spatiotemporal analysis of event related fmri data using partial least squares partial least squares pls has proven to be a important multivariate analytic tool for positron emission tomographic and more recently event related potential erp data the application to erp incorporates the ability to analyze space and time together a feature that has obvious appeal for event related functional magnetic resonance imaging fmri data this paper presents the extension of spatiotemporal pls st pls to fmri explaining the theoretical foundation and application to an fmri study of auditory and visual perceptual memory analysis of activation effects with st pls was compared with conventional univariate random effects analysis showing general consensus for both methods but several unique observations by st pls including enhanced statistical power the application of st pls for assessment of task dependent brain behavior relationships is also presented singular features of st pls include 1 no assumptions about the shape of the hemodynamic response functions hrfs 2 robust statistical assessment at the image level through permutation tests 3 protection against outlier influences at the voxel level through bootstrap resampling 4 flexible analytic configurations that allow assessment of activation difference brain behavior relations and functional connectivity these features enable st pls to act as an important complement to other multivariate and univariate approaches used in neuroimaging research

a pearson random walk with steps of uniform orientation and dirichlet distributed lengths a constrained diffusive random walk of n steps in d and a random flight in d which are equivalent were investigated independently in recent papers j stat phys 127 813 2007 j theor probab 20 769 2007 and j stat phys 131 1039 2008 the n steps of the walk are independent and identically distributed random vectors of exponential length and uniform orientation conditioned on the sum of their lengths being equal to a given value l closed form expressions for the distribution of the endpoint of the walk were obtained altogether for any n for d 1 2 4 uniform distributions of the endpoint inside a ball of radius l were evidenced for a walk of three steps in 2d and of two steps in 4d

a new family of solvable pearson dirichlet random walks an n step pearson gamma random walk in d starts at the origin and consists of n independent steps with gamma distributed lengths and uniform orientations the gamma distribution of each step length has a shape parameter q 0 constrained random walks of n steps in d are obtained from the latter walks by imposing that the sum of the step lengths is equal to a fixed value simple closed form expressions were obtained in particular for the distribution of the endpoint of such constrained walks for any d d0 and any n 2 when q is either q frac d 2 1 d0 3 or q d 1 d0 2 le caer in j stat phys 140 728 751 2010 when the total walk length is chosen without loss of generality to be equal to 1 then the constrained step lengths have a dirichlet distribution whose parameters are all equal to q and the associated walk is thus named a pearson dirichlet random walk the density of the endpoint position of a n step planar walk of this type n 2 with q d 2 was shown recently to be a weighted mixture of 1 floor n 2 endpoint densities of planar pearson dirichlet walks with q 1 beghin and orsingher in stochastics 82 201 229 2010 the previous result is generalized to any walk space dimension and any number of steps n 2 when the parameter of the pearson dirichlet random walk is q d 1 we rely on the connection between an unconstrained random walk and a constrained one which have both the same n and the same q d to obtain a closed form expression of the endpoint density the latter is a weighted mixture of 1 floor n 2 densities with simple forms equivalently expressed as a product of a power and a gauss hypergeometric function the weights are products of factors which depends both on d and n and bessel numbers independent of d

the ubiquitous kronecker product abstract the kronecker product has a rich and very pleasing algebra that supports a wide range of fast elegant and practical algorithms several trends in scientific computing suggest that this important matrix operation will have an increasingly greater role to play in the future first the application areas where kronecker products abound are all thriving these include signal processing image processing semidefinite programming and quantum computing second sparse factorizations and kronecker products are proving to be a very effective way to look at fast linear transforms researchers have taken the kronecker methodology as developed for the fast fourier transform and used it to build exciting alternatives third as computers get more powerful researchers are more willing to entertain problems of high dimension and this leads to kronecker products whenever low dimension techniques are tensored together

bayesian robust tensor factorization for incomplete multiway data we propose a generative model for robust tensor factorization in the presence of both missing data and outliers the objective is to explicitly infer the underlying low candecomp parafac cp rank tensor capturing the global information and a sparse tensor capturing the local information also considered as outliers thus providing the robust predictive distribution over missing entries the low cp rank tensor is modeled by multilinear interactions between multiple latent factors on which the column sparsity is enforced by a hierarchical prior while the sparse tensor is modeled by a hierarchical view of student t distribution that associates an individual hyperparameter with each element independently for model learning we develop an efficient variational inference under a fully bayesian treatment which can effectively prevent the overfitting problem and scales linearly with data size in contrast to existing related works our method can perform model selection automatically and implicitly without the need of tuning parameters more specifically it can discover the groundtruth of cp rank and automatically adapt the sparsity inducing priors to various types of outliers in addition the tradeoff between the low rank approximation and the sparse representation can be optimized in the sense of maximum model evidence the extensive experiments and comparisons with many state of the art algorithms on both synthetic and real world data sets demonstrate the superiorities of our method from several perspectives

random motions at finite speed in higher dimensions we present a general method of studying the transport process bold x t t 0 in the euclidean space m m 2 based on the analysis of the integral transforms of its distributions we show that the joint characteristic functions of bold x t are connected with each other by a convolution type recurrent relation this enables us to prove that the characteristic function fourier transform of bold x t in any dimension m 2 satisfies a convolution type volterra integral equation of second kind we give its solution and obtain the characteristic function of bold x t in terms of the multiple convolutions of the kernel of the equation with itself an explicit form of the laplace transform of the characteristic function in any dimension is given the complete solution of the problem of finding the initial conditions for the governing partial differential equations is given

exponential convergence to quasi stationary distribution and q process for general almost surely absorbed markov processes we obtain necessary and sufficient conditions for exponential convergence to a unique quasi stationary distribution in the total variation norm these conditions also ensure the existence and exponential ergodicity of the q process the process conditioned to never be absorbed we apply these results to one dimensional birth and death processes with catastrophes multi dimensional birth and death processes infinite dimensional population models with brownian mutations and neutron transport dynamics absorbed at the boundary of a bounded domain

