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

extracting low dimensional latent structure from time series in the presence of delays noisy high dimensional time series observations can often be described by a set of low dimensional latent variables commonly used methods to extract these latent variables typically assume instantaneous relationships between the latent and observed variables in many physical systems changes in the latent variables manifest as changes in the observed variables after time delays techniques that do not account for these delays can recover a larger number of latent variables than are present in the system thereby making the latent representation more difficult to interpret in this work we introduce a novel probabilistic technique time delay gaussian process factor analysis td gpfa that performs dimensionality reduction in the presence of a different time delay between each pair of latent and observed variables we demonstrate how using a gaussian process to model the evolution of each latent variable allows us to tractably learn these delays over a continuous domain additionally we show how td gpfa combines temporal smoothing and dimensionality reduction into a common probabilistic framework we present an expectation conditional maximization either ecme algorithm to learn the model parameters our simulations demonstrate that when time delays are present td gpfa is able to correctly identify these delays and recover the latent space we then applied td gpfa to the activity of tens of neurons recorded simultaneously in the macaque motor cortex during a reaching task td gpfa is able to better describe the neural activity using a more parsimonious latent space than gpfa a method that has been used to interpret motor cortex data but does not account for time delays more broadly td gpfa can help to unravel the mechanisms underlying high dimensional time series data by taking into account physical delays in the system

cartan decomposition subgroups of 2 for g sl 3 r and g so 2 n we give explicit practical conditions that determine whether or not a closed connected subgroup h of g has the property that there exists a compact subset c of g with chc g to do this we fix a cartan decomposition g ka k of g and then carry out an approximate calculation of khk n a for each closed connected subgroup h of g

iterative tree automata abstract the iterative tree automaton is introduced as a binary tree connected network with sequential input and output at the root of the tree the real and linear time computational power of this type of systolic system as a language acceptor is studied it is shown that for real time computations the arity of the tree is essential while this is not the case for linear time computations our main result is that every t n time nondeterministic turing machine can be simulated by an ita in deterministic ct n time a number of properties of real and linear time ita are proved

finiteness of the basic intersection cohomology of a killing foliation we prove that the basic intersection cohomology mathbb h overline p m mathcal f where mathcal f is the singular foliation determined by an isometric action of a lie group g on the compact manifold m is finite dimensional

linked independent component analysis for multimodal data fusion in recent years neuroimaging studies have increasingly been acquiring multiple modalities of data and searching for task or disease related changes in each modality separately a major challenge in analysis is to find systematic approaches for fusing these differing data types together to automatically find patterns of related changes across multiple modalities when they exist independent component analysis ica is a popular unsupervised learning method that can be used to find the modes of variation in neuroimaging data across a group of subjects when multimodal data is acquired for the subjects ica is typically performed separately on each modality leading to incompatible decompositions across modalities using a modular bayesian framework we develop a novel linked ica model for simultaneously modelling and discovering common features across multiple modalities which can potentially have completely different units signal and contrast to noise ratios voxel counts spatial smoothnesses and intensity distributions furthermore this general model can be configured to allow tensor ica or spatially concatenated ica decompositions or a combination of both at the same time linked ica automatically determines the optimal weighting of each modality and also can detect single modality structured components when present this is a fully probabilistic approach implemented using variational bayes we evaluate the method on simulated multimodal data sets as well as on a real data set of alzheimer s patients and age matched controls that combines two very different types of structural mri data morphological data grey matter density and diffusion data fractional anisotropy mean diffusivity and tensor mode

