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

expectation consistent approximate inference we propose a novel framework for approximations to intractable probabilistic models which is based on a free energy formulation the approximation can be understood as replacing an average over the original intractable distribution with a tractable one it requires two tractable probability distributions which are made consistent on a set of moments and encode different features of the original intractable distribution in this way we are able to use gaussian approximations for models with discrete or bounded variables which allow us to include non trivial correlations these are neglected in many other methods we test the framework on toy benchmark problems for binary variables on fully connected graphs and 2d grids and compare with other methods such as loopy belief propagation good performance is already achieved by using single nodes as tractable substructures significant improvements are obtained when a spanning tree is used instead

the segal bargmann coherent state transform for compact lie groups abstract let k be an arbitrary compact connected lie group we describe on k an analog of the segal bargmann coherent state transform and we prove theorem 1 that this generalized coherent state transform maps l 2 k isometrically onto the space of holomorphic functions in l 2 g where g is the complexification of k and where is an appropriate heat kernel measure on g the generalized coherent state transform is defined in terms of the heat kernel on the compact group k and its analytic continuation to the complex group g we also define a k averaged version of the coherent state transform and we prove a similar result for it theorem 2 in the appendix we describe the coherent states which motivate the definitions of these transforms in section 9 we obtain inversion formulas for both the coherent state transform and the k averaged coherent state transform theorems 3 4 as a consequence we obtain formulas for certain bergman kernels on the complex group g theorems 5 6 in section 10 we derive a general paley wiener theorem for k and we exhibit a family of convolution transforms each of which is an isometric isomorphism of l 2 k onto a certain l 2 space of holomorphic functions on g these transforms include the k averaged coherent state transform as a special case we also discuss the analogous results for r 1 the results of section 10 are theorems 7 10 finally in section 11 we discuss the case of quotient spaces we prove theorem 11 that both the coherent state transform and the k averaged coherent state transform pass in a natural way from k to k h where h is a closed connected subgroup of k

some rigorous results for yang mills theories on a cylinder using an approach based on the canonical formalism the yang mills theories on a cylinder are rigorously analyzed in this way the moduli space a g can be explicitly described with a being the space of connections and g the group of gauge transformations in particular a g0 g0 being the group of the pointed gauge transformations is diffeomorphic to the structure group of the theory g whereas a g is g modulo the group of inner automorphisms it is also proven that a g is a principal fiber bundle with structure group g0

on quantum gauge theories in two dimensions two dimensional quantum yang mills theory is studied from three points of view i by standard physical methods ii by relating it to the largek limit of three dimensional chern simons theory and two dimensional conformal field theory iii by relating its weak coupling limit to the theory of reidemeister ray singer torsion the results obtained from the three points of view agree and give formulas for the volumes of the moduli spaces of representations of fundamental groups of two dimensional surfaces

the inverse segal bargmann transform for compact lie groups in this paper i give a new inversion formula theorem 1 for the generalized segal bargmann transform introduced in b c hall j funct anal 122 1994 103 151 the inversion formula may be viewed as a formula for theinverseheat operator for a compact lie group i then use this formula to give a new direct proof of the unitary of thek invariant form of the segal bargmann transform theorem 2 the proof of the inversion formula relies on an identity theorem 5 which relates the laplacian for a compact lie groupkto the laplacian for the non compact dual symmetric space kc k

classifications of bundle connection pairs by parallel translation and lassos let m be a connected manifold and g be a closed lie subgroup of gl v the general linear group on a finite dimensional vector space v denote by m the space of h1 loops in m starting at a fixed point m let m be the set of p c m g modulo conjugation by an element of g such that p p p for m is the concatenation of the loops and p p if is reparameterization of and the differential of p satisfies a locality condition it is shown that the bundle connection pairs e up to equivalence with structure group g and fiber v are in one to one correspondence with m a similar result has been announced by kobayashi the correspondence is induced by the parallel translation operators of connections furthermore if the manifold m is simply connected then the space of bundle connection pairs can be classified by a collection of lie algebra valued 1 forms on the manifold m called integrated lassos these 1 forms are related to the differentials of elements of m this last result generalizes weil s characterization of u 1 line bundle connection pairs by the curvature 2 form it is also a generalization of gross results to base manifolds m other than rn

