rank of 3 tensors with 2 slices and kronecker canonical forms tensor type data are becoming important recently in various application fields we determine the rank of a 3 tensor with 2 slices in comparison with its kronecker canonical form over the complex and real number field

eeg extended source localization tensor based vs conventional methods the localization of brain sources based on eeg measurements is a topic that has attracted a lot of attention in the last decades and many different source localization algorithms have been proposed however their performance is limited in the case of several simultaneously active brain regions and low signal to noise ratios to overcome these problems tensor based preprocessing can be applied which consists in constructing a space time frequency stf or space time wave vector stwv tensor and decomposing it using the canonical polyadic cp decomposition in this paper we present a new algorithm for the accurate localization of extended sources based on the results of the tensor decomposition furthermore we conduct a detailed study of the tensor based preprocessing methods including an analysis of their theoretical foundation their computational complexity and their performance for realistic simulated data in comparison to conventional source localization algorithms such as sloreta cortical loreta cloreta and 4 exso music our objective consists on the one hand in demonstrating the gain in performance that can be achieved by tensor based preprocessing and on the other hand in pointing out the limits and drawbacks of this method finally we validate the stf and stwv techniques on real measurements to demonstrate their usefulness for practical applications

large deviations hypotheses testing and source coding for finite markov chains let x n n geq 1 be a finite markov chain with transition probability matrix of strictly positive entries a large deviation theorem is proved for the empirical transition count matrix and is used to get asymptotically optimal critical regions for testing simple hypotheses about the transition matrix as a corollary the error exponent in the source coding theorem for x n is obtained these results generalize the corresponding results for the independent and identically distributed case

the growth of iterates of multivariate generating functions the vector valued function m of a p vector has components m1 m2 mp for eachi exp mi is the multivariate laplace transform of a discrete measure concentrated on 0 1 p with only a nite number of atoms the main objective is to give conditions for the functional iteratesm n ofm to grow like n for a suitable 1 the initial stimulus was provided by results of miller and o sullivan 1992 on enumeration issues in context free languages results which can be improved using the theory devel oped here the theory also allow certain results in jones 2004 on multitype branching to be proved under signicantly weaker conditions

functional parallel factor analysis for functions of one and two dimensional arguments parallel factor analysis parafac is a useful multivariate method for decomposing three way data that consist of three different types of entities simultaneously this method estimates trilinear components each of which is a low dimensional representation of a set of entities often called a mode to explain the maximum variance of the data functional parafac permits the entities in different modes to be smooth functions or curves varying over a continuum rather than a collection of unconnected responses the existing functional parafac methods handle functions of a one dimensional argument e g time only in this paper we propose a new extension of functional parafac for handling three way data whose responses are sequenced along both a two dimensional domain e g a plane with x and y axis coordinates and a one dimensional argument technically the proposed method combines parafac with basis function expansion approximations using a set of piecewise quadratic finite element basis functions for estimating two dimensional smooth functions and a set of one dimensional basis functions for estimating one dimensional smooth functions in a simulation study the proposed method appeared to outperform the conventional parafac we apply the method to eeg data to demonstrate its empirical usefulness

three region inequalities for the second order elliptic equation with discontinuous coefficients and size estimate abstract in this paper we would like to derive a quantitative uniqueness estimate the three region inequality for the second order elliptic equation with jump discontinuous coefficients the derivation of the inequality relies on the carleman estimate proved in our previous work 5 we then apply the three region inequality to study the size estimate problem with one boundary measurement

multiway array decomposition of eeg spectrum implications of its stability for the exploration of large scale brain networks multiway array decomposition methods have been shown to be promising statistical tools for identifying neural activity in the eeg spectrum they blindly decompose the eeg spectrum into spatial temporal spectral patterns by taking into account inherent relationships among signals acquired at different frequencies and sensors our study evaluates the stability of spatial temporal spectral patterns derived by one particular method parallel factor analysis parafac we focused on patterns stability over time and in population and divided the complete data set containing data from 50 healthy subjects into several subsets our results suggest that the patterns are highly stable in time as well as among different subgroups of subjects further we show with simultaneously acquired fmri data that power fluctuations of some patterns have stable correspondence to hemodynamic fluctuations in large scale brain networks we did not find such correspondence for power fluctuations in standard frequency bands the common way of dealing with eeg data altogether our results suggest that parafac is a suitable method for research in the field of large scale brain networks and their manifestation in eeg signal

