pairwise sparsity preserving embedding for unsupervised subspace learning and classification two novel unsupervised dimensionality reduction techniques termed sparse distance preserving embedding sdpe and sparse proximity preserving embedding sppe are proposed for feature extraction and classification sdpe and sppe perform in the clean data space recovered by sparse representation and enhanced euclidean distances over noise removed data are employed to measure pairwise similarities of points in extracting informative features sdpe and sppe aim at preserving pairwise similarities between data points in addition to preserving the sparse characteristics this paper calculates the sparsest representation of all vectors jointly by a convex optimization the sparsest codes enable certain local information of data to be preserved and can endow sdpe and sppe a natural discriminating power adaptive neighborhood and robust characteristic against noise and errors in delivering low dimensional embeddings we also mathematically show sdpe and sppe can be effectively extended for discriminant learning in a supervised manner the validity of sdpe and sppe is examined by extensive simulations comparison with other related state of the art unsupervised algorithms show that promising results are delivered by our techniques

worst case linear discriminant analysis as scalable semidefinite feasibility problems in this paper we propose an efficient semidefinite programming sdp approach to worst case linear discriminant analysis wlda compared with the traditional lda wlda considers the dimensionality reduction problem from the worst case viewpoint which is in general more robust for classification however the original problem of wlda is non convex and difficult to optimize in this paper we reformulate the optimization problem of wlda into a sequence of semidefinite feasibility problems to efficiently solve the semidefinite feasibility problems we design a new scalable optimization method with a quasi newton method and eigen decomposition being the core components the proposed method is orders of magnitude faster than standard interior point sdp solvers experiments on a variety of classification problems demonstrate that our approach achieves better performance than standard lda our method is also much faster and more scalable than standard interior point sdp solvers based wlda the computational complexity for an sdp with m constraints and matrices of size d by d is roughly reduced from mathcal o m 3 md 3 m 2 d 2 to mathcal o d 3 m d in our case

constrained large margin local projection algorithms and extensions for multimodal dimensionality reduction a constrained large margin local projection cmlp technique for multimodal dimensionality reduction is proposed we elaborate the criterion of cmlp from a pairwise constrained marginal perspective four effective cmlp solution schemes are presented and the corresponding comparative analyses are given an equivalent weighted least squares formulation for cmlp is also detailed cmlp is originated from the criterion of locality preserving projections lpp but cmlp offers a number of attractive advantages over lpp to keep the intrinsic proximity relations of inter class and intra class similarity pairs the localized pairwise cannot link and must link constraints are applied to specify the types of those neighboring pairs by utilizing the cmlp criterion margins between inter and intra class clusters are significantly enlarged as a result multimodal distributions are effectively preserved to further optimize the cmlp criterion one feasible improvement strategy is described with kernel methods we present the kernelized extensions of our approaches mathematical comparisons and analyses between this work and the related works are also detailed extensive simulations including multivariate manifold visualization and classification on the benchmark ucl orl yale umist mit cbcl and usps datasets are conducted to verify the efficiency of our techniques the presented results reveal that our methods are highly competitive with and even outperform some widely used state of the art algorithms

spectral embedded clustering a framework for in sample and out of sample spectral clustering spectral clustering sc methods have been successfully applied to many real world applications the success of these sc methods is largely based on the manifold assumption namely that two nearby data points in the high density region of a low dimensional data manifold have the same cluster label however such an assumption might not always hold on high dimensional data when the data do not exhibit a clear low dimensional manifold structure e g high dimensional and sparse data the clustering performance of sc will be degraded and become even worse than k means clustering in this paper motivated by the observation that the true cluster assignment matrix for high dimensional data can be always embedded in a linear space spanned by the data we propose the spectral embedded clustering sec framework in which a linearity regularization is explicitly added into the objective function of sc methods more importantly the proposed sec framework can naturally deal with out of sample data we also present a new laplacian matrix constructed from a local regression of each pattern and incorporate it into our sec framework to capture both local and global discriminative information for clustering comprehensive experiments on eight real world high dimensional datasets demonstrate the effectiveness and advantages of our sec framework over existing sc methods and k means based clustering methods our sec framework significantly outperforms sc using the nystrom algorithm on unseen data

