the helmholtz machine discovering the structure inherent in a set of patterns is a fundamental aim of statistical inference or learning one fruitful approach is to build a parameterized stochastic generative model independent draws from which are likely to produce the patterns for all but the simplest generative models each pattern can be generated in exponentially many ways it is thus intractable to adjust the parameters to maximize the probability of the observed patterns we describe a way of finessing this combinatorial explosion by maximizing an easily computed lower bound on the probability of the observations our method can be viewed as a form of hierarchical self supervised learning that may relate to the function of bottom up and top down cortical processing pathways

a construction of homologically area minimizing hypersurfaces with higher dimensional singular sets we show that a large variety of singular sets can occur for homologically area minimizing codimenision one surfaces in a riemarinian manifold in particular as a result of theorem a if n is smooth compact ni 1 dimensional manifold n 7 and if s is an embedded orientable submanifold of dimension n then we cornstruct metrics on n such that the homologically area minimizing hypersurface m homologous to s has a singular set equal to a prescribed number of spheres and tori of codimension less than n 7 near each component e of the singular set m is isometric to a product c x e where c is any prescribed strictly stable strictly minimizing cone in theorem b other singular examples are constructed

asymptotic integration of some nonlinear differential equations with fractional time derivative we establish that under some simple integral conditions regarding the nonlinearity the 1 order fractional differential equation 0d t x f t x 0 t 0 has a solution with which can be expanded asymptotically as a bt o t 1 when t for given real numbers a b our arguments are based on fixed point theory here 0d t designates the riemann liouville derivative of order 0 1

finding minimum entropy codes to determine whether a particular sensory event is a reliable predictor of reward or punishment it is necessary to know the prior probability of that event if the variables of a sensory representation normally occur independently of each other then it is possible to derive the prior probability of any logical function of the variables from the prior probabilities of the individual variables without any additional knowledge hence such a representation enormously enlarges the scope of definable events that can be searched for reliable predictors finding a minimum entropy code is a possible method of forming such a representation and methods for doing this are explored in this paper the main results are 1 to show how to find such a code when the probabilities of the input states form a geometric progression as is shown to be nearly true for keyboard characters in normal text 2 to show how a minimum entropy code can be approximated by repeatedly recoding pairs triples etc of an original 7 bit code for keyboard characters 3 to prove that in some cases enlarging the capacity of the output channel can lower the entropy

minimal graphic functions on manifolds of nonnegative ricci curvature we study minimal graphic functions on complete riemannian manifolds with nonnegative ricci curvature euclidean volume growth and quadratic curvature decay we derive global bounds for the gradients for minimal graphic functions of linear growth only on one side then we can obtain a liouville type theorem with such growth via splitting for tangent cones of at infinity when in contrast we do not impose any growth restrictions for minimal graphic functions we also obtain a liouville type theorem under a certain nonradial ricci curvature decay condition on in particular the borderline for the ricci curvature decay is sharp by our example in the last section 2014 wiley periodicals inc

recurrent sampling models for the helmholtz machine many recent analysis by synthesis density estimation models of cortical learning and processing have made the crucial simplifying assumption that units within a single layer are mutually independent given the states of units in the layer below or the layer above in this article we suggest using either a markov random field or an alternative stochastic sampling architecture to capture explicitly particular forms of dependence within each layer we develop the architectures in the context of real and binary helmholtz machines recurrent sampling can be used to capture correlations within layers in the generative or the recognition models and we also show how these can be combined

the non standard finite difference scheme for linear fractional pdes in fluid mechanics a non standard finite difference scheme is developed to solve the linear partial differential equations with time and space fractional derivatives the grunwald letnikov method is used to approximate the fractional derivatives numerical illustrations that include the linear inhomogeneous time fractional equation linear space fractional telegraph equation linear inhomogeneous fractional burgers equation and the fractional wave equation are investigated to show the pertinent features of the technique numerical results are presented graphically and reveal that the non standard finite difference scheme is very effective and convenient for solving linear partial differential equations of fractional order

mechanics with fractional derivatives lagrangian and hamiltonian mechanics can be formulated to include derivatives of fractional order f riewe phys rev 53 1890 1996 lagrangians with fractional derivatives lead directly to equations of motion with nonconservative classical forces such as friction the present work continues the development of fractional derivative mechanics by deriving a modified hamilton s principle introducing two types of canonical transformations and deriving the hamilton jacobi equation using generalized mechanics with fractional and higher order derivatives the method is illustrated with a frictional force proportional to velocity in contrast to conventional mechanics with integer order derivatives quantization of a fractional derivative hamiltonian cannot generally be achieved by the traditional replacement of momenta with coordinate derivatives instead a quantum mechanical wave equation is proposed that follows from the hamilton jacobi equation by application of the correspondence principle