shift invariant multilinear decomposition of neuroimaging data we present an algorithm for multilinear decomposition that allows for arbitrary shifts along one modality the method is applied to neural activity arranged in the three modalities space time and trial thus the algorithm models neural activity as a linear superposition of components with a fixed time course that may vary across either trials or space in its overall intensity and latency its utility is demonstrated on simulated data as well as actual eeg and fmri data we show how shift invariant multilinear decompositions of multiway data can successfully cope with variable latencies in data derived from neural activity a problem that has caused degenerate solutions especially in modeling neuroimaging data with instantaneous multilinear decompositions our algorithm is available for download at www erpwavelab org

solving the eeg inverse problem based on space time frequency structured sparsity constraints abstract we introduce stout spatio temporal unifying tomography a novel method for the source analysis of electroencephalograpic eeg recordings which is based on a physiologically motivated source representation our method assumes that only a small number of brain sources are active throughout a measurement where each of the sources exhibits focal smooth but localized characteristics in space time and frequency this structure is enforced through an expansion of the source current density into appropriate spatio temporal basis functions in combination with sparsity constraints this approach combines the main strengths of two existing methods namely sparse basis field expansions haufe et al 2011 and time frequency mixed norm estimates gramfort et al 2013 by adjusting the ratio between two regularization terms stout is capable of trading temporal for spatial reconstruction accuracy and vice versa depending on the requirements of specific analyses and the provided data due to allowing for non stationary source activations stout is particularly suited for the localization of event related potentials erp and other evoked brain activity we demonstrate its performance on simulated erp data for varying signal to noise ratios and numbers of active sources our analysis of the generators of visual and auditory evoked n200 potentials reveals that the most active sources originate in the temporal and occipital lobes in line with the literature on sensory processing

tensor decompositions and applications this survey provides an overview of higher order tensor decompositions their applications and available software a tensor is a multidimensional or n way array decompositions of higher order tensors i e n way arrays with n geq 3 have applications in psycho metrics chemometrics signal processing numerical linear algebra computer vision numerical analysis data mining neuroscience graph analysis and elsewhere two particular tensor decompositions can be considered to be higher order extensions of the matrix singular value decomposition candecomp parafac cp decomposes a tensor as a sum of rank one tensors and the tucker decomposition is a higher order form of principal component analysis there are many other tensor decompositions including indscal parafac2 candelinc dedicom and paratuck2 as well as nonnegative variants of all of the above the n way toolbox tensor toolbox and multilinear engine are examples of software packages for working with tensors

optimal sparse singular value decomposition for high dimensional high order data in this article we consider the sparse tensor singular value decomposition which aims for dimension reduction on high dimensional high order data with certain sparsity structure a method named s

decomposing eeg data into space time frequency components using parallel factor analysis abstract finding the means to efficiently summarize electroencephalographic data has been a long standing problem in electrophysiology a popular approach is identification of component modes on the basis of the time varying spectrum of multichannel eeg recordings in other words a space frequency time atomic decomposition of the time varying eeg spectrum previous work has been limited to only two of these dimensions principal component analysis pca and independent component analysis ica have been used to create space time decompositions suffering an inherent lack of uniqueness that is overcome only by imposing constraints of orthogonality or independence of atoms conventional frequency time decompositions ignore the spatial aspects of the eeg framing of the data being as a three way array indexed by channel frequency and time allows the application of a unique decomposition that is known as parallel factor analysis parafac each atom is the tri linear decomposition into a spatial spectral and temporal signature we applied this decomposition to the eeg recordings of five subjects during the resting state and during mental arithmetic common to all subjects were two atoms with spectral signatures whose peaks were in the theta and alpha range these signatures were modulated by physiological state increasing during the resting stage for alpha and during mental arithmetic for theta furthermore we describe a new method source spectra imaging or ssi to estimate the location of electric current sources from the eeg spectrum the topography of the theta atom is frontal and the maximum of the corresponding ssi solution is in the anterior frontal cortex the topography of the alpha atom is occipital with maximum of the ssi solution in the visual cortex we show that the proposed decomposition can be used to search for activity with a given spectral and topographic profile in new recordings and that the method may be useful for artifact recognition and removal