isoperimetric inequality under measure contraction property abstract we prove that if x d m is an essentially non branching metric measure space with m x 1 having ricci curvature bounded from below by k and dimension bounded above by n 1 understood as a synthetic condition called measure contraction property then a sharp isoperimetric inequality a la levy gromov holds true measure theoretic rigidity is also obtained

compound rank k projections for bilinear analysis in many real world applications data are represented by matrices or high order tensors despite the promising performance the existing 2 d discriminant analysis algorithms employ a single projection model to exploit the discriminant information for projection making the model less flexible in this paper we propose a novel compound rank k projection crp algorithm for bilinear analysis the crp deals with matrices directly without transforming them into vectors and it therefore preserves the correlations within the matrix and decreases the computation complexity different from the existing 2 d discriminant analysis algorithms objective function values of crp increase monotonically in addition the crp utilizes multiple rank k projection models to enable a larger search space in which the optimal solution can be found in this way the discriminant ability is enhanced we have tested our approach on five data sets including uuim cvl pointing 04 usps and coil20 experimental results show that the performance of our proposed crp performs better than other algorithms in terms of classification accuracy

on the euler lagrange equation for a variational problem the general case ii in this paper we study the existence of a solution in l infty rm loc omega to the euler lagrange equation for the variational problem inf bar u w 1 infty 0 omega int limits omega bf i d nabla u g u dx quad quad quad quad quad 0 1 with d convex closed subset of mathbb r n with non empty interior by means of a disintegration theorem we next show that the euler lagrange equation can be reduced to an ode along characteristics and we deduce that there exists a solution to euler lagrange different from 0 a e and satisfies a uniqueness property these results prove a conjecture on the existence of variations on vector fields stated in bertone and cellina on the existence of variations to appear on control optimisation and calculus of variations

adaptive maximum margin analysis for image recognition abstract most existing discriminant manifold learning methods aim to maximize the margin among nearby data which is determined in the high dimensional original space as such they do not necessarily best maximize the margin between different classes in the low dimensional space which is a critically important property for image classification to handle this problem we propose an adaptive maximum margin analysis amma for feature extraction amma aims to seek a projection matrix that best maximize the margin which is calculated in the low dimensional space it uses sparse representation to adaptively construct the intrinsic and penalty graphs finally an iterative algorithm is developed to solve the projection matrix extensive experimental results on several image databases illustrate the effectiveness of the proposed approach

constructing optimal maps for monge s transport problem as a limit of strictly convex costs the monge kantorovich problem is to move one distribution of mass onto another as efficiently as possible where monge s original criterion for efficiency 19 was to minimize the average distance transported subsequently studied by many authors it was not until 1976 that sudakov showed solutions to be realized in the original sense of monge i e as mappings from rn to rn 23 a second proof of this existence result formed the subject of a recent monograph by evans and gangbo 7 who avoided sudakov s measure decomposition results by using a partial differential equations approach in the present manuscript we give a third existence proof for optimal mappings which has some advantages and disadvantages relative to existing approaches it requires no continuity or separation of the mass distributions yet our explicit construction yields more geometrical control than the abstract method of sudakov indeed this control turns out to be essential for addressing a gap which has recently surfaced in sudakov s approach to the problem in dimensions n 3 see the remarks at the end of this section it is also shorter and more flexible than either and can be adapted to handle transportation on riemannian manifolds or around obstacles as we plan to show in a subsequent work 13 the problem considered here is the classical one problem 1 monge fix a norm d x y llx yll on rn and two densities non negative borel functions f fe ll rn satisfying the mass balance condition

