the geometry of the universal teichmuller space and the euler weil petersson equation on the identity component of the universal teichmuller space endowed with the takhtajan teo topology the geodesics of the weil petersson metric are shown to exist for all time this component is naturally a subgroup of the quasisymmetric homeomorphisms of the circle viewed this way the regularity of its elements is shown to be h3 2 epsilon for all epsilon 0 the evolutionary pde associated to the spatial representation of the geodesics of the weil petersson metric is derived using multiplication and composition below the critical sobolev index 3 2 geodesic completeness is used to introduce special classes of solutions of this pde analogous to peakons our setting is used to prove that there exists a unique geodesic between each two shapes in the plane in the context of the application of the weil petersson metric in imaging c 2015 elsevier inc all rights reserved

a k nearest neighbor based method for the restoration of damaged images abstract the present paper deals with the problem of restoring a digital image starting from its random sampling or its damaged version the method is adaptive and uses local evaluation of the kernel parameters applications of the method on simulated and real data are presented and its performance on the reconstruction of very sparse images is shown

on douady earle extension and the contractibility of the vmo teichmuller space abstract douady earle extension provides a quasiconformal extension of a quasisymmetric homeomorphism to the unit disk and induces a continuous self map of beltrami coefficients in this note we show that is also continuous under a stronger topology on those beltrami coefficients which induce vanishing carleson measures and thus project to the vmo teichmuller space an immediate consequence is that the vmo teichmuller space is contractible

generalized iterative relief for supervised distance metric learning the relief algorithm is a popular approach for feature weighting many extensions of the relief algorithm are developed and i relief is one of the famous extensions in this paper i relief is generalized for supervised distance metric learning to yield a mahananobis distance function the proposed approach is justified by showing that the objective function of the generalized i relief is closely related to the expected leave one out nearest neighbor classification rate in addition the relationships among the generalized i relief the neighbourhood components analysis and graph embedding are also pointed out experimental results on various data sets all demonstrate the superiority of the proposed approach

weil petersson teichmuller space revisited abstract in our previous paper 23 it was proved that a sense preserving homeomorphism g on the unit circle s 1 belongs to the weil petersson class wp s 1 namely g can be extended to a quasiconformal mapping to the unit disk whose beltrami coefficient is square integrable in the poincare metric if and only if g is absolutely continuous such that log g belongs to the sobolev class h 1 2 in this sequel to 23 we show that the smooth hilbert manifold structure on wp s 1 inherited from h 1 2 by the pullback g log g is compatible with the standard hilbert manifold structure introduced by takhtajan teo 27 this enables us to give a fast approach to some results in our previous papers 23 and 24

general entropy criteria for inverse problems with applications to data compression pattern classification and cluster analysis minimum distance approaches are considered for the reconstruction of a real function from finitely many linear functional values an optimal class of distances satisfying an orthogonality condition analogous to that enjoyed by linear projections in hilbert space is derived these optimal distances are related to measures of distances between probability distributions recently introduced by c r rao and t k nayak 1985 and possess the geometric properties of cross entropy useful in speech and image compression pattern classification and cluster analysis several examples from spectrum estimation and image processing are discussed

a boosting approach for supervised mahalanobis distance metric learning determining a proper distance metric is often a crucial step for machine learning in this paper a boosting algorithm is proposed to learn a mahalanobis distance metric similar to most boosting algorithms the proposed algorithm improves a loss function iteratively in particular the loss function is defined in terms of hypothesis margins and a metric matrix base learner specific to the boosting framework is also proposed experimental results show that the proposed approach can yield effective mahalanobis distance metrics for a variety of data sets and demonstrate the feasibility of the proposed approach

an optimal global nearest neighbor metric a quadratic metric dao x y x y t ao x y is proposed which minimizes the mean squared error between the nearest neighbor asymptotic risk and the finite sample risk under linearity assumptions a heuristic argument is given which indicates that this metric produces lower mean squared error than the euclidean metric a nonparametric estimate of ao is developed if samples appear to come from a gaussian mixture an alternative parametrically directed distance measure is suggested for nearness decisions within a limited region of space examples of some two class gaussian mixture distributions are included