on bounded pseudodifferential operators in wiener spaces we aim at extending the definition of the weyl calculus to an infinite dimensional setting by replacing the phase space r2n by b2 where i h b is an abstract wiener space a first approach is to generalize the integral definition using the wigner function the symbol is then a function defined on b2 and belonging to a l1 space for a gaussian measure the weyl operator is defined as a quadratic form on a dense subspace of l2 b for example the symbol can be the stochastic extension on b2 in the sense of l gross of a function f which is continuous and bounded on h2 in the second approach this function f defined on h2 satisfies differentiability conditions analogous to the finite dimensional ones one needs to introduce hybrid operators acting as weyl operators on the variables of finite dimensional subset of h and as anti wick operators on the rest of the variables the final weyl operator is then defined as a limit and it is continuous on a l2 space under rather weak conditions it is an extension of the operator defined by the first approach we give examples of monomial symbols linking this construction to the classical pseudodifferential operators theory and other examples related to other fields or previous works on this subject

an alternative perspective on adaptive independent component analysis algorithms this article develops an extended independent component analysis algorithm for mixtures of arbitrary subgaussian and supergaussian sources the gaussian mixture model of pearson is employed in deriving a closedform generic score function for strictly subgaussian sources this is combined with the score function for a unimodal supergaussian density to provide a computationally simple yet powerful algorithm for performing independent component analysis on arbitrary mixtures of nongaussian sources

coherent state transforms and vector bundles on elliptic curves abstract we extend the coherent state transform cst of hall to the context of the moduli spaces of semistable holomorphic vector bundles with fixed determinant over elliptic curves we show that by applying the cst to appropriate distributions we obtain the space of level k rank n and genus one non abelian theta functions with the unitarity of the cst transform being preserved furthermore the shift in the level k k n appears in a natural way in this finite dimensional framework

brown measure support and the free multiplicative brownian motion abstract the free multiplicative brownian motion b t is the large n limit of brownian motion b t n on the general linear group gl n c we prove that the brown measure for b t which is an analog of the empirical eigenvalue distribution for matrices is supported on the closure of a certain domain t in the plane the domain t was introduced by biane in the context of the large n limit of the segal bargmann transform associated to gl n c we also consider a two parameter version b s t the large n limit of a related family of diffusion processes on gl n c introduced by the second author we show that the brown measure of b s t is supported on the closure of a certain planar domain s t generalizing t introduced by ho in the process we introduce a new family of spectral domains related to any operator in a tracial von neumann algebra the l p n spectrum for n n and p 1 a subset of the ordinary spectrum defined relative to potentially unbounded inverses we show that in general the support of the brown measure of an operator is contained in its l 2 2 spectrum

state space models from the em algorithm to a gradient approach slow convergence is observed in the em algorithm for linear state space models we propose to circumvent the problem by applying any off the shelf quasi newton type optimizer which operates on the gradient of the log likelihood function such an algorithm is a practical alternative due to the fact that the exact gradient of the log likelihood function can be computed by recycling components of the expectation maximization em algorithm we demonstrate the efficiency of the proposed method in three relevant instances of the linear state space model in high signal to noise ratios where em is particularly prone to converge slowly we show that gradient based learning results in a sizable reduction of computation time

coherent state transforms and abelian varieties we extend the coherent state transform cst of hall to the context of abelian varieties by considering them as quotients of the complexification of the abelian group k u 1 g we show that this transform applied to appropriate distributions on k gives all classical theta functions and that by defining on this space of theta functions an inner product related to the k averaged heat kernel the unitarity of the cst transform is still preserved

the two parameter free unitary segal bargmann transform and its biane gross malliavin identification abstract motivated by a conditional expectation interpretation of the segal bargmann transform we derive the integral kernel for the large n limit of the two parameter segal bargmann hall transform over the unitary group u n and explore its limiting behavior we also extend the notion of circular systems to more general elliptic systems in order to give an alternate construction of our new two parameter free unitary segal bargmann hall transform via a biane gross malliavin type theorem

a generalized koszul theory and its relation to the classical theory abstract let a i 0 a i be a graded locally finite k algebra where a 0 is a finite dimensional algebra whose finitistic dimension is 0 in this paper we develop a generalized koszul theory preserving many classical results and show an explicit correspondence between this generalized theory and the classical theory applications in representations of certain categories and extension algebras of standard modules of standardly stratified algebras are described

nonlinear and noisy extension of independent component analysis theory and its application to a pitch sensation model in this letter we propose a noisy nonlinear version of independent component analysis ica assuming that the probability density function p d f of sources is known a learning rule is derived based on maximum likelihood estimation mle our model involves some algorithms of noisy linear ica e g bermond cardoso 1999 or noise free nonlinear ica e g lee koehler orglmeister 1997 as special cases especially when the nonlinear function is linear the learning rule derived as a generalized expectation maximization algorithm has a similar form to the noisy ica algorithm previously presented by douglas cichocki and amari 1998 moreover our learning rule becomes identical to the standard noise free linear ica algorithm in the noiseless limit while existing mle based noisy ica algorithms do not rigorously include the noise free ica we trained our noisy nonlinear ica by using acoustic signals such as speech and music the model after learning successfully simulates virtual pitch phenomena and the existence region of virtual pitch is qualitatively similar to that observed in a psychoacoustic experiment although a linear transformation hypothesized in the central auditory system can account for the pitch sensation our model suggests that the linear transformation can be acquired through learning from actual acoustic signals since our model includes a cepstrum analysis in a special case it is expected to provide a useful feature extraction method that has often been given by the cepstrum analysis