graph embedding and extensions a general framework for dimensionality reduction a large family of algorithms supervised or unsupervised stemming from statistics or geometry theory has been designed to provide different solutions to the problem of dimensionality reduction despite the different motivations of these algorithms we present in this paper a general formulation known as graph embedding to unify them within a common framework in graph embedding each algorithm can be considered as the direct graph embedding or its linear kernel tensor extension of a specific intrinsic graph that describes certain desired statistical or geometric properties of a data set with constraints from scale normalization or a penalty graph that characterizes a statistical or geometric property that should be avoided furthermore the graph embedding framework can be used as a general platform for developing new dimensionality reduction algorithms by utilizing this framework as a tool we propose a new supervised dimensionality reduction algorithm called marginal fisher analysis in which the intrinsic graph characterizes the intraclass compactness and connects each data point with its neighboring points of the same class while the penalty graph connects the marginal points and characterizes the interclass separability we show that mfa effectively overcomes the limitations of the traditional linear discriminant analysis algorithm due to data distribution assumptions and available projection directions real face recognition experiments show the superiority of our proposed mfa in comparison to lda also for corresponding kernel and tensor extensions

the monge problem for distance cost in geodesic spaces we address the monge problem in metric spaces with a geodesic distance x d is a polish space and dl is a geodesic borel distance which makes x dl a non branching geodesic space we show that under the assumption that geodesics are d continuous and locally compact we can reduce the transport problem to 1 dimensional transport problems along geodesics we introduce two assumptions on the transport problem which imply that the conditional probabilities of the first marginal on each geodesic are continuous or absolutely continuous w r t the 1 dimensional hausdorff distance induced by dl it is known that this regularity is sufficient for the construction of a transport map we study also the dynamics of transport along the geodesic the stability of our conditions and show that in this setting dl cyclical monotonicity is not sufficient for optimality

measure rigidity of ricci curvature lower bounds abstract the measure contraction property mcp for short is a weak ricci curvature lower bound conditions for metric measure spaces the goal of this paper is to understand which structural properties such assumption or even weaker modifications implies on the measure on its support and on the geodesics of the space we start our investigation from the euclidean case by proving that if a positive radon measure m over r d is such that r d m verifies a weaker variant of mcp then its support spt m must be convex and m has to be absolutely continuous with respect to the relevant hausdorff measure of spt m this result is then used as a starting point to investigate the rigidity of mcp in the metric framework we introduce the new notion of reference measure for a metric space and prove that if x d m is essentially non branching and verifies mcp and is an essentially non branching mcp reference measure for spt m d then m is absolutely continuous with respect to on the set of points where an inversion plan exists as a consequence an essentially non branching mcp reference measure enjoys a weak type of uniqueness up to densities we also prove a stability property for reference measures under measured gromov hausdorff convergence provided an additional uniform bound holds in the final part we present concrete examples of metric spaces with reference measures both in smooth and non smooth setting the main example will be the hausdorff measure over an alexandrov space then we prove that the following are reference measures over smooth spaces the volume measure of a riemannian manifold the hausdorff measure of an alexandrov space with bounded curvature and the haar measure of the subriemannian heisenberg group

towards robust discriminative projections learning via non greedy l2 1 norm minmax linear discriminant analysis lda is one of the most successful supervised dimensionality reduction methods and has been widely used in many real world applications however the l 2 norm is employed as the distance measure in objective of lda which is sensitive to outliers many previous works improve the robustness of lda by using l1 norm distance however the robustness against outliers is limited and the solver of l1 norm are mostly based on greedy search strategy which is time consuming and easy to get stuck in a local optimum in this paper we propose a novel robust lda measured by l 2 1 norm to learn robust discriminative projections since the proposed model needs to minimize and maximize minmax l 2 1 norm terms simultaneously it is challenging to solve as a result we first systematically derive an efficient iterative optimization algorithm to solve a general ratio minimization problem and rigorously prove its convergence more importantly an efficient non greedy iterative re weighted optimization algorithm is developed based on preceding approach for solving proposed l 2 1 norm minmax problem besides an optimal weighted mean mechanism is driven according to designed objective and solver which can be applied to other approach for robustness improvement extensive experimental results on several real world datasets show the effectiveness of proposed method