an approach to n mode components analysis as an extension of lastovicka s four mode components analysis ann mode components analysis is developed using a convenient notation both a canonical and a least squares solution are derived the relation between both solutions and their computational aspects are discussed

uniqueness of three mode factor models with sparse cores the 3 x 3 x 3 case three mode factor analysis 3mfa and parafac are methods to describe three way data both methods employ models with components for the three modes of a three way array the 3mfa model also uses a three way core array for linking all components to each other the use of the core array makes the 3mfa model more general than the parafac model thus allowing a better fit but also more complicated moreover in the 3mfa model the components are not uniquely determined and it seems hard to choose among all possible solutions a particularly interesting feature of the parafac model is that it does give unique components the present paper introduces a class of 3mfa models in between 3mfa and parafac that share the good properties of the 3mfa model and the parafac model they fit almost as well as the 3mfa model they are relatively simple and they have the same uniqueness properties as the parafac model

residence time and collision statistics for exponential flights the rod problem revisited many random transport phenomena such as radiation propagation chemical biological species migration or electron motion can be described in terms of particles performing exponential flights for such processes we sketch a general approach based on the feynman kac formalism that is amenable to explicit expressions for the moments of the number of collisions and the residence time that the walker spends in a given volume as a function of the particle equilibrium distribution we then illustrate the proposed method in the case of the so called rod problem a one dimensional system and discuss the relevance of the obtained results in the context of monte carlo estimators

finite dimensional representations of periplectic lie superalgebras abstract we study categories of finite dimensional modules over the periplectic lie superalgebras and obtain a bgg type reciprocity in particular we prove that these categories have only finitely many blocks we also compute the characters for irreducible modules over periplectic lie superalgebras of ranks 2 and 3 and obtain explicit description of the blocks for ranks 2 3 and 4

exploring transient transfer entropy based on a group wise ica decomposition of eeg data this paper presents a data driven pipeline for studying asymmetries in mutual interdependencies between distinct components of eeg signal due to volume conductance estimating coherence between scalp electrodes may lead to spurious results a group based independent component analysis ica which is conducted across all subjects and conditions simultaneously is an alternative representation of the eeg measurements within this approach the extracted components are independent in a global sense while short lived or transient interdependencies may still be present between the components in this paper functional roles of the ica components are specified through a partial least squares pls analysis of task effects within the time course of the derived components functional integration is estimated within the information theoretic approach using transfer entropy analysis based on asymmetries in mutual interdependencies of reconstructed phase dynamics a secondary pls analysis is performed to assess robust task specific changes in transfer entropy estimates between functionally specific components

random walk models for search with particular reference to a bounded region abstract the position of a target moving randomly in an area is described the motion is a sequence of finite legs traversed at constant speed with exponentially distributed lengths and randomly oriented directions with respect to each other following kluyver 7 a new expression is derived for the distribution of the radius vector where the walk is of pearson type the relative directions of successive legs being u 0 2 this is followed by a study of such a walk within a circular area the purpose here is to investigate the effect of reflection policies at the boundary on the distribution of the position of the target within the area and on the coverage of the area by the walk

linking functional and structural brain images with multivariate network analyses a novel application of the partial least square method abstract in this article we introduce a multimodal multivariate network analysis to characterize the linkage between the patterns of information from the same individual s complementary brain images and illustrate its potential by showing its ability to distinguish older from younger adults with greater power than several previously established methods our proposed method uses measurements from every brain voxel in each person s complementary co registered images and uses the partial least square pls algorithm to form a combined latent variable that maximizes the covariance among all of the combined variables it represents a new way to calculate the singular value decomposition from the high dimensional covariance matrix in a computationally feasible way analyzing fluorodeoxyglucose positron emission tomography pet and volumetric magnetic resonance imaging mri images this method distinguished 14 older adults from 15 younger adults p 4e 12 with no overlap between groups no need to correct for multiple comparisons and greater power than the univariate statistical parametric mapping spm multimodal spm or multivariate pls analysis of either imaging modality alone this technique has the potential to link patterns of information among any number of complementary images from an individual to use other kinds of complementary complex datasets besides brain images and to characterize individual state or trait dependent brain patterns in a more powerful way

on uniqueness in candecomp parafac one of the basic issues in the analysis of three way arrays by candecomp parafac cp has been the question of uniqueness of the decomposition kruskal 1977 has proved that uniqueness is guaranteed when the sum of thek ranks of the three component matrices involved is at least twice the rank of the solution plus 2 since then little has been achieved that might further qualify kruskal s sufficient condition attempts to prove that it is also necessary for uniqueness except for rank 1 or 2 have failed but counterexamples to necessity have not been detected the present paper gives a method for generating the class of all solutions or at least a subset of that class given a cp solution that satisfies certain conditions this offers the possibility to examine uniqueness for a great variety of specific cp solutions it will be shown that kruskal s condition is necessary and sufficient when the rank of the solution is three but that uniqueness may hold even if the condition is not satisfied when the rank is four or higher

collision statistics for random flights with anisotropic scattering and absorption for a broad class of random walks with anisotropic scattering kernels and absorption we derive explicit formulas that allow expressing the moments of the collision number n v performed in a volume v as a function of the particle equilibrium distribution our results apply to arbitrary domains v and boundary conditions and allow assessing the hitting statistics for systems where the typical displacements are comparable to the domain size so that the diffusion limit is possibly not attained an example is discussed for one dimensional random flights with exponential displacements where analytical calculations can be carried out