phase space bounds for quantum mechanics on a compact lie group let k be a compact connected lie group and k bbb c its complexification i consider the hilbert space cal h l 2 left k bbb c nu t right of holomorphic functions introduced in h1 where the parameter t is to be interpreted as planck s constant in light of l s the complex group k bbb c may be identified canonically with the cotangent bundle of k using this identification i associate to each f in cal h l 2 left k bbb c nu t right a phase space probability density the main result of this paper is theorem 1 which provides an upper bound on this density which holds uniformly over all f and all points in phase space specifically the phase space probability density is at most a t left 2 pi t right n where n dim k and a t is a constant which tends to one exponentially fast as t tends to zero at least for small t this bound cannot be significantly improved

lattice independent component analysis for functional magnetic resonance imaging we introduce a lattice independent component analysis lica unsupervised scheme to functional magnetic resonance imaging fmri data analysis lica is a non linear alternative to independent component analysis ica such that ica s statistical independent sources correspond to lica s lattice independent sources in this paper lica uses an incremental lattice source induction algorithm ilsia to induce the lattice independent sources from the input dataset the ilsia computes a set of strongly lattice independent vectors using properties of lattice associative memories regarding lattice independence and chebyshev best approximation the lattice independent sources constitute a set of affine independent vectors that define a simplex covering the input data lica carries out data linear unmixing based on the lattice independent sources basis therefore lica is a hybrid combination of a non linear lattice based component and a linear unmixing component the principal advantage over ica is that lica does not impose any probabilistic model assumptions on the data sources we compare lica with ica in two case studies firstly including simulated fmri data lica discovers the spatial location of meaningful sources with less ambiguity than ica secondly including real data from an auditory stimulation experiment lica improves over some state of the art ica variants discovering the activation patterns detected by statistical parametric mapping spm on the same data

irreducibility and reducibility for the energy representation of the group of mappings of a riemannian manifold into a compact semisimple lie group abstract the irreducibility of the energy representation of the group of smooth mappings from a riemannian manifold of dimension d 3 into a compact semisimple lie group is proven the nonequivalence of the representations associated with different riemann structures is also proven for d 3 the case d 2 is examined and irreducibility and nonequivalence results are also given the reducibility in the case d 1 is pointed out in this case the commutant contains a representation equivalent with the energy representation

coherent state transforms for spaces of connections abstract the segal bargmann transform plays an important role in quantum theories of linear fields recently hall obtained a non linear analog of this transform for quantum mechanics on lie groups given a compact connected lie groupgwith its normalized haar measure h the hall transform is an isometric isomorphism froml2 g h to h g c l2 g c whereg c the complexification ofg h g c the space of holomorphic functions ong c and an appropriate heat kernel measure ong c we extend the hall transform to the infinite dimensional context of non abelian gauge theories by replacing the lie groupgby a certain extension of the space a g of connections modulo gauge transformations the resulting coherent state transform provides a holomorphic representation of the holonomyc algebra of real gauge fields this representation is expected to play a key role in a non perturbative canonical approach to quantum gravity in 4 dimensions

operator algebraic construction of gauge theories and jones actions of thompson s groups using ideas from jones lattice gauge theory and loop quantum gravity we construct 1 1 dimensional gauge theories on a spacetime cylinder given a separable compact group g we construct localized time zero fields on the spatial torus as a net of c algebras together with an action of the gauge group that is an infinite product of g over the dyadic rationals and using a recent machinery of jones an action of thompson s group t as a replacement of the spatial diffeomorphism group adding a family of probability measures on the unitary dual of g we construct a state and obtain a net of von neumann algebras carrying a state preserving gauge group action for abelian g we provide a very explicit description of our algebras for a single measure on the dual of g we have a state preserving action of thompson s group and semi finite von neumann algebras for g mathbf s the circle group together with a certain family of heat kernel states providing the measures we obtain hyperfinite type iii factors with a normal faithful state providing a nontrivial time evolution via tomita takesaki theory kms condition in the latter case we additionally have a non singular action of the group of rotations with dyadic angles as a subgroup of thompson s group t for geometrically motivated choices of families of heat kernel states