multiscale distance matrix for fast plant leaf recognition in this brief we propose a novel contour based shape descriptor called the multiscale distance matrix to capture the shape geometry while being invariant to translation rotation scaling and bilateral symmetry the descriptor is further combined with a dimensionality reduction to improve its discriminative power the proposed method avoids the time consuming pointwise matching encountered in most of the previously used shape recognition algorithms it is therefore fast and suitable for real time applications we applied the proposed method to the task of plan leaf recognition with experiments on two data sets the swedish leaf data set and the icl leaf data set the experimental results clearly demonstrate the effectiveness and efficiency of the proposed descriptor

nonnegative sparse locality preserving hashing it is a np hard problem to optimize the objective function of hash based similarity search algorithms such as spectral hashing and self taught hashing to make the problem solvable existing methods have relaxed the constraints on hash codes from binary values discrete to real values continuous then eigenvalue decomposition was employed to achieve the relaxed real solution the main problem is that the signs of the relaxed continuous solution are mixed such results may deviate severely from the true solution which has lead to significant semantic loss moreover eigenvalue decomposition confronts singularity problem when the dimension of the data is larger than the sample size to address these problems we propose a novel method named nonnegative sparse locality preserving hashing nslph nonnegative and sparse constraints are imposed for a more accurate solution which preserves semantic information well then we have applied nonnegative quadratic programming and multiplicative updating to solve the optimization problem which successfully avoids the singularity problem of the eigenvalue decomposition the extensive experiments presented in this paper demonstrate that the proposed approach outperforms the state of the art algorithms

monge problem in metric measure spaces with riemannian curvature dimension condition abstract we prove the existence of solutions for the monge minimization problem addressed in a metric measure space x d m enjoying the riemannian curvature dimension condition rcd k n with n for the first marginal measure we assume that 0 m as a corollary we obtain that the monge problem and its relaxed version the monge kantorovich problem attain the same minimal value moreover we prove a structure theorem for d cyclically monotone sets neglecting a set of zero m measure they do not contain any branching structures that is they can be written as the disjoint union of the image of a disjoint family of geodesics

sharp and rigid isoperimetric inequalities in metric measure spaces with lower ricci curvature bounds we prove that if x mathsf d mathfrak m is a metric measure space with mathfrak m x 1 having in a synthetic sense ricci curvature bounded from below by k 0 and dimension bounded above by n in 1 infty then the classic levy gromov isoperimetric inequality together with the recent sharpening counterparts proved in the smooth setting by milman for any k in mathbb r n ge 1 and upper diameter bounds holds i e the isoperimetric profile function of x mathsf d mathfrak m is bounded from below by the isoperimetric profile of the model space moreover if equality is attained for some volume v in 0 1 and k is strictly positive then the space must be a spherical suspension and in this case we completely classify the isoperimetric regions finally we also establish the almost rigidity if the equality is almost attained for some volume v in 0 1 and k is strictly positive then the space must be mgh close to a spherical suspension to our knowledge this is the first result about isoperimetric comparison for non smooth metric measure spaces satisfying ricci curvature lower bounds examples of spaces fitting our assumptions include measured gromov hausdorff limits of riemannian manifolds satisfying ricci curvature lower bounds alexandrov spaces with curvature bounded from below finsler manifolds endowed with a strongly convex norm and satisfying ricci curvature lower bounds the result seems new even in these celebrated classes of spaces

trace ratio problem revisited dimensionality reduction is an important issue in many machine learning and pattern recognition applications and the trace ratio tr problem is an optimization problem involved in many dimensionality reduction algorithms conventionally the solution is approximated via generalized eigenvalue decomposition due to the difficulty of the original problem however prior works have indicated that it is more reasonable to solve it directly than via the conventional way in this brief we propose a theoretical overview of the global optimum solution to the tr problem via the equivalent trace difference problem eigenvalue perturbation theory is introduced to derive an efficient algorithm based on the newton raphson method theoretical issues on the convergence and efficiency of our algorithm compared with prior literature are proposed and are further supported by extensive empirical results