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

anechoic blind source separation using wigner marginals blind source separation problems emerge in many applications where signals can be modeled as superpositions of multiple sources many popular applications of blind source separation are based on linear instantaneous mixture models if specific invariance properties are known about the sources for example translation or rotation invariance the simple linear model can be extended by inclusion of the corresponding transformations when the sources are invariant against translations spatial displacements or time shifts the resulting model is called an anechoic mixing model we present a new algorithmic framework for the solution of anechoic problems in arbitrary dimensions this framework is derived from stochastic time frequency analysis in general and the marginal properties of the wigner ville spectrum in particular the method reduces the general anechoic problem to a set of anechoic problems with non negativity constraints and a phase retrieval problem the first type of subproblem can be solved by existing algorithms for example by an appropriate modification of non negative matrix factorization nmf the second subproblem is solved by established phase retrieval methods we discuss and compare implementations of this new algorithmic framework for several example problems with synthetic and real world data including music streams natural 2d images human motion trajectories and two dimensional shapes

efficient simulations of simple models of parallel computation by time bounded atms and space bounded tms abstract we present efficient time bounded atm and space bounded tm simulations of one way conglomerates owcs which are interconnected networks of finite state machines that allow only one way communication between adjacent nodes in particular we show that owcs with depth d n and operating in time t n can be simulated by atms in time o d n log t n and hence by a tm with the same amount of space this extends ruzzo s result that boolean circuits of depth d n can be simulated by o d n time bounded atms and refines gold schlager s result that two way conglomerates operating in t n time can be simulated by t n space bounded tms by exploiting the regularity of interconnections in some owcs we obtain more efficient space bounded tm simulations for example using the atm result a k dimensional one way mesh array of nk nodes would require n2 space on a tm such an array can run in cn time in the worst case we show that the 7space can be reduced to n 2 1 k

bilinear discriminant component analysis factor analysis and discriminant analysis are often used as complementary approaches to identify linear components in two dimensional data arrays for three dimensional arrays which may organize data in dimensions such as space time and trials the opportunity arises to combine these two approaches a new method bilinear discriminant component analysis bdca is derived and demonstrated in the context of functional brain imaging data for which it seems ideally suited the work suggests to identify a subspace projection which optimally separates classes while ensuring that each dimension in this space captures an independent contribution to the discrimination

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

triangle cellular automata this paper introduces a special class of cellular automata called triangle cellular automata which accept as input strings of length a power of two in particular we study the special case in which state information can only move upward through a complete binary tree of finite state automata it is shown that this class of cellular automata can accept various string languages in o log n time where n is the length of the input string defining the initial states of the leaf vertices in the tree extensions to two dimensions defining a pyramid cellular automaton are also given

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

higher order spectral regression discriminant analysis hosrda a tensor feature reduction method for erp detection abstract tensors are valuable tools to represent electroencephalogram eeg data tucker decomposition is the most used tensor decomposition in multidimensional discriminant analysis and tensor extension of linear discriminant analysis lda called higher order discriminant analysis hoda is a popular tensor discriminant method used for analyzing event related potentials erp in this paper we introduce a new tensor based feature reduction technique named higher order spectral regression discriminant analysis hosrda for use in a classification framework for erp detection the proposed method hosrda is a tensor extension of spectral regression discriminant analysis srda and casts the eigenproblem of hoda to a regression problem the formulation of hosrda can open a new framework for adding different regularization constraints in higher order feature reduction problem additionally when the dimension and number of samples is very large the regression problem can be solved via efficient iterative algorithms we applied hosrda on data of a p300 speller from bci competition iii and reached average character detection accuracy of 96 5 for the two subjects hosrda outperforms almost all of other reported methods on this dataset additionally the results of our method are fairly comparable with those of other methods when 5 and 10 repetitions are used in the p300 speller paradigm

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

degeneracy in candecomp parafac and indscal explained for several three sliced arrays with a two valued typical rank the candecomp parafac cp method decomposes a three way array into a prespecified number r of rank 1 arrays by minimizing the sum of squares of the residual array the practical use of cp is sometimes complicated by the occurrence of so called degenerate sequences of solutions in which several rank 1 arrays become highly correlated in all three modes and some elements of the rank 1 arrays become arbitrarily large we consider the real valued cp decomposition of all known three sliced arrays i e of size p q 3 with a two valued typical rank these are the 5 3 3 and 8 4 3 arrays and the 3 3 4 and 3 3 5 arrays with symmetric 3 3 slices in the latter two cases cp is equivalent to the indscal model for a typical rank of m m 1 we consider the cp decomposition with r m of an array of rank m 1 we show that in most cases the cp objective function does not have a minimum but an infimum moreover any sequence of feasible cp solutions in which the objective value approaches the infimum will become degenerate we use the tools developed in stegeman 2006 who considers p p 2 arrays and present a framework of analysis which is of use to the future study of cp degeneracy related to a two valued typical rank moreover our examples show that cp uniqueness is not necessary for degenerate solutions to occur