singular homologically area minimizing surfaces of codimension one in riemannian manifolds for n 7 it is shown how to construct examples of smooth compact riemannian manifolds n n 1 g with non trivial n dimensional integer homology such that for some h n n z the hypersurface n current m which minimizes area among all hypersurfaces representing has singularities the singular set of m consists of two isolated points and the tangent cone at these points can be prescribed as any strictly stable strictly minimizing regular cone to my knowledge these are the first examples of codimension one homological minimizers with singularities

variational learning in nonlinear gaussian belief networks we view perceptual tasks such as vision and speech recognition as inference problems where the goal is to estimate the posterior distribution over latent variables e g depth in stereo vision given the sensory input the recent flurry of research in independent component analysis exemplifies the importance of inferring the continuousvalued latent variables of input data the latent variables found by this method are linearly related to the input but perception requires nonlinear inferences such as classification and depth estimation in this paper we present a unifying framework for stochastic neural networks with nonlinear latent variables nonlinear units are obtained by passing the outputs of linear gaussian units through various nonlinearities we present a general variational method that maximizes a lower bound on the likelihood of a training set and give results on two visual feature extraction problems we also show how the variational method can be used for pattern classification and compare the performance of these nonlinear networks with other methods on the problem of handwritten digit recognition

the fractional finite hankel transform and its applications in fractal space in the present work a generalized finite hankel transform is derived which is useful in solving equations in fractal dimension df and involving a fractal diffusion coefficient d0r the corresponding inversion formula is established and some properties are given then the transform is successfully used to solve a class of time fractional diffusion equations in fractional spatial dimension with an absorbent term and schrodinger equation in fractional dimensional space green s functions and exact wave function of the above problems are found

competition and multiple cause models if different causes can interact on any occasion to generate a set of patterns then systems modeling the generation have to model the interaction too we discuss a way of combining multiple causes that is based on the integrated segmentation and recognition architecture of keeler et al 1991 it is more cooperative than the scheme embodied in the mixture of experts architecture which insists that just one cause generate each output and more competitive than the noisy or combination function which was recently suggested by saund 1994a b simulations confirm its efficacy

on lp solutions for a class of sequential fractional differential equations abstract under some simple conditions on the coefficient a t we establish that the initial value problem 0 d t x a t x 0 t 0 lim t 0 t 1 x t 0 has no solution in l p 1 r where p 1 p 1 p and 0 d t designates the riemann liouville derivative of order our result might be useful for developing a non integer variant of h weyl s limit circle limit point classification of differential equations

a multiple cause mixture model for unsupervised learning this paper presents a formulation for unsupervised learning of clusters reflecting multiple causal structure in binary data unlike the hard k means clustering algorithm and the soft mixture model each of which assumes that a single hidden event generates each data point a multiple cause model accounts for observed data by combining assertions from many hidden causes each of which can pertain to varying degree to any subset of the observable dimensions we employ an objective function and iterative gradient descent learning algorithm resembling the conventional mixture model a crucial issue is the mixing function for combining beliefs from different cluster centers in order to generate data predictions whose errors are minimized both during recognition and learning the mixing function constitutes a prior assumption about underlying structural regularities of the data domain we demonstrate a weakness inherent to the popular weighted sum followed by sigmoid squashing and offer alternative forms of the nonlinearity for two types of data domain results are presented demonstrating the algorithm s ability successfully to discover coherent multiple causal representations in several experimental data sets

minimal hypersurfaces and bordism of positive scalar curvature metrics let y g be a compact riemannian manifold of positive scalar curvature psc it is well known due to schoen yau that any closed stable minimal hypersurface of y also admits a psc metric we establish an analogous result for stable minimal hypersurfaces with free boundary furthermore we combine this result with tools from geometric measure theory and conformal geometry to study psc bordism for instance assume y 0 g 0 and y 1 g 1 are closed psc manifolds equipped with stable minimal hypersurfaces x 0 subset y 0 and x 1 subset y 1 under natural topological conditions we show that a psc bordism z bar g y 0 g 0 rightsquigarrow y 1 g 1 gives rise to a psc bordism between x 0 and x 1 equipped with the psc metrics given by the schoen yau construction