ica based artifact correction improves spatial localization of adaptive spatial filters in meg abstract beamformers are one of the most common inverse models currently used in the estimation of source activity from magnetoencephelography meg data they rely on a minimization of total power while constraining the gain in the voxel of interest resulting in the suppression of background noise nonetheless in cases where background noise is strong compared to the source of interest or when many sources are present the ability of the beamformer to detect and accurately localize weak sources is reduced in visual paradigms two main background sources can substantially impact an accurate estimation of weaker sources ocular artifacts are orders of magnitude higher than neural sources making it difficult for the beamformer to effectively suppress them primary visual activations also result in strong signals that can impede localization of weak sources in this paper we systematically evaluated how neural visual and non neural eye heart sources affect the localization accuracy of frontal and medial temporal sources in visual tasks these sources are of tremendous interest in learning and memory studies as well as in clinical settings alzheimer s epilepsy and are typically difficult to localize robustly in meg empirical data from two tasks active learning and control were used to evaluate our analysis techniques global field power calculations showed multiple time periods where active learning was significantly different from response selection with dominant sources converging to the eyes extensive leakage of eye activity into frontal and visual that evoked responses into parietal cortices was also observed contributions from ocular activity to the reconstructed time series were indiscernible from task based recruitment of frontal sources in the original data removing artifacts eye movements cardiac and muscular by means of independent component analysis ica led to a significant improvement in detection and localization of frontal and medial temporal sources we verified our results by using simulations of sources placed in frontal and medial temporal regions with various types of background noise eye heart and visual we report that the detection and localization accuracy of frontal and medial temporal sources with beamformer techniques is highly dependent on the magnitude and location of background sources and that removing artifacts can substantially improve the beamformer s performance

symmetric lie superalgebras and deformed quantum calogero moser problems abstract the representation theory of symmetric lie superalgebras and corresponding spherical functions are studied in relation with the theory of the deformed quantum calogero moser systems in the special case of symmetric pair g gl n 2 m k osp n 2 m we establish a natural bijection between projective covers of spherically typical irreducible g modules and the finite dimensional generalised eigenspaces of the algebra of calogero moser integrals d n m acting on the corresponding laurent quasi invariants a n m

greater robustness of second order statistics than higher order statistics algorithms to distortions of the mixing matrix in blind source separation of human eeg implications for single subject and group analyses a mandatory assumption in blind source separation bss of the human electroencephalogram eeg is that the mixing matrix remains invariant i e that the sources electrodes and geometry of the head do not change during the experiment actually this is not often the case for instance it is common that some electrodes slightly move during eeg recording this issue is even more critical for group independent component analysis gica a method of growing interest in which only one mixing matrix is estimated for several subjects indeed because of interindividual anatomo functional variability this method violates the mandatory principle of invariance here using simulated experiments 1 and 2 and real experiment 3 eeg data we test how eleven current bss algorithms undergo distortions of the mixing matrix we show that this usual kind of perturbation creates non gaussian features that are virtually added to all sources impairing the estimation of real higher order statistics hos features of the actual sources by hos algorithms e g ext infomax fastica hos based methods are likely to identify more components with similar properties than actual neurological sources a problem frequently encountered by bss users in practice the quality of the recovered signal and the efficiency of subsequent source localization are substantially impaired performing dimensionality reduction before applying hos based bss does not seem to be a safe strategy to circumvent the problem second order statistics sos based bss methods belonging to the less popular sobi family class are much less sensitive to this bias

characterizing the response of pet and fmri data using multivariate linear models this paper presents a new method for characterizing brain responses in both pet and fmri data the aim is to capture the correlations between the scans of an experiment and a set of external predictor variables that are thought to affect the scans such as type intensity or shape of stimulus response its main feature is a canonical variates analysis cva of the estimated effects of the predictors from a multivariate linear model mlm the advantage of this over current methods is that temporal correlations can be incorporated into the model making the mlm method suitable for fmri as well as pet data moreover tests for the presence of any correlation and inference about the number of canonical variates needed to capture that correlation can be based on standard multivariate statistics rather than simulations when applied to an fmri data set previously analyzed by another cva method the mlm method reveals a pattern of responses that is closer to that detected in an earlier non cva analysis

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

explicit candecomp parafac solutions for a contrived 2 2 2 array of rank three kruskal harshman and lundy have contrived a special 2 2 2 array to examine formal properties of degenerate candecomp parafac solutions it is shown that for this array the candecomp parafac loss has an infimum of 1 in addition the array will be used to challenge the tradition of fitting indscal and related models by means of the candecomp parafac process

tensor envelope partial least squares regression abstractpartial least squares pls is a prominent solution for dimension reduction and high dimensional regressions recent prevalence of multidimensional tensor data has led to several tensor versions of the pls algorithms however none offers a population model and interpretation and statistical properties of the associated parameters remain intractable in this article we first propose a new tensor partial least squares algorithm then establish the corresponding population interpretation this population investigation allows us to gain new insight on how the pls achieves effective dimension reduction to build connection with the notion of sufficient dimension reduction and to obtain the asymptotic consistency of the pls estimator we compare our method both analytically and numerically with some alternative solutions we also illustrate the efficacy of the new method on simulations and two neuroimaging data analyses supplementary materials for this article are available online

a case of extreme simplicity of the core matrix in three mode principal components analysis in three mode principal components analysis thep q r core matrixg can be transformed to simple structure before it is interpreted it is well known that whenp qr g can be transformed to the identity matrix which implies that all elements become equal to values specified a priori in the present paper it is shown that whenp qr 1 g can be transformed to have nearly all elements equal to values spectified a priori a cllsed form solution for this transformation is offered theoretical and practical implications of this simple structure transformation ofg are discussed

a classification of multiplicity free actions abstract the action of a complex algebraic groupgon an affine varietyvis said to be multiplicity free if each irreducible representation ofgoccurs at most once in the ring c v of regular functions onv this paper concerns the classical setting wherevis a vector space andgacts linearly onv the irreducible linear multiplicity free actions have been classified by v kac the focus here is on nonirreducible linear multiplicity free actions we exhibit many such actions which do not decompose as direct products of irreducible actions our main results provide a classification for all linear multiplicity free actions on a vector space

