3,417 research outputs found
Learning with Algebraic Invariances, and the Invariant Kernel Trick
When solving data analysis problems it is important to integrate prior
knowledge and/or structural invariances. This paper contributes by a novel
framework for incorporating algebraic invariance structure into kernels. In
particular, we show that algebraic properties such as sign symmetries in data,
phase independence, scaling etc. can be included easily by essentially
performing the kernel trick twice. We demonstrate the usefulness of our theory
in simulations on selected applications such as sign-invariant spectral
clustering and underdetermined ICA
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
A general adaptive algorithm for nonGaussian source separation without any constraint
This paper deals with the blind source separation. The task consists in separating some independent and linearly mixed signals called sources. After some general remarks, the model is recalled and our approach based on the Maximum-Likelihood principle and on the higher-order statistics (HOS) is introduced. The main stages of the calculation are presented leading to the criterion of the separation based on a sum of squared cumulants of the sources at the fourth order. The second part is devoted to the adaptive implementation which is in opposition to the block treatment. The procedure using the gradient calculus is described.
Some results obtained in simulations are shown, they correspond to the case of a mixture of two real valued sources. Finally, an example of a possible integration in a communications system based on multidimensional beamformers is briefly shown. But some tests on real data should be carried out beforehand.Peer ReviewedPostprint (published version
Tensor Networks for Dimensionality Reduction and Large-Scale Optimizations. Part 2 Applications and Future Perspectives
Part 2 of this monograph builds on the introduction to tensor networks and
their operations presented in Part 1. It focuses on tensor network models for
super-compressed higher-order representation of data/parameters and related
cost functions, while providing an outline of their applications in machine
learning and data analytics. A particular emphasis is on the tensor train (TT)
and Hierarchical Tucker (HT) decompositions, and their physically meaningful
interpretations which reflect the scalability of the tensor network approach.
Through a graphical approach, we also elucidate how, by virtue of the
underlying low-rank tensor approximations and sophisticated contractions of
core tensors, tensor networks have the ability to perform distributed
computations on otherwise prohibitively large volumes of data/parameters,
thereby alleviating or even eliminating the curse of dimensionality. The
usefulness of this concept is illustrated over a number of applied areas,
including generalized regression and classification (support tensor machines,
canonical correlation analysis, higher order partial least squares),
generalized eigenvalue decomposition, Riemannian optimization, and in the
optimization of deep neural networks. Part 1 and Part 2 of this work can be
used either as stand-alone separate texts, or indeed as a conjoint
comprehensive review of the exciting field of low-rank tensor networks and
tensor decompositions.Comment: 232 page
Tensor Networks for Dimensionality Reduction and Large-Scale Optimizations. Part 2 Applications and Future Perspectives
Part 2 of this monograph builds on the introduction to tensor networks and
their operations presented in Part 1. It focuses on tensor network models for
super-compressed higher-order representation of data/parameters and related
cost functions, while providing an outline of their applications in machine
learning and data analytics. A particular emphasis is on the tensor train (TT)
and Hierarchical Tucker (HT) decompositions, and their physically meaningful
interpretations which reflect the scalability of the tensor network approach.
Through a graphical approach, we also elucidate how, by virtue of the
underlying low-rank tensor approximations and sophisticated contractions of
core tensors, tensor networks have the ability to perform distributed
computations on otherwise prohibitively large volumes of data/parameters,
thereby alleviating or even eliminating the curse of dimensionality. The
usefulness of this concept is illustrated over a number of applied areas,
including generalized regression and classification (support tensor machines,
canonical correlation analysis, higher order partial least squares),
generalized eigenvalue decomposition, Riemannian optimization, and in the
optimization of deep neural networks. Part 1 and Part 2 of this work can be
used either as stand-alone separate texts, or indeed as a conjoint
comprehensive review of the exciting field of low-rank tensor networks and
tensor decompositions.Comment: 232 page
Convexity in source separation: Models, geometry, and algorithms
Source separation or demixing is the process of extracting multiple
components entangled within a signal. Contemporary signal processing presents a
host of difficult source separation problems, from interference cancellation to
background subtraction, blind deconvolution, and even dictionary learning.
Despite the recent progress in each of these applications, advances in
high-throughput sensor technology place demixing algorithms under pressure to
accommodate extremely high-dimensional signals, separate an ever larger number
of sources, and cope with more sophisticated signal and mixing models. These
difficulties are exacerbated by the need for real-time action in automated
decision-making systems.
Recent advances in convex optimization provide a simple framework for
efficiently solving numerous difficult demixing problems. This article provides
an overview of the emerging field, explains the theory that governs the
underlying procedures, and surveys algorithms that solve them efficiently. We
aim to equip practitioners with a toolkit for constructing their own demixing
algorithms that work, as well as concrete intuition for why they work
Blind separation of complex-valued satellite-AIS data for marine surveillance: a spatial quadratic time-frequency domain approach
In this paper, the problem of the blind separation of complex-valued Satellite-AIS data for marine surveillance is addressed. Due to the specific properties of the sources under consideration: they are cyclo-stationary signals with two close cyclic frequencies, we opt for spatial quadratic time-frequency domain methods. The use of an additional diversity, the time delay, is aimed at making it possible to undo the mixing of signals at the multi-sensor receiver. The suggested method involves three main stages. First, the spatial generalized mean Ambiguity function of the observations across the array is constructed. Second, in the Ambiguity plane, Delay-Doppler regions of high magnitude are determined and Delay-Doppler points of peaky values are selected. Third, the mixing matrix is estimated from these Delay-Doppler regions using our proposed non-unitary joint zero-(block) diagonalization algorithms as to perform separation
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