Overview of Constrained PARAFAC Models

In this paper, we present an overview of constrained PARAFAC models where the constraints model linear dependencies among columns of the factor matrices of the tensor decomposition, or alternatively, the pattern of interactions between different modes of the tensor which are captured by the equivalent core tensor. Some tensor prerequisites with a particular emphasis on mode combination using Kronecker products of canonical vectors that makes easier matricization operations, are first introduced. This Kronecker product based approach is also formulated in terms of the index notation, which provides an original and concise formalism for both matricizing tensors and writing tensor models. Then, after a brief reminder of PARAFAC and Tucker models, two families of constrained tensor models, the co-called PARALIND/CONFAC and PARATUCK models, are described in a unified framework, for $N^{th}$ order tensors. New tensor models, called nested Tucker models and block PARALIND/CONFAC models, are also introduced. A link between PARATUCK models and constrained PARAFAC models is then established. Finally, new uniqueness properties of PARATUCK models are deduced from sufficient conditions for essential uniqueness of their associated constrained PARAFAC models

Nonlinear Blind Identification with Three-Dimensional Tensor Analysis

This paper deals with the analysis of a third-order tensor composed of a fourth-order output cumulants used for blind identification of a second-order Volterra-Hammerstein series. It is demonstrated that this nonlinear identification problem can be converted in a multivariable system with multiequations having the form of +=. The system may be solved using several methods. Simulation results with the Iterative Alternating Least Squares (IALS) algorithm provide good performances for different signal-to-noise ratio (SNR) levels. Convergence issues using the reversibility analysis of matrices and are addressed. Comparison results with other existing algorithms are carried out to show the efficiency of the proposed algorithm

Non-iterative solution for PARAFAC with a Toeplitz matrix factor

International audienceRecently, tensor signal processing has received an increased attention, particularly in the context of wireless communication applications. The so-called PARAllel FACtor (PARAFAC) decomposition is certainly the most used tensor tool. In general, the parameter estimation of a PARAFAC decomposition is carried out by means of the iterative ALS algorithm, which exhibits the following main drawbacks: convergence towards local minima, a high number of iterations for convergence, and difficulty to take, optimally, special matrix structures into account. In this paper, we propose a non-iterative parameter estimation method for a PARAFAC decomposition when one matrix factor has a Toeplitz structure, a situation that is commonly encountered in signal processing applications. We illustrate the proposed method by means of simulation results

Modelling and Analysis of Drosophila Early Visual System A Systems Engineering Approach

Over the past century or so Drosophila has been established as an ideal model organism to study, among other things, neural computation and in particular sensory processing. In this respect there are many features that make Drosophila an ideal model organism, especially the fact that it offers a vast amount of genetic and experimental tools for manipulating and interrogating neural circuits. Whilst comprehensive models of sensory processing in Drosophila are not yet available, considerable progress has been made in recent years in modelling the early stages of sensory processing. When it comes to visual processing, accurate empirical and biophysical models of the R1-R6 photoreceptors were developed and used to characterize nonlinear processing at photoreceptor level and to demonstrate that R1-R6 photoreceptors encode phase congruency. A limitation of the latest photoreceptor models is that these do not account explicitly for the modulation of photoreceptor responses by the network of interneurones hosted in the lamina. As a consequence, these models cannot describe in a unifying way the photoreceptor response in the absence of the feedback from the downstream neurons and thus cannot be used to elucidate the role of interneurones in photoreceptor adaptation. In this thesis, electrophysiological photoreceptor recordings acquired in-vivo from wild- type and histamine defficient mutant fruit flies are used to develop and validate new com- prehensive models of R1-R6 photoreceptors, which not only predict the response of these photoreceptors in wild-type and mutant fruit flies, over the entire environmental range of light intensities but also characterize explicitly the contribution of lamina neurons to photore- ceptor adaptation. As a consequence, the new models provide suitable building blocks for assembling a complete model of the retina which takes into account the true connectivity between photoreceptors and downstream interneurones. A recent study has demonstrated that R1-R6 photoreceptors employ nonlinear processing to selectively encode and enhance temporal phase congruency. It has been suggested that this processing strategy achieves an optimal trade-off between the two competing goals of minimizing distortion in decoding behaviourally relevant stimuli features and minimizing the information rate, which ultimately enables more efficient downstream processing of spatio-temporal visual stimuli for edge and motion detection.Using rigorous information theoretic tools, this thesis derives and analyzes the rate-distortion characteristics associated with the linear and nonlinear transformations performed by photoreceptors on a stimulus generated by a signal source with a well defined distribution