a hierarchical word merging algorithm with class separability measure in image recognition with the bag of features model a small sized visual codebook is usually preferred to obtain a low dimensional histogram representation and high computational efficiency such a visual codebook has to be discriminative enough to achieve excellent recognition performance to create a compact and discriminative codebook in this paper we propose to merge the visual words in a large sized initial codebook by maximally preserving class separability we first show that this results in a difficult optimization problem to deal with this situation we devise a suboptimal but very efficient hierarchical word merging algorithm which optimally merges two words at each level of the hierarchy by exploiting the characteristics of the class separability measure and designing a novel indexing structure the proposed algorithm can hierarchically merge 10 000 visual words down to two words in merely 90 seconds also to show the properties of the proposed algorithm and reveal its advantages we conduct detailed theoretical analysis to compare it with another hierarchical word merging algorithm that maximally preserves mutual information obtaining interesting findings experimental studies are conducted to verify the effectiveness of the proposed algorithm on multiple benchmark data sets as shown it can efficiently produce more compact and discriminative codebooks than the state of the art hierarchical word merging algorithms especially when the size of the codebook is significantly reduced

existence of optimal transport maps for crystalline norms we show the existence of optimal transport maps in the case when the cost function is the distance induced by a crystalline norm in n assuming that the initial distribution of mass is absolutely continuous with respect to mathcal l n the proof is based on a careful decomposition of the space in transport rays induced by a secondary variational problem having the euclidean distance as cost function moreover improving a construction by larman we show the existence of a nikodym set in 3 having full measure in the unit cube intersecting each element of a family of pairwise disjoint open lines only in one point this example can be used to show that the regularity of the decomposition in transport rays plays an essential role in sudakov type arguments for proving the existence of optimal transport maps

quantitative isoperimetry a la levy gromov on a riemannian manifold with a positive lower bound on the ricci tensor the distance of isoperimetric sets from geodesic balls is quantitatively controlled in terms of the gap between the isoperimetric profile of the manifold and that of a round sphere of suitable radius the deficit between the diameters of the manifold and of the corresponding sphere is bounded likewise these results are actually obtained in the more general context of possibly non smooth metric measure spaces with curvature dimension conditions through a quantitative analysis of the transport rays decompositions obtained by the localization method

network comparison embeddings and interiors this paper presents methods to compare networks where relationships between pairs of nodes in a given network are defined we define such network distance by searching for the optimal method to embed one network into another network prove that such distance is a valid metric in the space of networks modulo permutation isomorphisms and examine its relationship with other network metrics the network distance defined can be approximated via multidimensional scaling however the lack of structure in networks results in suboptimal approximations to alleviate such a problem we consider methods to define the interiors of networks we show that comparing interiors induced from a pair of networks yields the same result as the actual network distance between the original networks practical implications are explored by showing the ability to discriminate networks generated by different models

metrics in the space of high order networks this paper presents methods to compare high order networks defined as weighted complete hypergraphs collecting relationship functions between elements of tuples they can be considered as generalizations of conventional networks where only relationship functions between pairs are defined important properties between relationships of tuples of different lengths are established particularly when relationships encode dissimilarities or proximities between nodes two families of distances are then introduced in the space of high order networks the distances measure differences between networks we prove that they are valid metrics in the spaces of high order dissimilarity and proximity networks modulo permutation isomorphisms practical implications are explored by comparing the coauthorship networks of two popular signal processing researchers the metrics succeed in identifying their respective collaboration patterns