weil petersson teichmuller space iii dependence of riemann mappings for weil petersson curves the classical riemann mapping theorem implies that there exists a so called riemann mapping which takes the upper half plane onto the left domain bounded by a jordan curve in the extended complex plane the primary purpose of the paper is to study the basic problem how does a riemann mapping depend on the corresponding jordan curve we are mainly concerned with those jordan curves in the weil petersson class namely the corresponding riemann mappings can be quasiconformally extended to the whole plane with beltrami coefficients being square integrable under the poincare metric after giving a geometric characterization of a weil petersson curve we endow the space of all normalized weil petersson curves with a new real hilbert manifold structure in a geometric manner and show that this new structure is topologically equivalent to the standard complex hilbert manifold structure which implies that an appropriately chosen riemann mapping depends continuously on a weil petersson curve and vice versa this can be considered as the first result about the continuous dependence of riemann mappings on non smooth jordan curves

two dimensional minimum variance and self tuning controllers the problem of controlling by a minimum variance law a given two dimensional 2d stochastic process is formulated and solved two different algorithms are derived a the optimal controller for a known process model and b the self tuning controller for a process model with unknown parameters both are based on 2d predictors for 2d armax models also introduced procedures for finding solutions to these problems are illustrated by numerical examples

self tuning prediction and control for two dimensional processes part 1 fixed parameter algorithms least squares optimal prediction minimum variance control and generalized minimum variance control algorithms for a two dimensional carma process are developed each algorithm involves the algebraic solution of a two dimensional diophantine equation and may be embedded within classical two dimensional systems theory we show how the algorithms must be modified for any practical implementation to take into account the edges of the data field in this case we show how we may analyse the process using multivariable theory and explore the linkages between multivariable representations and two dimensional systems

adaptive local linear regression with application to printer color management local learning methods such as local linear regression and nearest neighbor classifiers base estimates on nearby training samples neighbors usually the number of neighbors used in estimation is fixed to be a global ldquooptimalrdquo value chosen by cross validation this paper proposes adapting the number of neighbors used for estimation to the local geometry of the data without need for cross validation the term enclosing neighborhood is introduced to describe a set of neighbors whose convex hull contains the test point when possible it is proven that enclosing neighborhoods yield bounded estimation variance under some assumptions three such enclosing neighborhood definitions are presented natural neighbors natural neighbors inclusive and enclosing k nn the effectiveness of these neighborhood definitions with local linear regression is tested for estimating lookup tables for color management significant improvements in error metrics are shown indicating that enclosing neighborhoods may be a promising adaptive neighborhood definition for other local learning tasks as well depending on the density of training samples

the optimal distance measure for nearest neighbor classification a local distance measure is shown to optimize the performance of the nearest neighbor two class classifier for a finite number of samples the difference between the finite sample error and the asymptotic error is used as the criterion of improvement this new distance measure is compared to the well known euclidean distance an algorithm for practical implementation is introduced this algorithm is shown to be computationally competitive with the present nearest neighbor procedures and is illustrated experimentally a closed form for the corresponding second order moment of this criterion is found finally the above results are extended to

on qk teichmuller spaces abstract recently wulan and ye 36 introduced the q k teichmuller spaces and proved some theorems about the theory of q k teichmuller spaces in this note we continue to consider this theory and obtain some results on the q k teichmuller spaces in particular we establish some new characterizations of the little q k teichmuller spaces

on the tangent space to the bmo teichmuller space abstract in a previous paper 31 we established the complex analytic manifold theory of the bmo teichmuller space in this paper we identify the function space which is the tangent space to the bmo teichmuller space

