natural gradient learning for over and under complete bases in ica independent component analysis or blind source separation is a new technique of extracting independent signals from mixtures it is applicable even when the number of independent sources is unknown and is larger or smaller than the number of observed mixture signals this article extends the natural gradient learning algorithm to be applicable to these overcomplete and undercomplete cases here the observed signals are assumed to be whitened by preprocessing so that we use the natural riemannian gradient in stiefel manifolds

learning overcomplete representations in an overcomplete basis the number of basis vectors is greater than the dimensionality of the input and the representation of an input is not a unique combination of basis vectors overcomplete representations have been advocated because they have greater robustness in the presence of noise can be sparser and can have greater flexibility in matching structure in the data overcomplete codes have also been proposed as a model of some of the response properties of neurons in primary visual cortex previous work has focused on finding the best representation of a signal using a fixed overcomplete basis or dictionary we present an algorithm for learning an overcomplete basis by viewing it as probabilistic model of the observed data we show that overcomplete bases can yield a better approximation of the underlying statistical distribution of the data and can thus lead to greater coding efficiency this can be viewed as a generalization of the technique of independent component analysis and provides a method for bayesian reconstruction of signals in the presence of noise and for blind source separation when there are more sources than mixtures

thermodynamics as a theory of decision making with information processing costs perfectly rational decision makers maximize expected utility but crucially ignore the resource costs incurred when determining optimal actions here we propose a thermodynamically inspired formalization of bounded rational decision making where information processing is modelled as state changes in thermodynamic systems that can be quantified by differences in free energy by optimizing a free energy bounded rational decision makers trade off expected utility gains and information processing costs measured by the relative entropy as a result the bounded rational decision making problem can be rephrased in terms of well known variational principles from statistical physics in the limit when computational costs are ignored the maximum expected utility principle is recovered we discuss links to existing decision making frameworks and applications to human decision making experiments that are at odds with expected utility theory since most of the mathematical machinery can be borrowed from statistical physics the main contribution is to re interpret the formalism of thermodynamic free energy differences in terms of bounded rational decision making and to discuss its relationship to human decision making experiments

stochastic optimal control via forward and backward stochastic differential equations and importance sampling abstract the aim of this work is to present a novel sampling based numerical scheme designed to solve a certain class of stochastic optimal control problems utilizing forward and backward stochastic differential equations fbsdes by means of a nonlinear version of the feynman kac lemma we obtain a probabilistic representation of the solution to the nonlinear hamilton jacobi bellman equation expressed in the form of a system of decoupled fbsdes this system of fbsdes can be solved by employing linear regression techniques the proposed framework relaxes some of the restrictive conditions present in recent sampling based methods within the linearly solvable optimal control framework and furthermore addresses problems in which the time horizon is not prespecified to enhance the efficiency of the proposed scheme when treating more complex nonlinear systems we then derive an iterative algorithm based on girsanov s theorem on the change of measure which features importance sampling this scheme is shown to be capable of learning the optimal control without requiring an initial guess

nonlinear stochastic receding horizon control stability robustness and monte carlo methods for control approximation this work considers the stability of nonlinear stochastic receding horizon control when the optimal controller is only computed approximately a number of general classes of controller approximation error are analysed including deterministic and probabilistic errors and even controller sample and hold errors in each case it is shown that the controller approximation errors do not accumulate even over an infinite time frame and the process converges exponentially fast to a small neighbourhood of the origin in addition to this analysis an approximation method for receding horizon optimal control is proposed based on monte carlo simulation this method is derived via the feynman kac formula which gives a stochastic interpretation for the solution of a hamilton jacobi bellman equation associated with the true optimal controller it is shown and it is a prime motivation for this study that this particular controller approximation method practically stabilises the underlying nonlinear process

an equivalence between sparse approximation and support vector machines in the first part of this paper we show a similarity between the principle of structural risk minimization principle srm vapnik 1982 and the idea of sparse approximation as defined in chen donoho and saunders 1995 and olshausen and field 1996 then we focus on two specific approximate implementations of srm and sparse approximation which have been used to solve the problem of function approximation for srm we consider the support vector machine technique proposed by v vapnik and his team at at t bell labs and for sparse approximation we consider a modification of the basis pursuit de noising algorithm proposed by chen donoho and saunders 1995 we show that under certain conditions these two techniques are equivalent they give the same solution and they require the solution of the same quadratic programming problem