quivers floer cohomology and braid group actions we consider the derived categories of modules over a certain family a m of graded rings and floer cohomology of lagrangian intersections in the symplectic manifolds which are the milnor fibres of simple singularities of type a m we show that each of these two rather different objects encodes the topology of curves on an m 1 punctured disc we prove that the braid group b m 1 acts faithfully on the derived category of a m modules and that it injects into the symplectic mapping class group of milnor fibers the philosophy behind our results is as follows using floer cohomology one should be able to associate to the milnor fibre a triangulated category its construction has not been carried out in detail this triangulated category should contain a full subcategory which is equivalent up to a slight difference in the grading to the derived category of a m modules the full embedding would connect the two occurrences of the braid group thus explaining the similarity between them

bgg algebras and the bgg reciprocity principle the bgg reciprocity principle is a famous result in the representation theary of complex semisimple lie algebras proved by bernstein gelfand and gelfand in bgg 21 it may be viewed as an analogue for the category 6 of the brauer reciprocity principle in the modular representation theory of finite groups and of humphreys reciprocity principle in the representation theory of restricted lie algebras indeed the result of humphreys served as motivation for the bgg result similar results have since been proved in a number of other settings by various approaches for all these results one works in a suitable category of modules in which each simple module l has a projective cover p in addition there is a family of intermediate modules mi such that each p has a filtration with m s as quotients then the appropriate reciprocity principle states that m occurs as a filtration quotient in p as often as l occurs as a composition factor in mi in the case of the category 0 the simple modules are the simple highest weight modules the category is introduced as a setting for projective covers to exist and the intermediate modules are the verma modules the proof of bgg reciprocity in bgg 21 is fairly straightforward once the basic facts about the category 0 are developed the crucial point is the construction of enough projective modules in 0 once these are constructed the existence of projective covers the existence of suitable filtrations on these and the reciprocity principle are ail proved with reference to the construction however another proof of reciprocity exists which is more natural in the sense that it makes no further reference to the construction of projective modules this proof can be used in some of the other settings as well although in any given setting no drastic simplifications result i believe it is still worthwhile to draw attention to this more conceptual approach the properties of the category 0 which are needed for the proof may be

linear state space models for blind source separation we apply a type of generative modelling to the problem of blind source separation in which prior knowledge about the latent source signals such as time varying auto correlation and quasi periodicity are incorporated into a linear state space model in simulations we show that in terms of signal to error ratio the sources are inferred more accurately as a result of the inclusion of strong prior knowledge we explore different schemes of maximum likelihood optimization for the purpose of learning the model parameters the expectation maximization algorithm which is often considered the standard optimization method in this context results in slow convergence when the noise variance is small in such scenarios quasi newton optimization yields substantial improvements in a range of signal to noise ratios we analyze the performance of the methods on convolutive mixtures of speech signals

on a common generalization of koszul duality and tilting equivalence abstract we propose a new definition of koszulity for graded algebras where the degree zero part has finite global dimension but is not necessarily semi simple the standard koszul duality theorems hold in this setting

the complex time segal bargmann transform abstract we introduce a new form of the segal bargmann transform for a connected lie group k of compact type we show that the heat kernel t x t 0 x k has a space time analytic continuation to a holomorphic function c z re 0 z k c where k c is the complexification of k the new transform is defined by the integral b f z k c z k 1 f k d k z k c if s 0 and d s s the disk of radius s centered at s this integral defines a holomorphic function on k c for each f l 2 k s we construct a heat kernel density s on k c such that for all s as above b s b l 2 k s is an isometric isomorphism from l 2 k s onto the space of holomorphic functions in l 2 k c s when t s the transform b t t coincides with the one introduced by the second author for compact groups and extended by the first author to groups of compact type when t 0 2 s the transform b s t coincides with the one introduced by the first two authors

koszul duality patterns in representation theory the aim of this paper is to work out a concrete example as well as to provide the general pattern of applications of koszul duality to repre sentation theory the paper consists of three parts relatively independent of each other the first part gives a reasonably selfcontained introduction to koszul rings and koszul duality koszul rings are certain z graded rings with particularly nice homological properties which involve a kind of duality thus to a koszul ring one associates naturally the dual koszul ring the second part is devoted to an application to representation theory of semisimple lie algebras we show license or copyright restrictions may apply to redistribution see http www ams org journal terms of use koszul duality patterns 527 that the block of the bernstein gelfand gelfand category o that corresponds to any fixed central character is governed by the koszul ring moreover the dual of that ring governs a certain subcategory of the category o again this generalizes the selfduality theorem conjectured by beilinson and ginsburg in 1986 and proved by soergel in 1990 in the third part we study certain cate gories of mixed perverse sheaves on a variety stratified by affine linear spaces we provide a general criterion for such a category to be governed by a koszul ring in the flag variety case this reduces to the setup of part two in the more general case of affine flag manifolds and affine grassmannians the criterion should yield interesting results about representations of quantum groups and affine lie algebras department of mathematics massachusetts institute of technology cambridge massachusetts 02139 e mail address sasha math mit edu department of mathematics the university of chicago chicago illinois 60637 e mail address ginzburg math uchicago edu max planck institut fur mathematik gottfried claren strase 26 d 53 bonn 3 germany current address mathematisches institut universitat freiburg albertstrase 23b d 79104 freiburg germany e mail address soergel sun1 mathematik uni freiburg de license or copyright restrictions may apply to redistribution see http www ams org journal terms of use