universal teichmuller space and bmo abstract we first give some new characterizations on bmoa teichmuller space and various characterizations on vmoa teichmuller space as well in particular we prove that a quasisymmetric conformal welding h corresponds to an asymptotically smooth curve in the sense of pommerenke 1978 32 precisely when h is absolutely continuous with log h vmo we then show that these bmo teichmuller spaces have natural complex structures

diff s 1 and the teichmuller spaces precisely two of the homogeneous spaces that appear as coadjoint orbits of the group of string reparametrizations widehat diff s 1 1 carry in a natural way the structure of infinite dimensional holomorphically homogeneous complex analytic kahler manifolds these aren diff s1 rot s1 andm diff s1 mob s1 note thatn is a holomorphic disc fiber space overm now m can be naturally considered as embedded in the classical universal teichmuller spacet 1 simply by noting that a diffeomorphism ofs1 is a quasisymmetric homeomorphism t 1 is itself a homomorphically homogeneous complex banach manifold we prove in the first part of the paper that the inclusion ofm int 1 iscomplex analytic

weil petersson teichmuller space ii smoothness of flow curves of h32 vector fields abstract given a continuous vector field t of sobolev class h 3 2 on the unit circle s 1 the flow maps g t of the differential equation d d t t 0 are known to be quasisymmetric homeomorphisms very recently gay balmaz ratiu 15 conjectured that the flow curve g t is in the weil petersson class wp s 1 and is continuously differentiable with respect to the hilbert manifold structure of wp s 1 introduced by takhtajan teo 40 the first assertion had already been demonstrated in our previous paper 36 in this sequel to 36 we will continue to deal with the weil petersson class wp s 1 and completely solve this conjecture in the affirmative

the multi class metric problem in nearest neighbour discrimination rules abstract nearest neighbour rules classify a new data point on the basis of the class of the closest design set point a prerequisite for this is a choice of metric by which distance may be calculated metrics which have been suggested for the two class case fall into two classes global and local in this paper local metrics are described and are generalized to the multi class case properties of various possible generalizations are compared and the results of a simulation study are described

the 2 nn rule for more accurate nn risk estimation by proper design of a nearest neighbor nn rule it is possible to reduce effects of sample size in nn risk estimation the 2 nn rule for the two class problem eliminates the first order effects of sample size since its asymptotic value is exactly half that of the 1 nn rule it is possible to substitute the 2 nn rule for the 1 nn rule with a resultant increase in accuracy for further stabilization of the risk estimate with respect to sample size 2 nn polarization is suggested examples are included the 2 nn approach is extended to m class and 2k nn

on flows of h 3 2 vector fields on the circle in this paper we study the regularity of flow maps of h 3 2 vector fields on the circle in terms of fractional sobolev spaces this problem is motivated by the understanding of the geometry of bers s universal teichmuller space

some notes on quasisymmetric flows of zygmund vector fields abstract in an important paper 24 reimann showed that the flow mappings of a continuous vector field of zygmund class are quasisymmetric homeomorphisms in this paper we will discuss the flow mappings when the vector field belongs to the smooth zygmund class or the sobolev class h 3 2

self tuning filters and predictors for two dimensional systems part 1 algorithms the filtering of two dimensional 2 d signals is treated using a self tuning technique based on a truncated innovations model of the data the resultant algorithms offer two key advantages over their fixed coefficient counterparts first the self tuning filters quickly and automatically set their own coefficients thus avoiding the normal off line design cycle secondly self tuning filters can function in an adaptive manner such that the filter retunes to track time variations in the two dimensional data the self tuning algorithms are formulated in terms of input output models and thus complement the more usual state space approach to the 2 d filtering problem

generative local metric learning for nearest neighbor classification we consider the problem of learning a local metric in order to enhance the performance of nearest neighbor classification conventional metric learning methods attempt to separate data distributions in a purely discriminative manner here we show how to take advantage of information from parametric generative models we focus on the bias in the information theoretic error arising from finite sampling effects and find an appropriate local metric that maximally reduces the bias based upon knowledge from generative models as a byproduct the asymptotic theoretical analysis in this work relates metric learning to dimensionality reduction from a novel perspective which was not understood from previous discriminative approaches empirical experiments show that this learned local metric enhances the discriminative nearest neighbor performance on various datasets using simple class conditional generative models such as a gaussian