cesaro summability of double walsh fourier series we introduce quasi local operators these include operators of calder6n zygmund type a hybrid hardy space ho of functions of two variables and we obtain sufficient conditions for a quasi local maximal operator to be of weak type 1 as an application we show that cesatro means of the double walsh fourier series of a function f converge a e when f belongs to ho we also obtain the dyadic analogue of a summability result of marcienkiewicz and zygmund valid for all f e l1 provided summability takes place in some positive cone

a theory for learning by weight flow on stiefel grassman manifold recently we introduced the concept of neural network learning on stiefel grassman manifold for multilayer perceptron like networks contributions of other authors have also appeared in the scientific literature about this topic this article presents a general theory for it and illustrates how existing theories may be explained within the general framework proposed here

modularly weighted four dimensional matrix summability with application to korovkin type approximation theorem abstract this work is concerned with various weighted four dimensional matrix summability methods in modular function spaces associating with generalized difference operator involving p q gamma function following very recent results of kadak 2017 20 we first introduce a new type of difference operator on double function sequences and also define modularly weighted a statistical convergence and modularly statistically n a summability by means of weighted four dimensional regular matrix a we also present some important inclusion relations between newly proposed summability methods moreover based on the concept of modularly statistically n a summability we prove a korovkin type approximation theorem for functions of two variables in modular spaces finally in order to show that our proposed method and its applications to approximation results are stronger than the existing methods we display an illustrative example using bivariate p q bernstein kantorovich operators

an iterative method for nonlinear stochastic optimal control based on path integrals this paper proposes a new iterative solution method for nonlinear stochastic optimal control problems based on path integral analysis first we provide an iteration law for solving a stochastic hamilton jacobi bellman shjb equation associated to this problem which is a nonlinear partial differential equation pde of second order each iteration procedure of the proposed method is represented by a cauchy problem for a linear parabolic pde and its explicit solution is given by the feynman kac formula second we derive a suboptimal feedback controller at each iteration by using the path integral analysis third the convergence property of the proposed method is investigated here some conditions are provided so that the sequence of solutions for the proposed iteration converges and the shjb equation is satisfied finally numerical simulations demonstrate the effectiveness of the proposed method

a lowest order composite finite element exact sequence on pyramids abstract composite basis functions for pyramidal elements on the spaces h 1 h curl h div and l 2 are presented in particular we construct the lowest order composite pyramidal elements and show that they respect the de rham diagram i e we have an exact sequence and satisfy the commuting property moreover the finite elements are fully compatible with the standard finite elements for the lowest order raviart thomas nedelec sequence on tetrahedral and hexahedral elements that is to say the new elements have the same degrees of freedom on the shared interface with the neighbouring hexahedral or tetrahedra elements and the basis functions are conforming in the sense that they maintain the required level of continuity full tangential component normal component etc across the interface furthermore we study the approximation properties of the spaces as an initial partition consisting of tetrahedra hexahedra and pyramid elements are successively subdivided and show that the spaces result in the same optimal order of approximation in terms of the mesh size h as one would obtain using purely hexahedral or purely tetrahedral partitions

statistical dynamics of on line independent component analysis the learning dynamics of on line independent component analysis is analysed in the limit of large data dimension we study a simple hebbian learning algorithm that can be used to separate out a small number of non gaussian components from a high dimensional data set the de mixing matrix parameters are confined to a stiefel manifold of tall orthogonal matrices and we introduce a natural gradient variant of the algorithm which is appropriate to learning on this manifold for large input dimension the parameter trajectory of both algorithms passes through a sequence of unstable fixed points each described by a diffusion process in a polynomial potential choosing the learning rate too large increases the escape time from each of these fixed points effectively trapping the learning in a sub optimal state in order to avoid these trapping states a very low learning rate must be chosen during the learning transient resulting in learning time scales of o n2 or o n3 iterations where n is the data dimension escape from each sub optimal state results in a sequence of symmetry breaking events as the algorithm learns each source in turn this is in marked contrast to the learning dynamics displayed by related on line learning algorithms for multilayer neural networks and principal component analysis although the natural gradient variant of the algorithm has nice asymptotic convergence properties it has an equivalent transient dynamics to the standard hebbian algorithm