mean field approaches to independent component analysis we develop mean field approaches for probabilistic independent component analysis ica the sources are estimated from the mean of their posterior distribution and the mixing matrix and noise level is estimated by maximum a posteriori map the latter requires the computation of a good approximation to the correlations between sources for this purpose we investigate three increasingly advanced mean field methods the variational also known as naive mean field approach linear response corrections and an adaptive version of the thouless anderson and palmer 1977 tap mean field approach which is due to opper and winther 2001 the resulting algorithms are tested on a number of problems on synthetic data the advanced mean field approaches are able to recover the correct mixing matrix in cases where the variational mean field theory fails for handwritten digits sparse encoding is achieved using nonnegative source and mixing priors for speech the mean field method is able to separate in the underdetermined overcomplete case of two sensors and three sources one major advantage of the proposed method is its generality and algorithmic simplicity finally we point out several possible extensions of the approaches developed here

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

quasi hereditary algebras and generalized koszul duality abstract we present an easily applicable sufficient condition for standard koszul algebras to be koszul with respect to if a quasi hereditary algebra is koszul with respect to then and the yoneda extension algebra of are koszul dual in a sense explained below implying in particular that their bounded derived categories of finitely generated graded modules are equivalent we also prove that the extension algebra of is koszul in the classical sense

yang mills theory and the segal bargmann transform we use a variant of the segal bargmann transform to study canonically quantized yang mills theory on a space time cylinder with a compact structure group k the non existent lebesgue measure on the space of connections is approximated by a gaussian measure with large variance the segal bargmann transform is then a unitary map from the l 2 space over the space of connections to a holomorphicl 2 space over the space of complexified connections with a certain gaussian measure this transform is given roughly bye t1a 2 followed by analytic continuation here 1a is the laplacian on the space of connections and is the hamiltonian for the quantized theory on the gauge trivial subspace consisting of functions of the holonomy around the spatial circle the segal bargmann transform becomes e t1 k 2 followed by analytic continuation where 1k is the laplacian for the structure group k this result gives a rigorous meaning to the idea that 1a reduces to 1k on functions of the holonomy by letting the variance of the gaussian measure tend to infinity we recover the standard realization of the quantized yang mills theory on a space time cylinder namely 1 1k is the hamiltonian and l 2 k is the hilbert space as a byproduct of these considera tions we find a new one parameter family of unitary transforms from l 2 k to certain holomorphicl 2 spaces over the complexification of k this family of transformations interpolates between the two previously known unitary transformations our work is motivated by results of landsman and wren and uses probabilistic techniques similar to those of gross and malliavin

analytical method for blind binary signal separation the blind separation of multiple co channel binary digital signals using an antenna array involves finding a factorization of a data matrix x into x as where all entries of s are 1 or 1 it is shown that this problem can be solved exactly and noniteratively via a certain generalized eigenvalue decomposition as indicated by simulations the algorithm is robust in the presence of noise an interesting implication is that certain cluster segmentation problems can be solved using eigenvalue techniques

the segal bargmann transform for path groups letkbe a connected lie group of compact type and letw k denote the set of continuous paths ink starting at the identity and with time interval 0 1 thenw k forms an infinite dimensional group under the operation of pointwise multiplication let denote the wiener measure onw k we construct an analog of the segal bargmann transform forw k letkcbe the complexification ofk w kc the set of continuous paths inkcstarting at the identity and the wiener measure onw kc our transform is a unitary map ofl2 w k onto the holomorphic subspace ofl2 w kc by analogy with the classical transform our transform is given by convolution with the wiener measure followed by analytic continuation we prove that the transform forw k is nicely related by means of the ito map to the classical segal bargmann transform for the path space in the lie algebra ofk