the capelli problem for gl m n and the spectrum of invariant differential operators abstract the capelli problem for the symmetric pairs gl gl gl gl o and gl sp is closely related to the theory of jack polynomials and shifted jack polynomials for special values of the parameter see 12 15 14 18 in this paper we extend this connection to the lie superalgebra setting namely to the supersymmetric pairs g k gl m 2 n osp m 2 n and gl m n gl m n gl m n acting on w s 2 c m 2 n and c m n c m n to achieve this goal we first prove that the center of the universal enveloping algebra of the lie superalgebra g maps surjectively onto the algebra pd w g of g invariant differential operators on the superspace w thereby providing an affirmative answer to the abstract capelli problem for w our proof works more generally for gl m n acting on s 2 c m n and is new even for the ordinary cases m 0 or n 0 considered by howe and umeda in 9 we next describe a natural basis d of pd w g that we call the capelli basis using the above result on the abstract capelli problem we generalize the work of kostant and sahi 12 15 20 by showing that the spectrum of d is given by a polynomial c which is characterized uniquely by certain vanishing and symmetry properties we further show that the top homogeneous parts of the eigenvalue polynomials c coincide with the spherical polynomials d which arise as radial parts of k spherical vectors of finite dimensional g modules and which are super analogues of jack polynomials this generalizes results of knop and sahi 14 finally we make a precise connection between the polynomials c and the shifted super jack polynomials of sergeev and veselov 25 for special values of the parameter we show that the two families are related by a change of coordinates that we call the frobenius transform

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

optimizing methods for linking cinematic features to fmri data abstract one of the challenges of naturalistic neurosciences using movie viewing experiments is how to interpret observed brain activations in relation to the multiplicity of time locked stimulus features as previous studies have shown less inter subject synchronization across viewers of random video footage than story driven films new methods need to be developed for analysis of less story driven contents to optimize the linkage between our fmri data collected during viewing of a deliberately non narrative silent film at land by maya deren 1944 and its annotated content we combined the method of elastic net regularization with the model driven linear regression and the well established data driven independent component analysis ica and inter subject correlation isc methods in the linear regression analysis both ic and region of interest roi time series were fitted with time series of a total of 36 binary valued and one real valued tactile annotation of film features the elastic net regularization and cross validation were applied in the ordinary least squares linear regression in order to avoid over fitting due to the multicollinearity of regressors the results were compared against both the partial least squares pls regression and the un regularized full model regression non parametric permutation testing scheme was applied to evaluate the statistical significance of regression we found statistically significant correlation between the annotation model and 9 ics out of 40 ics regression analysis was also repeated for a large set of cubic rois covering the grey matter both ic and roi based regression analyses revealed activations in parietal and occipital regions with additional smaller clusters in the frontal lobe furthermore we found elastic net based regression more sensitive than pls and un regularized regression since it detected a larger number of significant ics and rois along with the isc ranking methods our regression analysis proved a feasible method for ordering the ics based on their functional relevance to the annotated cinematic features the novelty of our method is in comparison to the hypothesis driven manual pre selection and observation of some individual regressors biased by choice in applying data driven approach to all content features simultaneously we found especially the combination of regularized regression and ica useful when analyzing fmri data obtained using non narrative movie stimulus with a large set of complex and correlated features

counting statistics a feynman kac perspective by building upon a feynman kac formalism we assess the distribution of the number of collisions in a given region for a broad class of discrete time random walks in absorbing and nonabsorbing media we derive the evolution equation for the generating function of the number of collisions and we complete our analysis by examining the moments of the distribution and their relation to the walker equilibrium density some significant applications are discussed in detail in particular we revisit the gambler s ruin problem and generalize to random walks with absorption the arcsine law for the number of collisions on the half line

groupwise independent component decomposition of eeg data and partial least square analysis this paper focuses on two methodological developments for analysis of neuroimaging data the first is the derivation of robust spatiotemporal activity patterns across a group of subjects using a combination of principal component analysis pca and independent component analysis ica in applications to erp data the space dimension is typically represented in terms of scalp electrodes the signal recorded by high density electrode caps is known to be highly correlated due in part to volume conduction consequently this redundancy is also reflected in spatiotemporal patterns characterizing signal differences across experimental conditions we present an alternative spatial representation and signal compression based on pca for dimensionality reduction and ica conducted across all subjects and conditions simultaneously the second advancement is the use of partial least squares pls analysis to assess task dependent changes in the expression of the independent components in an application to empirical erp data we derive an efficient number of independent component maps comparative pls analysis on the independent components versus original electrode data shows that task effects are not only preserved under compression but also enhanced statistically

spatiotemporal nonlinearity in resting state fmri of the human brain in this work the spatiotemporal nonlinearity in resting state fmri data of the human brain was detected by nonlinear dynamics methods nine human subjects during resting state were imaged using single shot gradient echo planar imaging on a 1 5t scanner eigenvalue spectra for the covariance matrix correlation dimensions and spatiotemporal lyapunov exponents were calculated to detect the spatiotemporal nonlinearity in resting state fmri data by simulating adjusting and comparing the eigenvalue spectra of pure correlated noise with the corresponding real fmri data the intrinsic dimensionality was estimated the intrinsic dimensionality was used to extract the first few principal components from the real fmri data using principal component analysis which will preserve the correct phase dynamics while reducing both computational load and noise level of the data then the phase space was reconstructed using the time delay embedding method for their principal components and the correlation dimension was estimated by the grassberger procaccia algorithm of multiple variable series the spatiotemporal lyapunov exponents were calculated by using the method based on coupled map lattices through nonlinearity testing there are significant differences of correlation dimensions and spatiotemporal lyapunov exponents between fmri data and their surrogate data the fractal dimension and the positive spatiotemporal lyapunov exponents characterize the spatiotemporal nonlinear dynamics property of resting state fmri data therefore the results suggest that fluctuations presented in resting state may be an inherent model of basal neural activation of human brain cannot be fully attributed to noise