a general regularity theory for stable codimension 1 integral varifolds we give a necessary and sufficient geometric structural condition which we call the a structural hypothesis for a stable codimension 1 integral varifold on a smooth riemannian manifold to correspond to an embedded smooth hypersurface away from a small set of generally unavoidable singularities the a structural hypothesis says that no point of the support of the varifold has a neighborhood in which the support is the union of three or more embedded c 1 a hypersurfaces with boundary meeting only along their common boundary we establish that whenever a stable integral n varifold on a smooth n 1 dimensional riemannian manifold satisfies the a structural hypothesis for some a 0 1 2 its singular set is empty if n 6 discrete if n 7 and has hausdorff dimension n 7 if n 8 in view of well known examples this is the best possible general dimension estimate on the singular set of a varifold satisfying our hypotheses we also establish compactness of mass bounded subsets of the class of stable codimension 1 integral varifolds satisfying the a structural hypothesis for some a 0 1 2 the a structural hypothesis on an n varifold for any a 0 1 2 is readily implied by either of the following two hypotheses i the varifold corresponds to an absolutely area minimizing rectifiable current with no boundary ii the singular set of the varifold has vanishing n 1 dimensional hausdorff measure thus our theory subsumes the well known regularity theory for codimension 1 area minimizing rectifiable currents and settles the long standing question as to which weakest size hypothesis on the singular set of a stable minimal hypersurface guarantees the validity of the above regularity conclusions an optimal strong maximum principle for stationary codimension 1 integral varifolds follows from our regularity and compactness theorems

asymptotic integration of 1 α order fractional differential equations we establish the long time asymptotic formula of solutions to the 1 a order fractional differential equation 0 io t 1 ax a t x 0 t 0 under some simple restrictions on the functional coefficient a t where 0 io t 1 a is one of the fractional differential operators 0d t a x 0d t ax 0d t 1 ax and 0d t a tx x here 0d t a designates the riemann liouville derivative of order a 0 1 the asymptotic formula reads as b o 1 x s m a l l c x l a r g e as t for given b c r where x s m a l l and x l a r g e represent the eventually small and eventually large solutions that generate the solution space of the fractional differential equation 0 io t 1 ax 0 t 0

curvature estimates and sheeting theorems for weakly stable cmc hypersurfaces abstract weakly stable constant mean curvature cmc hypersurfaces are stable critical points of the area functional with respect to volume preserving deformations we establish a pointwise curvature estimate in the non singular dimensions and a sheeting theorem in all dimensions for weakly stable cmc hypersurfaces giving an effective version of the compactness theorem for weakly stable cmc hypersurfaces established in the recent work of the first and third named authors 6 our results generalize the curvature estimate and the sheeting theorem proven respectively by schoen simon yau and schoen simon for strongly stable hypersurfaces

building blocks for variational bayesian learning of latent variable models we introduce standardised building blocks designed to be used with variational bayesian learning the blocks include gaussian variables summation multiplication nonlinearity and delay a large variety of latent variable models can be constructed from these blocks including nonlinear and variance models which are lacking from most existing variational systems the introduced blocks are designed to fit together and to yield efficient update rules practical implementation of various models is easy thanks to an associated software package which derives the learning formulas automatically once a specific model structure has been fixed variational bayesian learning provides a cost function which is used both for updating the variables of the model and for optimising the model structure all the computations can be carried out locally resulting in linear computational complexity we present experimental results on several structures including a new hierarchical nonlinear model for variances and means the test results demonstrate the good performance and usefulness of the introduced method

minimal hypersurfaces with bounded index we prove a structural theorem that provides a precise local picture of how a sequence of closed embedded minimal hypersurfaces with uniformly bounded index and volume if the ambient dimension is greater than three in a riemannian manifold m n g 3 le n le 7 can degenerate loosely speaking our results show that embedded minimal hypersurfaces with bounded index behave qualitatively like embedded stable minimal hypersurfaces up to controlled errors several compactness finiteness theorems follow from our local picture

stability of hypersurfaces with constant mean curvature soit m n compacte orientable et soit x m n r n 1 une immersion a courbure moyenne constante non nulle alors x est stable si et seulement si x m r n 1 est une sphere s n r n 1

a nagumo like uniqueness theorem for fractional differential equations we extend to fractional differential equations a recent generalization of the nagumo uniqueness theorem for ordinary differential equations of first order