gaussian processes for classification mean field algorithms we derive a mean field algorithm for binary classification with gaussian processes that is based on the tap approach originally proposed in statistical physics of disordered systems the theory also yields an approximate leave one out estimator for the generalization error which is computed with no extra computational cost we show that from the tap approach it is possible to derive both a simpler naive mean field theory and support vector machines svms as limiting cases for both mean field algorithms and support vector machines simulation results for three small benchmark data sets are presented they show that one may get state of the art performance by using the leave one out estimator for model selection and the built in leave one out estimators are extremely precise when compared to the exact leave one out estimate the second result is taken as strong support for the internal consistency of the mean field approach

logarithmic sobolev inequalities for pinned loop groups abstract letgbe a connected compact type lie group equipped with anadg invariant inner product on the lie algebra g ofg given this data there is a well known left invariant h1 riemannian structure on l l g the infinite dimensional group of continuous based loops ing using this riemannian structure we define and construct a heat kernel t g0 associated to the laplace beltrami operator on l g heret 0 g0 l g and t g0 is a certain probability measure on l g for fixedg0 l g andt 0 we use the measure t g0 and the riemannian structure on l g to construct a classical pre dirichlet form the main theorem of this paper asserts that this pre dirichlet form admits a logarithmic sobolev inequality

on the kakutani ito segal gross and segal bargmann hall isomorphisms recently gross has shown that the kakutani ito segal isomorphism theorem has an extension from the setting of gaussian measure on a vector space to heat kernel measure pt on a simply connected lie group g of compact type the isomorphism relates l2 pt to a certain completion of the universal enveloping algebra of g lie g gross proves this result using the kakutani ito segal theorem and an infinite dimensional calculus associated to g valued brownian motion hijab has greatly simplified and clarified gross proof hiiab s proof avoids most but not all of the infinite dimensional analysis in the original proof in this paper we will build on hijab s proof to give a completely finite dimensional non probabilistic proof of gross isomorphism theorem the proof given here relies heavily on hall s beautiful extension of the segal bargmann transformation to the setting of compact lie groups this theorem relating l2 pt to a certain l2 space of holomorphic functions l2 gc h on the complexified lie group gc will also be generalized to lie groups of compact type in the process it is shown how to characterize in terms of summability conditions on all the derivatives at the identity in gc those holomorphic functions which are in l2 gc

isometry theorem for the segal bargmann transform on a noncompact symmetric space of the complex type we consider the segal bargmann transform on a noncompact symmetric space of the complex type we establish isometry and surjectivity theorems for the transform in a form as parallel as possible to the results in the dual compact case the isometry theorem involves integration over a tube of radius r in the complexification followed by analytic continuation with respect to r a cancellation of singularities allows the relevant integral to have a nonsingular extension to large r even though the function being integrated has singularities

unitarity in quantization commutes with reduction let m be a compact kahler manifold equipped with a hamiltonian action of a compact lie group g in this paper we study the geometric quantization of the symplectic quotient m g guillemin and sternberg invent math 67 515 538 1982 have shown under suitable regularity assumptions that there is a natural invertible map between the quantum hilbert space over m g and the g invariant subspace of the quantum hilbert space over m

on the equivalence of measures on loop space let k be a simply connected compact lie group equipped with an ad k invariant inner product on the lie algebra of k given this data there is a well known left invariant h 1 riemannian structure on l k the infinite dimensional group of continuous based loops in k as well as a heat kernel t k 0 associated with the laplace beltrami operator on l k here t 0 k 0 l k and t k 0 is a certain probability measure on l k in this paper we show that 1 e is equivalent to pinned wiener measure on k on s0 x t t 0 s 0 the algebra generated by truncated loops up to time s 0

on the relation between the isra and the em algorithm for positron emission tomography the image space reconstruction algorithm isra was proposed as a modification of the expectation maximization em algorithm based on physical considerations for application in volume emission computered tomography as a consequence of this modification isra searches for least squares solutions instead of maximizing poisson likelihoods as the em algorithm it is shown that both algorithms may be obtained from a common mathematical framework this fact is used to extend isra for penalized likelihood estimates

variational bayesian matrix factorization for bounded support data a novel bayesian matrix factorization method for bounded support data is presented each entry in the observation matrix is assumed to be beta distributed as the beta distribution has two parameters two parameter matrices can be obtained which matrices contain only nonnegative values in order to provide low rank matrix factorization the nonnegative matrix factorization nmf technique is applied furthermore each entry in the factorized matrices i e the basis and excitation matrices is assigned with gamma prior therefore we name this method as beta gamma nmf bg nmf due to the integral expression of the gamma function estimation of the posterior distribution in the bg nmf model can not be presented by an analytically tractable solution with the variational inference framework and the relative convexity property of the log inverse beta function we propose a new lower bound to approximate the objective function with this new lower bound we derive an analytically tractable solution to approximately calculate the posterior distributions each of the approximated posterior distributions is also gamma distributed which retains the conjugacy of the bayesian estimation in addition a sparse bg nmf can be obtained by including a sparseness constraint to the gamma prior evaluations with synthetic data and real life data demonstrate the good performance of the proposed method

