
natural generalizations of some commonly used signal processing paradigms, such
as canonical correlation and subspace techniques, signal separation, linear
regression, feature extraction and classification. We also cover computational
aspects, and point out how ideas from compressed sensing and scientific
computing may be used for addressing the otherwise unmanageable storage and
manipulation problems associated with big datasets. The concepts are supported
by illustrative real world case studies illuminating the benefits of the tensor
framework, as efficient and promising tools for modern signal processing, data
analysis and machine learning applications; these benefits also extend to
vector/matrix data through tensorization. Keywords: ICA, NMF, CPD, Tucker
decomposition, HOSVD, tensor networks, Tensor Train
Using the Proximal Gradient and the Accelerated Proximal Gradient as a Canonical Polyadic tensor decomposition Algorithms in difficult situations
Canonical Polyadic (CP) tensor decomposition is useful in many real-world applications due to its uniqueness, and the ease of interpretation of its factor matrices. This work addresses the problem of calculating the CP decomposition of tensors in difficult cases where the factor matrices in one or all modes are almost collinear - i.e. bottleneck or swamp problems arise. This is done by introducing a constraint on the coherences of the factor matrices that ensures the existence of a best low-rank approximation, which makes it possible to estimate these highly correlated factors. Two new algorithms optimizing the CP decomposition based on proximal methods are proposed. Simulation results are provided and demonstrate the good behaviour of these algorithms, as well as a better compromise between accuracy and convergence speed than other algorithms in the literature
Numerical performance of a tensor music algorithm based on HOSVD for a mixture of polarized sources
International audienceIn this paper, we develop an improved tensor MUSIC algorithm adapted to multidimensional data by means of multilinear algebra tools. This approach allows to preserve the multidimensional structure as the signal and the noise subspaces are estimated from the Higher Order Singular Value Decomposition (HOSVD) of the covariance tensor. The proposed algorithm is applied to a polarized source model. By computing the Mean Squared Error (MSE) for different scenarios, the performance of this method is compared to the classical MUSIC algorithm as well as the vector MUSIC algorithm that includes the polarization information. The simulations show that our algorithm outperforms the vector algorithms
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