adaptive online learning algorithms for blind separation maximum entropy and minimum mutual information there are two major approaches for blind separation maximum entropy me and minimum mutual information mmi both can be implemented by the stochastic gradient descent method for obtaining the demixing matrix the mi is the contrast function for blind separation the entropy is not to justify the me the relation between me and mmi is first elucidated by calculating the first derivative of the entropy and proving that the mean subtraction is necessary in applying the me and at the solution points determined by the mi the me will not update the demixing matrix in the directions of increasing the cross talking second the natural gradient instead of the ordinary gradient is introduced to obtain efficient algorithms because the parameter space is a riemannian space consisting of matrices the mutual information is calculated by applying the gram charlier expansion to approximate probability density functions of the outputs finally we propose an efficient learning algorithm that incorporates with an adaptive method of estimating the unknown cumulants it is shown by computer simulation that the convergence of the stochastic descent algorithms is improved by using the natural gradient and the adaptively estimated cumulants

a local projection operator for quadrilateral finite elements this note studies the approximation error of a local projection operator on polynomials of total degree k defined on quadrilaterals among other applications this projection operator permits to derive easily error estimates for quadrature formulas

tauberian theorems for power series methods applied to double sequences abstract we discuss the relations between power series methods weighted mean methods and ordinary convergence for double sequences in particular we study tauberian theorems for methods being products of the related one dimensional summability methods

estimating functions of independent component analysis for temporally correlated signals this article studies a general theory of estimating functions of independent component analysis when the independent source signals are temporarily correlated estimating functions are used for deriving both batch and on line learning algorithms and they are applicable to blind cases where spatial and temporal probability structures of the sources are unknown most algorithms proposed so far can be analyzed in the framework of estimating functions an admissible class of estimating functions is derived and related efficient on line learning algorithms are introduced we analyze dynamical stability and statistical efficiency of these algorithms different from the independently and identically distributed case the algorithms work even when only the second order moments are used the method of simultaneous diagonalization of cross covariance matrices is also studied from the point of view of estimating functions

natural gradient works efficiently in learning when a parameter space has a certain underlying structure the ordinary gradient of a function does not represent its steepest direction but the natural gradient does information geometry is used for calculating the natural gradients in the parameter space of perceptrons the space of matrices for blind source separation and the space of linear dynamical systems for blind source deconvolution the dynamical behavior of natural gradient online learning is analyzed and is proved to be fisher efficient implying that it has asymptotically the same performance as the optimal batch estimation of parameters this suggests that the plateau phenomenon which appears in the backpropagation learning algorithm of multilayer perceptrons might disappear or might not be so serious when the natural gradient is used an adaptive method of updating the learning rate is proposed and analyzed

tauberian theorems for double sequences that are statistically summable c 1 1 abstract let x jk j k 0 1 2 be a double sequence of real or complex numbers and set mn m 1 1 n 1 1 j 0 m k 0 n x jk for m n 0 1 2 we give necessary and sufficient conditions under which st lim mn implies st lim x jk where is a finite number these tauberian conditions are one sided when the xjk are real numbers and they are two sided when the xjk are complex numbers in particular these tauberian conditions are clearly satisfied if xjk is statistically slowly decreasing in the case of real sequences or if xjk is statistically slowly oscillating in the case of complex sequences

pointwise convergence of multiple trigonometric series abstract this is devoted to the study of pointwise convergence of the rectangular partial sums of multiple trigonometric series with coefficients satisfying certain conditions involving finite order differences three types of such conditions are posed convergence may be understood both in the unrestricted sense and in the restricted sense we prove that if the first arithmetic mean of an n fold trigonometric series of the above type converges then so does its rectangular partial sum as a corollary we obtain that any multiple fourier series of the above type converges restrictedly almost everywhere these results extend those of chen chen and hsieh moricz and moricz and waterman