geometric quantization and the generalized segal bargmann transform for lie groups of compact type let k be a connected lie group of compact type and let t k be its cotangent bundle this paper considers geometric quantization of t k first using the vertical polarization and then using a natural kahler polarization obtained by identifying t k with the complexified group k the first main result is that the hilbert space obtained by using the kahler polarization is naturally identifiable with the generalized segal bargmann space introduced by the author from a different point of view namely that of heat kernels the second main result is that the pairing map of geometric quantization coincides with the generalized segal bargmann transform introduced by the author this means that the pairing map in this case is a constant multiple of a unitary map for both results it is essential that the half form correction be included when using the kahler polarization

square integrable holomorphic functions on infinite dimensional heisenberg type groups we introduce a class of non commutative complex infinite dimensional heisenberg like lie groups based on an abstract wiener space the holomorphic functions which are also square integrable with respect to a heat kernel measure on these groups are studied in particular we establish a unitary equivalence between the square integrable holomorphic functions and a certain completion of the universal enveloping algebra of the lie algebra of this class of groups using quasi invariance of the heat kernel measure we also construct a skeleton map which characterizes globally defined functions from the l 2 closure of holomorphic polynomials by their values on the cameron martin subgroup

absolute continuity of heat kernel measure with pinned wiener measure on loop groups let t 0 k be a connected compact lie group equipped with an ad k invariant inner product on the lie algebra of k associated to this data are two measures 0 t and t 0 on l k the space of continuous loops based at e e k the measure 0 t is pinned wiener measure with variance t while the measure 0 t is a heat kernel measure on l k the measure 0 t is constructed using a k valued brownian motion while the measure 0 t is constructed using a l k valued brownian motion in this paper we show that 0 t is absolutely continuous with respect to 0 t and the radon nikodym derivative d 0 t d 0 t is bounded

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

analysis of mean field annealing in substactive interference cancellation in this contribution we derive the cost function corresponding to the linear complexity subtractive interference cancellation with tangent hyperbolic tentative decisions we use the cost function to analyse the fix points of solving the subtractive interference cancellation equations the analysis show that we can control the slope of the tangent hyperbolic functions so that the corresponding cost function is convex we also show that increasing the slope can make the cost non convex going from the convex regime into the non convex regime we prove that the bifurcation of the fix points for non singular signal correlation matrices consist of the fix point of interest together with a saddle node bifurcation this proves that tracking the solution from low slopes with a convex cost gradually increasing to higher slopes with non convex cost can bring us to the best solution being very close to the optimal determined by enumeration this tracking is the idea behind annealing we show monte carlo studies with a substantial signal to noise ratio gain compared to not using annealing we also show how annealing can be used to increase capacity at a given target bit error rate in fact this capacity gain is the same obtained by improved parallel interference cancellation making us believe that the latter includes a mechanism to avoid local minima similar to annealing

the large n limit of the segal bargmann transform on un we study the two parameter segal bargmann transform b n on the unitary group un for large n acting on matrix valued functions that are equivariant under the adjoint action of the group the transform has a meaningful limit gs t as n which can be identified as an operator on the space of complex laurent polynomials we introduce the space of trace polynomials and use it to give effective computational methods to determine the action of the heat operator and thus the segal bargmann transform we prove several concentration of measure and limit theorems giving a direct connection from the finite dimensional transform b n to its limit gs t we characterize the operator gs t through its inverse action on the standard polynomial basis finally we show that in the case s t the limit transform gt t is the free hall transform g t introduced by biane

the weyl extension algebra of gl2 f p abstract we compute the yoneda extension algebra of the collection of weyl modules for g l 2 over an algebraically closed field of characteristic p 0 by developing a theory of generalised koszul duality for certain 2 functors one of which controls the rational representation theory of g l 2 over such a field

efficient learning in boltzmann machines using linear response theory the learning process in boltzmann machines is computationally very expensive the computational complexity of the exact algorithm is exponential in the number of neurons we present a new approximate learning algorithm for boltzmann machines based on mean field theory and the linear response theorem the computational complexity of the algorithm is cubic in the number of neurons in the absence of hidden units we show how the weights can be directly computed from the fixed point equation of the learning rules thus in this case we do not need to use a gradient descent procedure for the learning process we show that the solutions of this method are close to the optimal solutions and give a significant improvement when correlations play a significant role finally we apply the method to a pattern completion task and show good performance for networks up to 100 neurons