a regularized discriminative framework for eeg analysis with application to brain computer interface we propose a framework for signal analysis of electroencephalography eeg that unifies tasks such as feature extraction feature selection feature combination and classification which are often independently tackled conventionally under a regularized empirical risk minimization problem the features are automatically learned selected and combined through a convex optimization problem moreover we propose regularizers that induce novel types of sparsity providing a new technique for visualizing eeg of subjects during tasks from a discriminative point of view the proposed framework is applied to two typical bci problems namely the p300 speller system and the prediction of self paced finger tapping in both datasets the proposed approach shows competitive performance against conventional methods while at the same time the results are easier accessible to neurophysiological interpretation note that our novel approach is not only applicable to brain imaging beyond eeg but also to general discriminative modeling of experimental paradigms beyond bci

factorization strategies for third order tensors abstract operations with tensors or multiway arrays have become increasingly prevalent in recent years traditionally tensors are represented or decomposed as a sum of rank 1 outer products using either the candecomp parafac cp or the tucker models or some variation thereof such decompositions are motivated by specific applications where the goal is to find an approximate such representation for a given multiway array the specifics of the approximate representation such as how many terms to use in the sum orthogonality constraints etc depend on the application in this paper we explore an alternate representation of tensors which shows promise with respect to the tensor approximation problem reminiscent of matrix factorizations we present a new factorization of a tensor as a product of tensors to derive the new factorization we define a closed multiplication operation between tensors a major motivation for considering this new type of tensor multiplication is to devise new types of factorizations for tensors which can then be used in applications specifically this new multiplication allows us to introduce concepts such as tensor transpose inverse and identity which lead to the notion of an orthogonal tensor the multiplication also gives rise to a linear operator and the null space of the resulting operator is identified we extend the concept of outer products of vectors to outer products of matrices all derivations are presented for third order tensors however they can be easily extended to the order p p 3 case we conclude with an application in image deblurring

structure seeking multilinear methods for the analysis of fmri data in comprehensive fmri studies of brain function the data structures often contain higher order ways such as trial task condition subject and group in addition to the intrinsic dimensions of time and space while multivariate bilinear methods such as principal component analysis pca have been used successfully for extracting information about spatial and temporal features in data from a single fmri run the need to unfold higher order data sets into bilinear arrays has led to decompositions that are nonunique and to the loss of multiway linkages and interactions present in the data these additional dimensions or ways can be retained in multilinear models to produce structures that are unique and which admit interpretations that are neurophysiologically meaningful multiway analysis of fmri data from multiple runs of a bilateral finger tapping paradigm was performed using the parallel factor parafac model a trilinear model was fitted to a data cube of dimensions voxels by time by run similarly a quadrilinear model was fitted to a higher way structure of dimensions voxels by time by trial by run the spatial and temporal response components were extracted and validated by comparison to results from traditional svd pca analyses based on scenarios of unfolding into lower order bilinear structures

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

an information maximization approach to blind separation and blind deconvolution we derive a new self organizing learning algorithm that maximizes the information transferred in a network of nonlinear units the algorithm does not assume any knowledge of the input distributions and is defined here for the zero noise limit under these conditions information maximization has extra properties not found in the linear case linsker 1989 the nonlinearities in the transfer function are able to pick up higher order moments of the input distributions and perform something akin to true redundancy reduction between units in the output representation this enables the network to separate statistically independent components in the inputs a higher order generalization of principal components analysis we apply the network to the source separation or cocktail party problem successfully separating unknown mixtures of up to 10 speakers we also show that a variant on the network architecture is able to perform blind deconvolution cancellation of unknown echoes and reverberation in a speech signal finally we derive dependencies of information transfer on time delays we suggest that information maximization provides a unifying framework for problems in blind signal processing

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