path integral control and state dependent feedback in this paper we address the problem of computing state dependent feedback controls for path integral control problems to this end we generalize the path integral control formula and utilize this to construct parametrized state dependent feedback controllers in addition we show a relation between control and importance sampling better control in terms of control cost yields more efficient importance sampling in terms of effective sample size the optimal control provides a zero variance estimate

bernstein bezier finite elements on tetrahedral hexahedral pyramidal partitions abstract a construction for high order continuous finite elements on partitions consisting of tetrahedra hexahedra and pyramids based on polynomial bernstein bezier shape functions is presented along with algorithms that allow the computation of the system matrices in optimal complexity o 1 per entry

adaptive importance sampling for control and inference path integral pi control problems are a restricted class of non linear control problems that can be solved formally as a feynman kac pi and can be estimated using monte carlo sampling in this contribution we review pi control theory in the finite horizon case we subsequently focus on the problem how to compute and represent control solutions we review the most commonly used methods in robotics and control within the pi theory the question of how to compute becomes the question of importance sampling efficient importance samplers are state feedback controllers and the use of these requires an efficient representation learning and representing effective state feedback controllers for non linear stochastic control problems is a very challenging and largely unsolved problem we show how to learn and represent such controllers using ideas from the cross entropy method we derive a gradient descent method that allows to learn feed back controllers using an arbitrary parametrisation we refer to this method as the path integral cross entropy method or pice we illustrate this method for some simple examples the pi control methods can be used to estimate the posterior distribution in latent state models in neuroscience these problems arise when estimating connectivity from neural recording data using em we demonstrate the pi control method as an accurate alternative to particle filtering

geometrical validity of curvilinear pyramidal finite elements a method to efficiently determine the geometrical validity of curvilinear finite elements of any order was recently proposed in 1 the method is based on the adaptive expansion of the jacobian determinant in a polynomial basis built using bezier functions that has both properties of boundedness and positivity while this technique can be applied to all usual finite elements triangles quadrangles tetrahedra hexahedra and prisms it cannot readily be applied to pyramids due to non polynomial nature of pyramidal finite element spaces in this short paper we extend the results from 1 to pyramidal elements by making use of the high order nodal pyramidal finite element proposed by bergot et al 10 which exhibits optimal convergence properties inh1 norm the paper is organized as follows we begin by briefly recalling the pyramidal finite element space in section 2 before constructing the function space of the jacobian determinant in section 3 section 4 then introduces a generalized bezier function basis which can be used to obtain adaptive bounds on the pyramidal jacobian determinant numerical results showing the sharpness of the estimates are given in section 5

tauberian conditions under which the original convergence of double sequences follows from the statistical convergence of their weighted means abstract in this paper we introduce a new type of slow oscillation and slow decrease conditions we prove that these or their variants are tauberian conditions from s m n s t s to s m n s we also prove that they are tauberian conditions from t m n 11 s t s to s m n s where t m n 11 are the weighted means of the double sequence s m n m n 0 our results not only generalize well known results but also solve the conjecture of moricz posed in f moricz tauberian theorems for double sequences that are statistically summable c 1 1 j math anal appl 286 2003 340 350

equivariant adaptive source separation source separation consists of recovering a set of independent signals when only mixtures with unknown coefficients are observed this paper introduces a class of adaptive algorithms for source separation that implements an adaptive version of equivariant estimation and is henceforth called equivariant adaptive separation via independence easi the easi algorithms are based on the idea of serial updating this specific form of matrix updates systematically yields algorithms with a simple structure for both real and complex mixtures most importantly the performance of an easi algorithm does not depend on the mixing matrix in particular convergence rates stability conditions and interference rejection levels depend only on the normalized distributions of the source signals closed form expressions of these quantities are given via an asymptotic performance analysis the theme of equivariance is stressed throughout the paper the source separation problem has an underlying multiplicative structure the parameter space forms a matrix multiplicative group we explore the favorable consequences of this fact on implementation performance and optimization of easi algorithms