a new lda based face recognition system which can solve the small sample size problem a new lda based face recognition system is presented in this paper linear discriminant analysis lda is one of the most popular linear projection techniques for feature extraction the major drawback of applying lda is that it may encounter the small sample size problem in this paper we propose a new lda based technique which can solve the small sample size problem we also prove that the most expressive vectors derived in the null space of the within class scatter matrix using principal component analysis pca are equal to the optimal discriminant vectors derived in the original space using lda the experimental results show that the new lda process improves the performance of a face recognition system signi cantly 2000 pattern recognition society published by elsevier science ltd all rights reserved

multi manifold discriminant isomap for visualization and classification isomap aims to recover the intrinsic geometric structure of manifold by preserving geodesic distances between all pairs of data points however it is an unsupervised dimensionality reduction method usually using class label information can increase the discriminating capability hence a new supervised isomap is proposed in this paper dubbed multi manifold discriminant isomap mmd isomap first data points are partitioned into different manifolds according to their class label information then mmd isomap aims at seeking an optimal nonlinear subspace to preserve the geometrical structure of each manifold according to the isomap criterion meanwhile to enhance the discriminating capability by maximizing the distances between data points of different manifolds finally the corresponding optimization problem is solved by using a majorization algorithm furthermore two new numerical metrics are designed to measure the performance of dimensionality reduction method in both visualization and classification experiments mmd isomap achieves improved performance over many state of the art methods preserve the geometrical structure and enhance the discriminating capability smacof algorithm is introduced to solve the optimization problem two metrics are designed to evaluate dimensionality reduction methods experiments show the effectiveness of the method in visualization and classification

efficient and robust feature extraction by maximum margin criterion in pattern recognition feature extraction techniques are widely employed to reduce the dimensionality of data and to enhance the discriminatory information principal component analysis pca and linear discriminant analysis lda are the two most popular linear dimensionality reduction methods however pca is not very effective for the extraction of the most discriminant features and lda is not stable due to the small sample size problem in this paper we propose some new linear and nonlinear feature extractors based on maximum margin criterion mmc geometrically feature extractors based on mmc maximize the average margin between classes after dimensionality reduction it is shown that mmc can represent class separability better than pca as a connection to lda we may also derive lda from mmc by incorporating some constraints by using some other constraints we establish a new linear feature extractor that does not suffer from the small sample size problem which is known to cause serious stability problems for lda the kernelized nonlinear counterpart of this linear feature extractor is also established in the paper our extensive experiments demonstrate that the new feature extractors are effective stable and efficient

regularized class specific subspace classifier in this paper we mainly focus on how to achieve the translated subspace representation for each class which could simultaneously indicate the distribution of the associated class and the differences from its complementary classes by virtue of the reconstruction problem the class specific subspace classifier cssc problem could be represented as a series of biobjective optimization problems which minimize and maximize the reconstruction errors of the related class and its complementary classes respectively besides the regularization term is specifically introduced to ensure the whole system s stability accordingly a regularized class specific subspace classifier rcssc method can be further proposed based on solving a general quadratic ratio problem the proposed rcssc method consistently converges to the global optimal subspace and translation under the variations of the regularization parameter furthermore the proposed rcssc method could be extended to the unregularized case which is known as unregularized cssc ucssc method via orthogonal decomposition technique as a result the effectiveness and the superiority of both proposed rcssc and ucssc methods can be verified analytically and experimentally

face recognition using lda based algorithms low dimensional feature representation with enhanced discriminatory power is of paramount importance to face recognition fr systems most of traditional linear discriminant analysis lda based methods suffer from the disadvantage that their optimality criteria are not directly related to the classification ability of the obtained feature representation moreover their classification accuracy is affected by the small sample size sss problem which is often encountered in fr tasks in this paper we propose a new algorithm that deals with both of the shortcomings in an efficient and cost effective manner the proposed method is compared in terms of classification accuracy to other commonly used fr methods on two face databases results indicate that the performance of the proposed method is overall superior to those of traditional fr approaches such as the eigenfaces fisherfaces and d lda methods