coherent states on spheres we describe a family of coherent states and an associated resolution of the identity for a quantum particle whose classical configuration space is the d dimensional sphere sd the coherent states are labeled by points in the associated phase space t sd these coherent states are not of perelomov type but rather are constructed as the eigenvectors of suitably defined annihilation operators we describe as well the segal bargmann representation for the system the associated unitary segal bargmann transform and a natural inversion formula although many of these results are in principle special cases of the results of hall and stenzel we give here a substantially different description based on ideas of thiemann and of kowalski and rembielinski all of these results can be generalized to a system whose configuration space is an arbitrary compact symmetric space we focus on the sphere case in order to carry out the calculations in a self contained and explicit way

independent factor analysis we introduce the independent factor analysis ifa method for recovering independent hidden sources from their observed mixtures ifa generalizes and unifies ordinary factor analysis fa principal component analysis pca and independent component analysis ica and can handle not only square noiseless mixing but also the general case where the number of mixtures differs from the number of sources and the data are noisy ifa is a two step procedure in the first step the source densities mixing matrix and noise covariance are estimated from the observed data by maximum likelihood for this purpose we present an expectation maximization em algorithm which performs unsupervised learning of an associated probabilistic model of the mixing situation each source in our model is described by a mixture of gaussians thus all the probabilistic calculations can be performed analytically in the second step the sources are reconstructed from the observed data by an optimal nonlinear estimator a variational approximation of this algorithm is derived for cases with a large number of sources where the exact algorithm becomes intractable our ifa algorithm reduces to the one for ordinary fa when the sources become gaussian and to an em algorithm for pca in the zero noise limit we derive an additional em algorithm specifically for noiseless ifa this algorithm is shown to be superior to ica since it can learn arbitrary source densities from the data beyond blind separation ifa can be used for modeling multidimensional data by a highly constrained mixture of gaussians and as a tool for nonlinear signal encoding

berezin toeplitz quantization on lie groups let k be a connected compact semisimple lie group and kc its complexification the generalized segal bargmann space for kc is a space of square integrable holomorphic functions on kc with respect to a k invariant heat kernel measure this space is connected to the schrodinger hilbert space l 2 k by a unitary map the generalized segal bargmann transform this paper considers certain natural operators on l 2 k namely mul tiplication operators and differential operators conjugated by the general ized segal bargmann transform the main results show that the resulting operators on the generalized segal bargmann space can be represented as toeplitz operators the symbols of these toeplitz operators are expressed in terms of a certain subelliptic heat kernel on kc i also examine some of the results from an infinite dimensional point of view based on the work of l gross and p malliavin

the segal bargmann transform for noncompact symmetric spaces of the complex type we consider the generalized segal bargmann transform defined in terms of the heat operator for a noncompact symmetric space of the complex type for radial functions we show that the segal bargmann transform is a unitary map onto a certain l 2 space of meromorphic functions for general functions we give an inversion formula for the segal bargmann transform involving integration against an unwrapped version of the heat kernel for the dual compact symmetric space both results involve delicate cancellations of singularities

the geometry of loop groups l espace g des boucles basees sur un groupe de lie compact admet une metrique de kahler on exprime sa courbure en termes d operateurs de toeplitz et on definit des classes de chern par analogie avec la theorie de chern weil aux dimensions finies il y a une structure de fredhohn naturelle sur g induite a partir de la famille des operateurs de toeplitz on obtient des formules explicites pour su n

uniqueness of ground states for schrodinger operators over loop groups abstract over the space l of based loops in a compact connected lie group g there is a natural gradient operator induced by the left action on l of the hilbert lie group k 0 which consists of finite energy elements of l denote by pinned brownian motion measure on l the schrodinger operator v acting in l 2 l is shown to have a unique ground state over each homotopy class in l the proof of uniqueness is reduced to the proof of ergodicity of the left action of k 0 on l for a simply connected group g of compact type ergodicity in turn is proved by first characterizing the ito expansion coefficients of k 0 invariant functions and then regularizing such functions via a smooth approximation of multiple stratonovich integrals as a byproduct of this method one obtains an isometric isomorphism of l 2 g heat kernel measure at time one with a natural completion of the universal enveloping algebra of g thus extending to the noncommutative case the well known isometry between l 2 r d gauss measure and the space of symmetric tensors over r d

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

harmonic analysis with respect to heat kernel measure this paper surveys developments over the last decade in harmonic analysis on lie groups relative to a heat kernel measure these include analogs of the hermite expansion the segal bargmann transform and the taylor ex pansion some of the results can be understood from the standpoint of geo metric quantization others are intimately related to stochastic analysis