cesaro summability of two dimensional walsh fourier series we introduce p quasi local operators and the two dimensional dyadic hardy spaces hp defined by the dyadic squares it is proved that if a sublinear operator t is p quasi local and bounded from l to l then it is also bounded from hp to lp 0 p 1 as an application it is shown that the maximal operator of the cesaro means of a martingale is bounded from hp to lp 1 2 p and is of weak type 1 1 provided that the supremum in the maximal operator is taken over a positive cone so we obtain the dyadic analogue of a summability result with respect to two dimensional trigonometric fourier series due to marcinkievicz and zygmund more exactly the cesaro means of a function f l1 converge a e to the function in question provided again that the limit is taken over a positive cone finally it is verified that if we take the supremum in a cone but for two powers only then the maximal operator of the cesaro means is bounded from hp to lp for every 0 p

variational learning of clusters of undercomplete nonsymmetric independent components we apply a variational method to automatically determine the number of mixtures of independent components in high dimensional datasets in which the sources may be nonsymmetrically distributed the data are modeled by clusters where each cluster is described as a linear mixture of independent factors the variational bayesian method yields an accurate density model for the observed data without overfitting problems this allows the dimensionality of the data to be identified for each cluster the new method was successfully applied to a difficult real world medical dataset for diagnosing glaucoma

dynamical movement primitives learning attractor models for motor behaviors nonlinear dynamical systems have been used in many disciplines to model complex behaviors including biological motor control robotics perception economics traffic prediction and neuroscience while often the unexpected emergent behavior of nonlinear systems is the focus of investigations it is of equal importance to create goal directed behavior e g stable locomotion from a system of coupled oscillators under perceptual guidance modeling goal directed behavior with nonlinear systems is however rather difficult due to the parameter sensitivity of these systems their complex phase transitions in response to subtle parameter changes and the difficulty of analyzing and predicting their long term behavior intuition and time consuming parameter tuning play a major role this letter presents and reviews dynamical movement primitives a line of research for modeling attractor behaviors of autonomous nonlinear dynamical systems with the help of statistical learning techniques the essence of our approach is to start with a simple dynamical system such as a set of linear differential equations and transform those into a weakly nonlinear system with prescribed attractor dynamics by means of a learnable autonomous forcing term both point attractors and limit cycle attractors of almost arbitrary complexity can be generated we explain the design principle of our approach and evaluate its properties in several example applications in motor control and robotics

a variational hexahedral grid generator with control metric a variational method of constructing a spatial structured grid composed of hexahedral cells is presented the method executes a minimization of the variational functional the integrand of the functional is a ratio of the orthogonal invariants the functional depends on two metrics one metric is induced by a curvilinear mesh generated in the physical domain and the other control metric given in the canonical domain is responsible for an additional cell shape control for instance for condensing the coordinate surfaces and orthogonalization of the grid lines towards the domain boundary generally defining the control metric allows to generate an arbitrary given non folded mesh in the physical domain for every cell the functional is discretized on ten tetrahedra forming two dodecahedrons with the same vertices which span the hexahedral cell the discrete functional possesses an infinite barrier on the boundary of the set of non folded dodecahedral cells that ensures the construction of the non folded grid composed of such cells in the most practical cases the hexahedral grid with the same nodes is non folded as well the method of boundary nodes redistribution is considered examples of the grid construction are reported

linear dimensionality reduction survey insights and generalizations linear dimensionality reduction methods are a cornerstone of analyzing high dimensional data due to their simple geometric interpretations and typically attractive computational properties these methods capture many data features of interest such as covariance dynamical structure correlation between data sets input output relationships and margin between data classes methods have been developed with a variety of names and motivations in many fields and perhaps as a result the connections between all these methods have not been highlighted here we survey methods from this disparate literature as optimization programs over matrix manifolds we discuss principal component analysis factor analysis linear multidimensional scaling fisher s linear discriminant analysis canonical correlations analysis maximum autocorrelation factors slow feature analysis sufficient dimensionality reduction undercomplete independent component analysis linear regression distance metric learning and more this optimization framework gives insight to some rarely discussed shortcomings of well known methods such as the suboptimality of certain eigenvector solutions modern techniques for optimization over matrix manifolds enable a generic linear dimensionality reduction solver which accepts as input data and an objective to be optimized and returns as output an optimal low dimensional projection of the data this simple optimization framework further allows straightforward generalizations and novel variants of classical methods which we demonstrate here by creating an orthogonal projection canonical correlations analysis more broadly this survey and generic solver suggest that linear dimensionality reduction can move toward becoming a blackbox objective agnostic numerical technology