propagation of smallness for an elliptic pde with piecewise lipschitz coefficients abstract in this paper we derive a propagation of smallness result for a scalar second elliptic equation in divergence form whose leading order coefficients are lipschitz continuous on two sides of a c 2 hypersurface that crosses the domain but may have jumps across this hypersurface our propagation of smallness result is in the most general form regarding the locations of domains which may intersect the interface of discontinuity at the end we also list some consequences of the propagation of smallness result including stability results for the associated cauchy problem a propagation of smallness result from sets of positive measure and a quantitative runge approximation property

three sphere inequality for second order elliptic equations with coefficients with jump discontinuity abstract this is a short note to complete the paper appeared in francini et al 2016 4 where a rough version of the classical well known hadamard three circle theorem for solution of an elliptic pde in divergence form has been proved precisely instead of circles the authors obtain a similar inequality in a more complicated geometry in this paper we clean the geometry and obtain a generalized version of the three circle inequality for elliptic equation with coefficients with discontinuity of jump type

carleman estimate for elliptic operators with coefficients with jumps at an interface in arbitrary dimension and application to the null controllability of linear parabolic equations in a bounded domain of r n 1 n geq 2 we consider a second order elliptic operator a d x 0 2 nabla x cdot c x nabla x where the scalar coefficient c x is piecewise smooth yet discontinuous across a smooth interface s we prove a local carleman estimate for a in the nhd of any point of the interface the observation region can be chosen independently of the sign of the jump of the coefficient c at the considered point the derivation of this estimate relies on the separation of the problem into three microlocal regions and the calderon projector technique following the method of lebeau and robbiano cite lr 95 we then prove the null controllability for the linear parabolic initial problem with dirichlet boundary conditions associated to the operator d t nabla x cdot c x nabla x

local and global carleman estimates for parabolic operators with coefficients with jumps at interfaces in 0 t open subset of n n 2 we consider a parabolic operator p t x t x x where the scalar coefficient t x is piecewise smooth in space yet discontinuous across a smooth interface s we prove a global in time local in space carleman estimate for p in the neighborhood of any point of the interface the observation region can be chosen independently of the sign of the jump of the coefficient at the considered point the derivation of this estimate relies on the separation of the problem into three microlocal regions related to high and low tangential frequencies at the interface in the high frequency regime we use calderon projectors in the low frequency regime we follow a more classical approach because of the parabolic nature of the problem we need to introduce weyl hormander anisotropic metrics symbol classes and pseudo differential operators each frequency regime and the associated technique require a different calculus a global in time and space carleman estimate on 0 t m m a manifold is also derived from the local result

entropic aspects of random fields on trees the existence of the entropy rate of shift invariant random fields on binary trees is proven alternative representations of and bounds for the entropy rate and surface entropy rate are obtained in terms of conditional entropy particular emphasis is placed on markov chain fields on trees explicit results are obtained some of which extend to a more complicated class of tree models

universal tree source coding using grammar based compression the problem of universal source coding for binary trees is considered zhang yang and kieffer derived upper bounds on the average case redundancy of codes based on directed acyclic graph dag compression for binary tree sources with certain properties in this paper a natural class of binary tree sources is presented such that the demanded properties are fulfilled moreover for both subclasses considered in the paper of zhang yang and kieffer their result is improved by deriving bounds on the maximal pointwise redundancy or worst case redundancy instead of the average case redundancy finally using context free tree grammars instead of dags upper bounds on the maximal pointwise redundancy for certain binary tree sources are derived this yields universal codes for new classes of binary tree sources

the error exponent for the noiseless encoding of finite ergodic markov sources a new approach to the classical fixed length noiseless source coding problem is proposed for the case of finite ergodic markov sources this approach is based on simple counting arguments the central notion of markov type a set containing all the source sequences having the same transition counts from letter to letter is introduced and the cardinality of such a set is evaluated via graph theoretical tools the error exponent is shown to be a weighted average of informational divergences and the universal character of the result is stressed as a corollary the classical source coding theorem determining the achievable rates is rederived