sufficient conditions for uniqueness in candecomp parafac and indscal with random component matrices a key feature of the analysis of three way arrays by candecomp parafac is the essential uniqueness of the trilinear decomposition we examine the uniqueness of the candecomp parafac and indscal decompositions in the latter the array to be decomposed has symmetric slices we consider the case where two component matrices are randomly sampled from a continuous distribution and the third component matrix has full column rank in this context we obtain almost sure sufficient uniqueness conditions for the candecomp parafac and indscal models separately involving only the order of the three way array and the number of components in the decomposition both uniqueness conditions are closer to necessity than the classical uniqueness condition by kruskal

spherical indecomposable representations of lie superalgebras abstract we present a classification of all spherical indecomposable representations of classical and exceptional lie superalgebras we also include information about stabilizers symmetric algebras and borels for which sphericity is achieved in one such computation the symmetric algebra of the standard module of osp m 2 n is computed which in particular gives the representation theoretic structure of polynomials on the complex supersphere

numerical operator calculus in higher dimensions when an algorithm in dimension one is extended to dimension d in nearly every case its computational cost is taken to the power d this fundamental difficulty is the single greatest impediment to solving many important problems and has been dubbed the curse of dimensionality for numerical analysis in dimension d we propose to use a representation for vectors and matrices that generalizes separation of variables while allowing controlled accuracy basic linear algebra operations can be performed in this representation using one dimensional operations thus bypassing the exponential scaling with respect to the dimension although not all operators and algorithms may be compatible with this representation we believe that many of the most important ones are we prove that the multiparticle schrodinger operator as well as the inverse laplacian can be represented very efficiently in this form we give numerical evidence to support the conjecture that eigenfunctions inherit this property by computing the ground state eigenfunction for a simplified schrodinger operator with 30 particles we conjecture and provide numerical evidence that functions of operators inherit this property in which case numerical operator calculus in higher dimensions becomes feasible

detecting spatiotemporal nonlinear dynamics in resting state of human brain based on fmri datasets in this work a nonlinear dynamics method coupled map lattices was applied to functional magnetic resonance imaging fmri datasets to examine the spatiotemporal properties of resting state blood oxygen level dependent bold fluctuations spatiotemporal lyapunov exponent sple was calculated to study the deterministic nonlinearity in resting state human brain of nine subjects based on fmri datasets the results show that there is nonlinearity and determinism in resting state human brain furthermore the results demonstrate that there is a spatiotemporal chaos phenomenon in resting state brain and suggest that fluctuations of fmri data in resting state brain cannot be fully attributed to nuclear magnetic resonance noise at the same time the spatiotemporal chaos phenomenon suggests that the correlation between voxels varies with time and there is a dynamic functional connection or network in resting state human brain

two step dirichlet random walks abstract random walks of n steps taken into independent uniformly random directions in a d dimensional euclidean space d 2 which are characterized by a sum of step lengths which is fixed and taken to be 1 without loss of generality are named dirichlet when this constraint is realized via a dirichlet law of step lengths the latter continuous multivariate distribution which depends on n positive parameters generalizes the beta distribution n 2 it is simply obtained from n independent gamma random variables with identical scale factors previous literature studies of these random walks dealt with symmetric dirichlet distributions whose parameters are all equal to a value q which takes half integer or integer values in the present work the probability density function of the distance from the endpoint to the origin is first made explicit for a symmetric dirichlet random walk of two steps it is valid for any positive value of q and for all d 2 the latter pdf is used in turn to express the related density of a random walk of two steps whose step length is distributed according to an asymmetric beta distribution which depends on two parameters namely q and q s where s is a positive integer

the diffusion limit of transport equations derived from velocity jump processes in this paper we study the diffusion approximation to a transport equation that describes the motion of individuals whose velocity changes are governed by a poisson process we show that under an appropriate scaling of space and time the asymptotic behavior of solutions of such equations can be approximated by the solution of a diffusion equation obtained via a regular perturbation expansion in general the resulting diffusion tensor is anisotropic and we give necessary and sufficient conditions under which it is isotropic we also give a method to construct approximations of arbitrary high order for large times in a second paper part ii we use this approach to systematically derive the limiting equation under a variety of external biases imposed on the motion depending on the strength of the bias it may lead to an anisotropic diffusion equation to a drift term in the flux or to both our analysis generalizes and simplifies previous derivations that lead to the classical patlak keller segel alt

measure projection analysis a probabilistic approach to eeg source comparison and multi subject inference a crucial question for the analysis of multi subject and or multi session electroencephalographic eeg data is how to combine information across multiple recordings from different subjects and or sessions each associated with its own set of source processes and scalp projections here we introduce a novel statistical method for characterizing the spatial consistency of eeg dynamics across a set of data records measure projection analysis mpa first finds voxels in a common template brain space at which a given dynamic measure is consistent across nearby source locations then computes local mean eeg measure values for this voxel subspace using a statistical model of source localization error and between subject anatomical variation finally clustering the mean measure voxel values in this locally consistent brain subspace finds brain spatial domains exhibiting distinguishable measure features and provides 3 d maps plus statistical significance estimates for each eeg measure of interest applied to sufficient high quality data the scalp projections of many maximally independent component ic processes contributing to recorded high density eeg data closely match the projection of a single equivalent dipole located in or near brain cortex we demonstrate the application of mpa to a multi subject eeg study decomposed using independent component analysis ica compare the results to k means ic clustering in eeglab sccn ucsd edu eeglab and use surrogate data to test mpa robustness a measure projection toolbox mpt plug in for eeglab is available for download sccn ucsd edu wiki mpt together mpa and ica allow use of eeg as a 3 d cortical imaging modality with near cm scale spatial resolution