pointwise convergence of double trigonometric series abstract the pointwise convergence problem of the rectangular partial sums of a certain type of double trigonometric series is considered this type of series obeys certain conditions on the finite order differences of its coefficients we prove that if the cesaro sums of the double series converge unrestrictedly then so do its partial sums it is pointed out that the converse of the last statement may not hold for the same kind of double trigonometric series as a corollary it is shown that the double fourier series of the mentioned type converges unrestrictedly almost everywhere generalizations of the above results to the restricted case are also established these results generalize the theorems of chen

a generalized path integral control approach to reinforcement learning with the goal to generate more scalable algorithms with higher efficiency and fewer open parameters reinforcement learning rl has recently moved towards combining classical techniques from optimal control and dynamic programming with modern learning techniques from statistical estimation theory in this vein this paper suggests to use the framework of stochastic optimal control with path integrals to derive a novel approach to rl with parameterized policies while solidly grounded in value function estimation and optimal control based on the stochastic hamilton jacobi bellman hjb equations policy improvements can be transformed into an approximation problem of a path integral which has no open algorithmic parameters other than the exploration noise the resulting algorithm can be conceived of as model based semi model based or even model free depending on how the learning problem is structured the update equations have no danger of numerical instabilities as neither matrix inversions nor gradient learning rates are required our new algorithm demonstrates interesting similarities with previous rl research in the framework of probability matching and provides intuition why the slightly heuristically motivated probability matching approach can actually perform well empirical evaluations demonstrate significant performance improvements over gradient based policy learning and scalability to high dimensional control problems finally a learning experiment on a simulated 12 degree of freedom robot dog illustrates the functionality of our algorithm in a complex robot learning scenario we believe that policy improvement with path integrals pi2 offers currently one of the most efficient numerically robust and easy to implement algorithms for rl based on trajectory roll outs

some tauberian theorems for statistical c 1 summability abstract schoenberg 1959 23 proved that if a sequence x k is bounded and statistically convergent to l then it is statistically summable c 1 to l however the converse is not true in this paper we prove necessary and sufficient tauberian conditions based on generator sequences under which statistical convergence follows from statistical summability c 1

the invertibility of the isoparametric mapping for pyramidal and prismatic finite elements we consider the isoparametric transformation which maps a given reference element onto a global element given by its vertices for multi linear finite elements on pyramids and prisms we present easily computable conditions on the position of the vertices which ensure that the isoparametric transformation is bijective

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

superefficiency in blind source separation blind source separation is the problem of extracting independent signals from their mixtures without knowing the mixing coefficients nor the probability distributions of source signals and may be applied to eeg and meg imaging of the brain it is already known that certain algorithms work well for the extraction of independent components the present paper is concerned with superefficiency of these based on the statistical and dynamical analysis in a statistical estimation using t examples the covariance of any two extracted independent signals converges to 0 of the order of 1 t on line dynamics shows that the covariance is of the order of spl eta when the learning rate spl eta is fixed to a small constant in contrast with the above general properties a surprising superefficiency holds in blind source separation under certain conditions where superefficiency implies that covariance decreases in the order of 1 t sup 2 or of spl eta sup 2 the paper uses the natural gradient learning algorithm and method of estimating functions to obtain superefficient procedures for both batch estimation and on line learning a standardized estimating function is introduced to this end superefficiency does not imply that the error variances of the extracted signals decrease in the order of 1 t sup 2 or spl eta sup 2 but implies that their covariances and independencies do