fitting time series models to nonstationary processes a general minimum distance estimation procedure is presented for nonstationary time series models that have an evolutionary spectral representation the asymptotic properties of the estimate are derived under the assumption of possible model misspecification for autoregressive processes with time varying coefficients the estimate is compared to the least squares estimate furthermore the behavior of estimates is explained when a stationary model is fitted to a nonstationary process

mr ntd manifold regularization nonnegative tucker decomposition for tensor data dimension reduction and representation with the advancement of data acquisition techniques tensor multidimensional data objects are increasingly accumulated and generated for example multichannel electroencephalographies multiview images and videos in these applications the tensor objects are usually nonnegative since the physical signals are recorded as the dimensionality of tensor objects is often very high a dimension reduction technique becomes an important research topic of tensor data from the perspective of geometry high dimensional objects often reside in a low dimensional submanifold of the ambient space in this paper we propose a new approach to perform the dimension reduction for nonnegative tensor objects our idea is to use nonnegative tucker decomposition ntd to obtain a set of core tensors of smaller sizes by finding a common set of projection matrices for tensor objects to preserve geometric information in tensor data we employ a manifold regularization term for the core tensors constructed in the tucker decomposition an algorithm called manifold regularization ntd mr ntd is developed to solve the common projection matrices and core tensors in an alternating least squares manner the convergence of the proposed algorithm is shown and the computational complexity of the proposed method scales linearly with respect to the number of tensor objects and the size of the tensor objects respectively these theoretical results show that the proposed algorithm can be efficient extensive experimental results have been provided to further demonstrate the effectiveness and efficiency of the proposed mr ntd algorithm

simultaneous estimation and testing of sources in multiple meg data sets the proposed extended couple dipole model ecdm is a trilinear component model that can be used to analyze multiple related meg data sets simultaneously related meg data sets are data sets that contain activity of the same sources or activity of sources that have proportional source amplitudes the simultaneous model uses a set of common sources and a set of common source time functions wave shapes to model the measured data in each data set the set of common sources contains all sources that are active in at least one of the data sets to be analyzed the number of common spatial and temporal components is specified by the user the model for each data set is a linear combination of these common spatial and temporal components this linear combination is estimated in a coupling matrix unlike the coupled dipole model where the user selects certain entries of the coupling matrix to be zero the entire coupling matrix is estimated in the ecdm this yields a more objective and statistically transparent estimation method of which the identifiability constraints do not depend on the user s chosen design as in the cdm crame spl grave r rao bounds are derived for the ecdm and the significance of the estimated source activity is computed and illustrated by confidence regions around estimated source time functions

entropies and combinatorics of random branching processes and context free languages the entropies and combinatorics of trees that branch according to fixed but finite numbers of rules are studied context free grammars are used to categorize the ways in which nodes branch to yield daughter nodes thus providing an organized setting to examine the entropies for random branching processes whose realizations are trees and whose probabilities are determined by probabilities associated with the substitution rules of the grammar normalized entropy rates h are derived for the critical branching rate and supercritical branching rate processes an equipartition theorem is proved for the supercritical processes a strong departure from classical theorems for markov sources occurs for supercritical branching processes as the typical sets have supergeometric growth rates the combinatorics of the set of all trees that can be generated from the context free substitution rules is also studied

two volume integral equations for the inhomogeneous and anisotropic forward problem in electroencephalography abstract this work presents two new volume integral equations for the electroencephalography eeg forward problem which differently from the standard integral approaches in the domain can handle heterogeneities and anisotropies of the head brain conductivity profiles the new formulations translate to the quasi static regime some volume integral equation strategies that have been successfully applied to high frequency electromagnetic scattering problems this has been obtained by extending to the volume case the two classical surface integral formulations used in eeg imaging and by introducing an extra surface equation in addition to the volume ones to properly handle boundary conditions numerical results corroborate theoretical treatments showing the competitiveness of our new schemes over existing techniques and qualifying them as a valid alternative to differential equation based methods