geometry and the complexity of matrix multiplication it has been known for some time that algebraic geometry and representation theory are useful for proving lower bounds on the complexity of the matrix multiplication operator in this talk i will explain how geometry can be used to prove both upper and lower bounds after a discussion of the problem and its history i will present very recent work with m michalek on border rank algorithms faculty host eric allender

simplicity of core arrays in three way principal component analysis and the typical rank of p q 2 arrays abstract interpreting the solution of a principal component analysis of a three way array is greatly simplified when the core array has a large number of zero elements the possibility of achieving this has recently been explored by rotations to simplicity or to simple targets on the one hand and by mathematical analysis on the other in the present paper it is shown that a p q 2 array with p q 2 can almost surely be transformed to have all but 2 q elements zero it is also shown that arrays of that form have three way rank p at most this has direct implications for the typical rank of p q 2 arrays also when p q when p 2q the typical rank is 2 q when q it is p and when p q the rank is typically almost surely p or p 1 these typical rank results pertain to the decomposition of real valued three way arrays in terms of real valued rank one arrays and do not apply in the complex setting where the typical rank of p q 2 arrays is also min p 2q when p q but it is p when p q

collision densities and mean residence times for d dimensional exponential flights in this paper we analyze some aspects of exponential flights a stochastic process that governs the evolution of many random transport phenomena such as neutron propagation chemical or biological species migration and electron motion we introduce a general framework for d dimensional setups and emphasize that exponential flights represent a deceivingly simple system where in most cases closed form formulas can hardly be obtained we derive a number of exact where possible or asymptotic results among which are the stationary probability density for two dimensional systems a long standing issue in physics and the mean residence time in a given volume bounded or unbounded domains as well as scattering or absorbing domains are examined and monte carlo simulations are performed so as to support our findings

low rank tensor train coefficient array estimation for tensor on tensor regression the tensor on tensor regression can predict a tensor from a tensor which generalizes most previous multilinear regression approaches including methods to predict a scalar from a tensor and a tensor from a scalar however the coefficient array could be much higher dimensional due to both high order predictors and responses in this generalized way compared with the current low candecomp parafac cp rank approximation based method the low tensor train tt approximation can further improve the stability and efficiency of the high or even ultrahigh dimensional coefficient array estimation in the proposed low tt rank coefficient array estimation for tensor on tensor regression we adopt a tt rounding procedure to obtain adaptive ranks instead of selecting ranks by experience besides an ell 2 constraint is imposed to avoid overfitting the hierarchical alternating least square is used to solve the optimization problem numerical experiments on a synthetic data set and two real life data sets demonstrate that the proposed method outperforms the state of the art methods in terms of prediction accuracy with comparable computational complexity and the proposed method is more computationally efficient when the data are high dimensional with small size in each mode

multisubject fmri studies and conjunction analyses abstract in this paper we present an approach to making inferences about generic activations in groups of subjects using fmri in particular we suggest that activations common to all subjects reflect aspects of functional anatomy that may be typical of the population from which that group was sampled these commonalities can be identified by a conjunction analysis of the activation effects in which the contrasts testing for an activation are specified separately for each subject a conjunction is the joint refutation of multiple null hypotheses in this instance of no activation in any subject the motivation behind this use of conjunctions is that fixed effect analyses are generally more sensitive than equivalent random effect analyses this is because fixed effect analyses can harness the large degrees of freedom and small scan to scan variability relative to the variability in responses from subject to subject when assessing the significance of an estimated response the price one pays for the apparent sensitivity of fixed effect analyses is that the ensuing inferences pertain to and only to the subjects studied however a conjunction analysis using a fixed effect model allows one to infer i that every subject studied activated and ii that at least a certain proportion of the population would have shown this effect the second inference depends upon a meta analytic formulation in terms of a confidence region for this proportion this approach retains the sensitivity of fixed effect analyses when the inference that only a substantial proportion of the population activates is sufficient

the border rank of the multiplication of 2 2 matrices is seven one of the leading problems of algebraic complexity theory is matrix multiplication the naive multiplication of two n n matrices uses n multiplications in 1969 strassen 20 presented an explicit algorithm for multiplying 2 2 matrices using seven multiplications in the opposite direction hopcroft and kerr 12 and winograd 22 proved independently that there is no algorithm for multiplying 2 2 matrices using only six multiplications the precise number of multiplications needed to execute matrix multiplication or any given bilinear map is called the rank of the bilinear map a related problem is to determine the border rank of matrix multiplication or any given bilinear map first introduced in 6 5 roughly speaking some bilinear maps may be approximated with arbitrary precision by less complicated bilinear maps and the border rank of a bilinear map is the complexity of arbitrarily small good perturbations of the map these perturbed maps can give rise to fast exact algorithms for matrix multiplication see 7 the border rank made appearances in the literature in the 1980s and early 1990s see e g 6 5 19 8 15 2 10 11 4 3 17 16 18 1 9 21 but to our knowledge there has not been much progress on the question since then more precisely for any complex projective variety x cp pv and point p pv define the x rank of p to be the smallest number r such that p is in the linear span of r points of x define r x the r th secant variety of x to be the zariski closure of the set of points of x rank r and define the x border rank of p to be the smallest r such that p r x the terminology is motivated by the case x seg pa 1 pb 1 p c c the segre variety of rank one matrices then the x rank of a matrix is just its usual rank let a b c be vector spaces and let f a b c be a bilinear map i e an element of a b c let x seg pa pb pc p a b c denote the segre variety of decomposable tensors in a b c the border rank of a bilinear map is its x border rank while for the segre product of two projective spaces border rank coincides with rank here they can be quite different in this paper we prove the theorem stated in the title let mmult c c c denote the matrix multiplication operator for 2 2 matrices strassen 18 showed that mmult 5 p p p our method of proof is to decompose