Contributions à l’analyse des systèmes en réseau

La dernière décennie a vu l’émergence des travaux autour des systèmes dynamiques interconnectés (systèmes en réseaux ou systèmes cyberphysiques). Dans cette habilitation à diriger des recherches, je donne un aperçu des contributions qui ont été les miennes durant la dernière décennie sur : l’analyse des systèmes en réseaux (problème de consensus, observabilité et application à la préservation de la vie privée), le traitement des données de grandes dimensions (analyse tensorielle pour l’identification des systèmes non-linéaires, décomposition distribuée de tenseurs de grandes dimensions), et l’application à la mobilité intelligente (navigation en milieu urbain, prédiction et estimation de trafic, estimation d’attitude pour la navigation pédestre). Une prospective est ensuite développée autour de la sécurité des systèmes en réseaux, en se basant sur la théorie des systèmes, et sur l’analyse des données de grandes dimensions organisées dans des tenseurs de données avec des applications sur la mobilité intelligente

Real-time Digital Simulation of Guitar Amplifiers as Audio Effects

Práce se zabývá číslicovou simulací kytarových zesilovačů, jakož to nelineárních analogových hudebních efektů, v reálném čase. Hlavním cílem práce je návrh algoritmů, které by umožnily simulaci složitých systémů v reálném čase. Tyto algoritmy jsou prevážně založeny na automatizované DK-metodě a aproximaci nelineárních funkcí. Kvalita navržených algoritmů je stanovana pomocí poslechových testů.The work deals with the real-time digital simulation of guitar amplifiers considered as nonlinear analog audio effects. The main aim is to design algorithms which are able to simulate complex systems in real-time. These algorithms are mainly based on the automated DK-method and the approximation of nonlinear functions. Quality of the designed algorithms is evaluated using listening tests.

Compositional nonlinear audio signal processing with Volterra series

We develop a compositional theory of nonlinear audio signal processing based on a categorification of the Volterra series. We augment the classical definition of the Volterra series to be functorial with respect to a base category whose objects are temperate distributions and whose morphisms are certain linear transformations. This leads to formulae describing how the outcomes of nonlinear transformations are affected if their input signals are first linearly processed. We then consider how nonlinear audio systems change, and introduce as a model thereof the notion of morphism of Volterra series. We show how morphisms can be parameterized and used to generate indexed families of Volterra series, which are well-suited to model nonstationary or time-varying nonlinear phenomena. We describe how Volterra series and their morphisms organize into a functor category, Volt, whose objects are Volterra series and whose morphisms are natural transformations. We exhibit the operations of sum, product, and series composition of Volterra series as monoidal products on Volt and identify, for each in turn, its corresponding universal property. We show, in particular, that the series composition of Volterra series is associative. We then bridge between our framework and a subject at the heart of audio signal processing: time-frequency analysis. Specifically, we show that an equivalence between a certain class of second-order Volterra series and the bilinear time-frequency distributions (TFDs) can be extended to one between certain higher-order Volterra series and the so-called polynomial TFDs. We end with prospects for future work, including the incorporation of nonlinear system identification techniques and the extension of our theory to the settings of compositional graph and topological audio signal processing.Comment: Master's thesi

Nonlinear stochastic system identification techniques for biological tissues/

Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Mechanical Engineering, 2010.Cataloged from PDF version of thesis.Includes bibliographical references (p. 205-212).This research develops a device capable of measuring the nonlinear dynamic mechanical properties of human tissue in vivo. The enabling technology is the use of nonlinear stochastic system identification techniques in conjunction with a high bandwidth actuator to perturb the tissue. The desktop and handheld instruments used for this investigation were custom-built Lorentz force actuators which were able to measure the dynamic compliance between the input force and the output displacement. The actuators have a nominal stroke of 32 mm and were actuated with forces under 15 N. The design includes custom electronics and user software which collects and analyses the information. This research also explores nonlinear stochastic system identification techniques that would be applicable to biological tissues. Several system identification techniques were used including linear, Wiener static nonlinear, Volterra kernel and partitioning techniques. Real time system identification and real time input generation schemes are also implemented. The mathematical formulation and implementation details of these techniques are also discussed. It was found that a simple linear stochastic system identification technique had a variance accounted for (VAF) of 70 to 75 %. More complicated representations using Volterra kernels or partitioning techniques had a VAF of 90 to 97 %. More complex nonlinear system identification techniques can not only capture more of the nonlinear dynamics but also capture those dynamics in an interpretable way. Indentation, extension, and surface mechanics experiments were conducted to investigate the nonlinear mechanical compliance of skin in vivo. The techniques and devices used in this research can be applied directly to consumer product efficacy analysis, medical diagnosis as well as research in biomechanical tissues.by Yi Chen.S.M