relaxational metric adaptation and its application to semi supervised clustering and content based image retrieval the performance of many supervised and unsupervised learning algorithms is very sensitive to the choice of an appropriate distance metric previous work in metric learning and adaptation has mostly been focused on classification tasks by making use of class label information in standard clustering tasks however class label information is not available in order to adapt the metric to improve the clustering results some background knowledge or side information is needed one useful type of side information is in the form of pairwise similarity or dissimilarity information recently some novel methods e g the parametric method proposed by xing et al for learning global metrics based on pairwise side information have been shown to demonstrate promising results in this paper we propose a nonparametric method called relaxational metric adaptation rma for the same metric adaptation problem while rma is local in the sense that it allows locally adaptive metrics it is also global because even patterns not in the vicinity can have long range effects on the metric adaptation process experimental results for semi supervised clustering based on both simulated and real world data sets show that rma outperforms xing et al s method under most situations besides applying rma to semi supervised learning we have also used it to improve the performance of content based image retrieval systems through metric adaptation experimental results based on two real world image databases show that rma significantly outperforms other methods in improving the image retrieval performance

locally linear metric adaptation with application to semi supervised clustering and image retrieval many computer vision and pattern recognition algorithms are very sensitive to the choice of an appropriate distance metric some recent research sought to address a variant of the conventional clustering problem called semi supervised clustering which performs clustering in the presence of some background knowledge or supervisory information expressed as pairwise similarity or dissimilarity constraints however existing metric learning methods for semi supervised clustering mostly perform global metric learning through a linear transformation in this paper we propose a new metric learning method that performs nonlinear transformation globally but linear transformation locally in particular we formulate the learning problem as an optimization problem and present three methods for solving it through some toy data sets we show empirically that our locally linear metric adaptation llma method can handle some difficult cases that cannot be handled satisfactorily by previous methods we also demonstrate the effectiveness of our method on some uci data sets besides applying llma to semi supervised clustering we have also used it to improve the performance of content based image retrieval systems through metric learning experimental results based on two real world image databases show that llma significantly outperforms other methods in boosting the image retrieval performance

nonparametric discriminant analysis a nonparametric method of discriminant analysis is proposed it is based on nonparametric extensions of commonly used scatter matrices two advantages result from the use of the proposed nonparametric scatter matrices first they are generally of full rank this provides the ability to specify the number of extracted features desired this is in contrast to parametric discriminant analysis which for an l class problem typically can determine at most l 1 features second the nonparametric nature of the scatter matrices allows the procedure to work well even for non gaussian data sets using the same basic framework a procedure is proposed to test the structural similarity of two distributions the procedure works in high dimensional space it specifies a linear decomposition of the original data space in which a relative indication of dissimilarity along each new basis vector is provided the nonparametric scatter matrices are also used to derive a clustering procedure which is recognized as a k nearest neighbor version of the nonparametric valley seeking algorithm the form which results provides a unified view of the parametric nearest mean reclassification algorithm and the nonparametric valley seeking algorithm

a kernel approach for semisupervised metric learning while distance function learning for supervised learning tasks has a long history extending it to learning tasks with weaker supervisory information has only been studied recently in particular some methods have been proposed for semisupervised metric learning based on pairwise similarity or dissimilarity information in this paper we propose a kernel approach for semisupervised metric learning and present in detail two special cases of this kernel approach the metric learning problem is thus formulated as an optimization problem for kernel learning an attractive property of the optimization problem is that it is convex and hence has no local optima while a closed form solution exists for the first special case the second case is solved using an iterative majorization procedure to estimate the optimal solution asymptotically experimental results based on both synthetic and real world data show that this new kernel approach is promising for nonlinear metric learning