new area minimizing lawson osserman cones abstract it has been 40 years since lawson and osserman introduced the three minimal cones associated with dirichlet problems in their 1977 acta paper 13 the first cone was shown area minimizing by harvey and lawson in the celebrated paper 10 in this paper we confirm that the other two are also area minimizing in fact we show that every lawson osserman cone of type n p 2 constructed in 26 is area minimizing

maximal causes for non linear component extraction we study a generative model in which hidden causes combine competitively to produce observations multiple active causes combine to determine the value of an observed variable through a max function in the place where algorithms such as sparse coding independent component analysis or non negative matrix factorization would use a sum this max rule can represent a more realistic model of non linear interaction between basic components in many settings including acoustic and image data while exact maximum likelihood learning of the parameters of this model proves to be intractable we show that efficient approximations to expectation maximization em can be found in the case of sparsely active hidden causes one of these approximations can be formulated as a neural network model with a generalized softmax activation function and hebbian learning thus we show that learning in recent softmax like neural networks may be interpreted as approximate maximization of a data likelihood we use the bars benchmark test to numerically verify our analytical results and to demonstrate the competitiveness of the resulting algorithms finally we show results of learning model parameters to fit acoustic and visual data sets in which max like component combinations arise naturally

qualitative and quantitative estimates for minimal hypersurfaces with bounded index and area we prove qualitative estimates on the total curvature of closed minimal hypersurfaces in closed riemannian manifolds in terms of their index and area restricting to the case where the hypersurface has dimension less than seven in particular we prove that if we are given a sequence of closed minimal hypersurfaces of bounded area and index the total curvature along the sequence is quantised in terms of the total curvature of some limit surface plus a sum of total curvatures of complete properly embedded minimal hypersurfaces in euclidean space all of which are finite thus we obtain qualitative control on the topology of minimal hypersurfaces in terms of index and area as a corollary

on the asymptotic integration of nonlinear differential equations the aim of the present paper is twofold firstly the paper surveys the literature concerning a specific topic in asymptotic integration theory of ordinary differential equations the class of second order equations with bihari like nonlinearity secondly some general existence results are established with regard to a condition that has been found recently to be of significant use in the theory of elliptic partial differential equations

stable solutions of the allen cahn equation in dimension 8 and minimal cones abstract for all n 1 we are interested in bounded solutions of the allen cahn equation u u u 3 0 which are defined in all r n 1 and whose zero set is asymptotic to a given minimal cone in particular in dimension n 1 8 we prove the existence of stable solutions of the allen cahn equation whose zero sets are not hyperplanes

effective versions of the positive mass theorem the study of stable minimal surfaces in riemannian 3 manifolds m g with non negative scalar curvature has a rich history in this paper we prove rigidity of such surfaces when m g is asymptotically flat and has horizon boundary as a consequence we obtain an effective version of the positive mass theorem in terms of isoperimetric or more generally closed volume preserving stable cmc surfaces that is appealing from both a physical and a purely geometric point of view we also include a proof of the following conjecture of schoen an asymptotically flat riemannian 3 manifold with non negative scalar curvature that contains an unbounded area minimizing surface is isometric to flat mathbb r 3

expectation truncation and the benefits of preselection in training generative models we show how a preselection of hidden variables can be used to efficiently train generative models with binary hidden variables the approach is based on expectation maximization em and uses an efficiently computable approximation to the sufficient statistics of a given model the computational cost to compute the sufficient statistics is strongly reduced by selecting for each data point the relevant hidden causes the approximation is applicable to a wide range of generative models and provides an interpretation of the benefits of preselection in terms of a variational em approximation to empirically show that the method maximizes the data likelihood it is applied to different types of generative models including a version of non negative matrix factorization nmf a model for non linear component extraction mca and a linear generative model similar to sparse coding the derived algorithms are applied to both artificial and realistic data and are compared to other models in the literature we find that the training scheme can reduce computational costs by orders of magnitude and allows for a reliable extraction of hidden causes

byy harmony learning independent state space and generalized apt financial analyses first the relationship between factor analysis fa and the well known arbitrage pricing theory apt for financial market is discussed comparatively with a number of to be improved problems listed an overview is made from a unified perspective on the related studies in the literatures of statistics control theory signal processing and neural networks next we introduce the fundamentals of the bayesian ying yang byy system and the harmony learning principle we further show that a specific case of the framework called byy independent state space iss system provides a general guide for systematically tackling various fa related learning tasks and the above to be improved problems for the apt analyses third on various specific cases of the byy iss system in three typical architectures adaptive algorithms regularization methods and model selection criteria are provided for either or both of parameter learning with automated model selection and parameter learning followed by model selection finally we introduce some other financial applications that are based on the underlying independent factors via the apt analyses