on kruskal s uniqueness condition for the candecomp parafac decomposition let x be a real valued three way array the candecomp parafac cp decomposition is written as x under bar y under bar 1 y under bar r e where y r are rank 1 arrays and e under bar is a rest term each rank 1 array is defined by the outer product of three vectors a r b r and c r i e y ijk r a i r b j r c k t these vectors make up the r columns of the component matrices a b and c if 2r 2 is less than or equal to the sum of the k ranks of a b and c then the fitted part of the decomposition is unique up to a change in the order of the rank 1 arrays and rescaling counterscaling of each triplet of vectors a r b r c r forming a rank 1 array this classical result was shown by kruskal his proof is however rather inaccessible and does not seem intuitive in order to contribute to a better understanding of cp uniqueness this paper provides an accessible and intuitive proof of kruskal s condition the proof is both self contained and compact and can easily be adapted for the complex valued cp decomposition c 2006 elsevier inc all rights reserved

algorithms for numerical analysis in high dimensions nearly every numerical analysis algorithm has computational complexity that scales exponentially in the underlying physical dimension the separated representation introduced previously allows many operations to be performed with scaling that is formally linear in the dimension in this paper we further develop this representation by i discussing the variety of mechanisms that allow it to be surprisingly efficient ii addressing the issue of conditioning iii presenting algorithms for solving linear systems within this framework and iv demonstrating methods for dealing with antisymmetric functions as arise in the multiparticle schrodinger equation in quantum mechanics numerical examples are given

statistical tests for fmri based on experimental randomization statistical parametric mapping spm analysis of fmri data requires specifying correctly the temporal and spatial noise covariance structure this is a difficult if not impossible task when these assumptions are not satisfied statistical inference can be invalid or inefficient permutation tests are free of strong assumptions on the distribution of signal noise we propose permutation tests of fmri data based on experimental randomization of the stimulus sequences smooth hemodynamic response curves are estimated using quadratic b splines we study fmri data obtained from event related potential erp oddball paradigm tests of two tone and three tone stimulus sequences are conducted to illustrate the application of the proposed method comparisons of the spm and proposed method show that a permutation test is more specific and less susceptible to artifacts

partial and total matrix multiplication in 1979 considerable progress was made in estimating the complexity of matrix multiplication here the new techniques and recent results are presented based upon the notion of approximate rank and the observation that certain patterns of partial matrix multiplication some of the entries of the matrices may be zero can efficiently be utilized to perform multiplication of large total matrices by combining pan s trilinear technique with a strong version of our compression theorem for the case of several disjoint matrix multiplications it is shown that multiplication of n times n matrices over arbitrary fields is possible in time o n beta where beta is a bit smaller than 3 ln 52 ln 110 approx 2 522

spherical representations of lie supergroups abstract the classical cartan helgason theorem characterises finite dimensional spherical representations of reductive lie groups in terms of their highest weights we generalise the theorem to the case of a reductive symmetric supergroup pair g k of even type along the way we compute the harish chandra c function of the symmetric superspace g k by way of an application we show that in type a iii a iii all spherical representations are self dual

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

branching exponential flights travelled lengths and collision statistics the evolution of several physical and biological systems ranging from neutron transport in multiplying media to epidemics or population dynamics can be described in terms of branching exponential flights a stochastic process which couples a galton watson birth death mechanism with random spatial displacements within this context one is often called to assess the length lv that the process travels in a given region v of the phase space or the number of visits nv to this same region in this paper we address this issue by resorting to the feynman kac formalism which allows characterizing the full distribution of lv and nv and in particular deriving explicit moment formulas some other significant physical observables associated to lv and nv such as the survival probability are discussed as well and results are illustrated by revisiting the classical example of the rod model in nuclear reactor physics

on representations of the lie superalgebra p n abstract we introduce a new way to study representations of the lie superalgebra p n since the center of the universal enveloping algebra u acts trivially on all irreducible representations we suggest to study the quotient algebra u by the radical of u we show that u has a large center which separates typical finite dimensional irreducible representations we give a description of u factored by a generic central character using this description we obtain character formulae of generic infinite dimensional irreducible representations we also describe some geometric properties of the supervariety spec gr u in the coadjoint representation

kernel partial least squares regression in reproducing kernel hilbert space a family of regularized least squares regression models in a reproducing kernel hilbert space is extended by the kernel partial least squares pls regression model similar to principal components regression pcr pls is a method based on the projection of input explanatory variables to the latent variables components however in contrast to pcr pls creates the components by modeling the relationship between input and output variables while maintaining most of the information in the input variables pls is useful in situations where the number of explanatory variables exceeds the number of observations and or a high level of multicollinearity among those variables is assumed motivated by this fact we will provide a kernel pls algorithm for construction of nonlinear regression models in possibly high dimensional feature spaces we give the theoretical description of the kernel pls algorithm and we experimentally compare the algorithm with the existing kernel pcr and kernel ridge regression techniques we will demonstrate that on the data sets employed kernel pls achieves the same results as kernel pcr but uses significantly fewer qualitatively different components

multiple tensor on tensor regression an approach for modeling processes with heterogeneous sources of data in recent years measurement or collection of heterogeneous sets of data such as those containing scalars waveform signals images and even structured point clouds has become more common statis

handwritten digit classification using higher order singular value decomposition in this paper we present two algorithms for handwritten digit classification based on the higher order singular value decomposition hosvd the first algorithm uses hosvd for construction of the class models and achieves classification results with error rate lower than 6 the second algorithm uses the hosvd for tensor approximation simultaneously in two modes classification results for the second algorithm are almost down at 5 even though the approximation reduces the original training data with more than 98 before the construction of the class models the actual classification in the test phase for both algorithms is conducted by solving a series least squares problems considering computational amount for the test presented the second algorithm is twice as efficient as the first one