transformation invariant clustering using the em algorithm clustering is a simple effective way to derive useful representations of data such as images and videos clustering explains the input as one of several prototypes plus noise in situations where each input has been randomly transformed e g by translation rotation and shearing in images and videos clustering techniques tend to extract cluster centers that account for variations in the input due to transformations instead of more interesting and potentially useful structure for example if images from a video sequence of a person walking across a cluttered background are clustered it would be more useful for the different clusters to represent different poses and expressions instead of different positions of the person and different configurations of the background clutter we describe a way to add transformation invariance to mixture models by approximating the nonlinear transformation manifold by a discrete set of points we show how the expectation maximization algorithm can be used to jointly learn clusters while at the same time inferring the transformation associated with each input we compare this technique with other methods for filtering noisy images obtained from a scanning electron microscope clustering images from videos of faces into different categories of identification and pose and removing foreground obstructions from video we also demonstrate that the new technique is quite insensitive to initial conditions and works better than standard techniques even when the standard techniques are provided with extra data

on the solution set for a class of sequential fractional differential equations we establish here that under some simple restrictions on the functional coefficient a t the solution set of the fractional differential equation 0d tx a t x 0 splits between eventually small and eventually large solutions as t where 0d t designates the riemann liouville derivative of the order 0 1

on large volume preserving stable cmc surfaces in initial data sets let m g be a complete 3 dimensional asymptotically flat manifold with everywhere positive scalar curvature we prove that given a compact subset k subset m all volume preserving stable constant mean curvature surfaces of sufficiently large area will avoid k this complements the results of g huisken and s t yau and of j qing and g tian on the uniqueness of large volume preserving stable constant mean curvature spheres in initial data sets that are asymptotically close to schwarzschild with mass m gt 0 the analysis in g huisken and s t yau and in j qing and g tian takes place in the asymptotic regime of m here we adapt ideas from the minimal surface proof of the positive mass theorem by r schoen and s t yau and develop geometric properties of volume preserving stable constant mean curvature surfaces to handle surfaces that run through the part of m that is far from euclidean

regularity at infinity for area minimizing hypersurfaces in hyperbolic space equipped with the hyperbolic metric y z dxz dy2 a standard compactification of ih involves adding the boundary r 0 w so that i i is simply the one point compactification of the euclidean closed half space r 0 suppose 0 7 any interior singularities of m must remain in a bounded region of hyperbolic space near points of f in the euclidean topology m w f may thus be described as the graph of a function this function is the solution of an interesting partial differential equation that becomes degenerate along the part of the boundary corresponding to f the second author has recently established l a boundary higher regularity result for this equation which implies in particular that m w f is r if f is cgk for k 2 3 finally in case f bounds a star

stability of hypersurfaces of constant mean curvature in riemannian manifolds hypersurfaces m n with constant mean curvature in a riemannian manifold overline m n 1 display many similarities with minimal hypersurfaces of overline m n 1 they are both solutions to the variational problem of minimizing the area function for certain variations in the first case however the admissible variations are only those that leave a certain volume function fixed for precise definitions see sect 2 this isoperimetric character of the variational problem associated to hypersurfaces of constant mean curvature introduces additional complications in the treatment of stability of such hypersurfaces

on shifted jacobi spectral approximations for solving fractional differential equations in this paper a new formula of caputo fractional order derivatives of shifted jacobi polynomials of any degree in terms of shifted jacobi polynomials themselves is proved we discuss a direct solution technique for linear multi order fractional differential equations fdes subject to nonhomogeneous initial conditions using a shifted jacobi tau approximation a quadrature shifted jacobi tau q sjt approximation is introduced for the solution of linear multi order fdes with variable coefficients we also propose a shifted jacobi collocation technique for solving nonlinear multi order fractional initial value problems the advantages of using the proposed techniques are discussed and we compare them with other existing methods we investigate some illustrative examples of fdes including linear and nonlinear terms we demonstrate the high accuracy and the efficiency of the proposed techniques

local removable singularity theorems for minimal laminations in this paper we prove a local removable singularity theorem for certain minimal laminations with isolated singularities in a riemannian three manifold this removable singularity theorem is the key result used in our proof that a complete embedded minimal surface in r 3 with quadratic decay of curvature has nite total curvature