a randomized algorithm for a tensor based generalization of the singular value decomposition an algorithm is presented and analyzed that when given as input a d mode tensor a computes an approximation a the approximation a is computed by performing the following for each of the d modes first form implicitly a matrix by unfolding the tensor along that mode then choose columns from the matrices thus generated and finally project the tensor along that mode onto the span of those columns an important issue affecting the quality of the approximation is the choice of the columns from the matrices formed by unfolding the tensor along each of its modes in order to address this issue two algorithms of independent interest are presented that given an input matrix a and a target rank k select columns that span a space close to the best rank k subspace of the matrix for example in one of the algorithms a number c that depends on k an error parameter and a failure probability of columns are chosen in c independent random trials according to a nonuniform probability distribution depending on the euclidean lengths of the columns when this algorithm for choosing columns is used in the tensor approximation then under appropriate assumptions bounds of the form

short hyperuniform random walks random walks of two steps with fixed sums of lengths of 1 taken into uniformly random directions in d dimensional euclidean spaces d ge 2 are investigated to construct continuous step length distributions which make them hyperuniform the endpoint positions of hyperuniform walks are spread out in the unit ball as the projections in the walk space of points uniformly distributed on the surface of the unit hypersphere of some k dimensional euclidean space k d unique symmetric continuous step length distributions exist for given d and k provided that d k 2d the walk becomes uniform on the unit ball when k d 2 the symmetric densities reduce to simple polynomials for uniform random walks and are mixtures of two pairs of asymmetric beta distributions

some uniqueness results for parafac2 whereas the unique axes properties of parafac1 have been examined extensively little is known about uniqueness properties for the parafac2 model for covariance matrices this paper is concerned with uniqueness in the rank two case of parafac2 for this case harshman and lundy have recently shown subject to mild assumptions that parafac2 is unique when five covariance matrices are analyzed in the present paper this result is sharpened parafac2 is shown to be usually unique with four matrices with three matrices it is not unique unless a certain additional assumption is introduced if for instance the diagonal matrices of weights are constrained to be non negative three matrices are enough to have uniqueness in the rank two case of parafac2

the representation theory of the exceptional lie superalgebras f 4 and g 3 this thesis is a resolution of three related problems proposed by yu i manin and v kac for the so called exceptional lie superalgebras f 4 and g 3 the first problem posed by kac 1978 is the problem of finding character and superdimension formulae for the simple modules the second problem posed by kac 1978 is the problem of classifying all indecomposable representations the third problem posed by manin 1981 is the problem of constructing the superanalogue of borel weil bott theorem

invariants of the orthosymplectic lie superalgebra and super pfaffians given a complex orthosymplectic superspace v the orthosymplectic lie superalgebra mathfrak osp v and general linear algebra mathfrak gl n both act naturally on the coordinate super ring mathcal s n of the dual space of v otimes mathbb c n and their actions commute hence the subalgebra mathcal s n mathfrak osp v of mathfrak osp v invariants in mathcal s n has a mathfrak gl n module structure sergeev has indicated how to define a pfaffian in this space and announced that together with the invariants of the group mathrm osp v it generates all invariants of mathfrak osp v we introduce a space of super pfaffians show it is a simple mathfrak gl n submodule of mathcal s n mathfrak osp v give an explicit formula for its highest weight vector and prove that the super pfaffians and the mathrm osp v invariants generate mathcal s n mathfrak osp v as an algebra the decomposition of mathcal s n mathfrak osp v as a direct sum of simple mathfrak gl n submodules is obtained and shown to be multiplicity free using howe s mathfrak gl v mathfrak gl n duality on mathcal s n we further analyse the module structure of that space these results also enable us to determine the mathfrak osp v invariants in the tensor powers v otimes r for all r

maximal rank of m x n x mn k tensors it is shown that the maximal rank of m times n times mn k tensors with k leqq min m 1 2 2 n 1 2 2 is greater than mn 4 sqrt 2k o 1

the exact probability distribution of a two dimensional random walk a calculation is made of the exact probability distribution of the two dimensional displacement of a particle at timet that starts at the origin moves in straight line paths at constant speed and changes its direction after exponentially distributed time intervals where the lengths of the straight line paths and the turn angles are independent the angles being uniformly distributed this random walk is the simplest model for the locomotion of microorganisms on surfaces its weak convergence to a wiener process is also shown

higher order partial least squares hopls a generalized multilinear regression method a new generalized multilinear regression model termed the higher order partial least squares hopls is introduced with the aim to predict a tensor multiway array y from a tensor x through projecting the data onto the latent space and performing regression on the corresponding latent variables hopls differs substantially from other regression models in that it explains the data by a sum of orthogonal tucker tensors while the number of orthogonal loadings serves as a parameter to control model complexity and prevent overfitting the low dimensional latent space is optimized sequentially via a deflation operation yielding the best joint subspace approximation for both x and y instead of decomposing x and y individually higher order singular value decomposition on a newly defined generalized cross covariance tensor is employed to optimize the orthogonal loadings a systematic comparison on both synthetic data and real world decoding of 3d movement trajectories from electrocorticogram signals demonstrate the advantages of hopls over the existing methods in terms of better predictive ability suitability to handle small sample sizes and robustness to noise

dimensionality reduction in higher order signal processing and rank r1 r2 rn reduction in multilinear algebra abstract in this paper we review a multilinear generalization of the singular value decomposition and the best rank r 1 r 2 r n approximation of higher order tensors we show that they are important tools for dimensionality reduction in higher order signal processing we discuss applications in independent component analysis simultaneous matrix diagonalization and subspace variants of algorithms based on higher order statistics

an alternating least squares algorithm for fitting the two and three way dedicom model and the idioscal model the dedicom model is a model for representing asymmetric relations among a set of objects by means of a set of coordinates for the objects on a limited number of dimensions the present paper offers an alternating least squares algorithm for fitting the dedicom model the model can be generalized to represent any number of sets of relations among the same set of objects an algorithm for fitting this three way dedicom model is provided as well based on the algorithm for the three way dedicom model an algorithm is developed for fitting the idioscal model in the least squares sense