monge s transport problem on a riemannian manifold monge s problem refers to the classical problem of optimally transporting mass given borel probability measures find the measurepreserving map s m m between them which minimizes the average distance transported set on a complete connected riemannian manifold m and assuming absolute continuity of an optimal map will be shown to exist aspects of its uniqueness are also established

inexact implementation using krylov subspace methods for large scale exponential discriminant analysis with applications to high dimensionality reduction problems abstract exponential discriminant analysis eda is a generalized discriminant analysis method based on matrix exponential it can essentially overcome the intrinsic difficulty of small sample size problem that exists in the classical linear discriminant analysis lda however for data with high dimensionality one has to solve a large matrix exponential eigenproblem in this method and the time complexity is dominated by the computation of exponential of large matrices in this paper we propose two inexact krylov subspace algorithms for solving the large matrix exponential eigenproblem efficiently the contribution of this work is threefold first we consider how to compute matrix exponential vector products efficiently which is the key step in the krylov subspace method second we compare the discriminant analysis criterion of eda and that of lda from a theoretical point of view third we establish the relationship between the accuracy of the approximate eigenvectors and the distance to nearest neighbor classifier and show why the matrix exponential eigenproblem can be solved approximately in practice numerical experiments on some real world databases show the superiority of our new algorithms over their original counterpart for face recognition

on the geometry of metric measure spaces ii we introduce a curvature dimension condition cd k n for metric measure spaces it is more restrictive than the curvature bound underline text curv left m text d m right k introduced in sturm k t 2006 on the geometry of metric measure spaces i acta math 196 65 131 which is recovered as the borderline case cd k the additional real parameter n plays the role of a generalized upper bound for the dimension for riemannian manifolds cd k n is equivalent to text ric m left xi xi right k left xi right 2 and dim m n the curvature dimension condition cd k n is stable under convergence for any triple of real numbers k n l the family of normalized metric measure spaces m d m with cd k n and diameter l is compact condition cd k n implies sharp version of the brunn minkowski inequality of the bishop gromov volume comparison theorem and of the bonnet myers theorem moreover it implies the doubling property and local scale invariant poincare inequalities on balls in particular it allows to construct canonical dirichlet forms with gaussian upper and lower bounds for the corresponding heat kernels

single site localization via maximum discrimination multipath fingerprinting a novel approach to single site localization based on maximum discrimination multipath fingerprinting is presented in contrast to the existing approach which extracts each fingerprint only from the data of that location the new approach uses also the data of all the other locations in the database and leverages it to extract a fingerprint that is as different as possible from the other fingerprints in the database the performance of this approach validated with both simulated and real data is superior to the existing approach demonstrating single site localization accuracy of 1 m in typical indoor environments the new approach has also a lower computational complexity

hierarchical clustering given confidence intervals of metric distances this paper considers metric the exact dissimilarities between pairs of points are not unknown but known to belong to some interval the goal is to study methods for the determination of hierarchical clusters i e a family of nested partitions indexed by a resolution parameter induced from the given distance intervals of the dissimilarities our construction of hierarchical clustering methods is based on defining admissible methods to be those methods that satisfy the axioms of value nodes in a metric space with two nodes are clustered together at the convex combination of the upper and lower bounds determined by a parameter and transformation when both distance bounds are reduced the output may become more clustered but not less two admissible methods are constructed and are shown to provide universal bounds in the space of admissible methods practical implications are explored by clustering moving points via snapshots and by clustering coauthorship networks representing collaboration between researchers from different communities the proposed clustering methods succeed in identifying underlying hierarchical clustering structures via the maximum and minimum distances in all snapshots as well as in differentiating collaboration patterns in journal publications between different research communities based on bounds of network distances

adaptive data embedding framework for multiclass classification the objective of this paper is the design of an engine for the automatic generation of supervised manifold embedding models it proposes a modular and adaptive data embedding framework for classification referred to as defc which realizes in different stages including initial data preprocessing relation feature generation and embedding computation for the computation of embeddings the concepts of friend closeness and enemy dispersion are introduced to better control at local level the relative positions of the intraclass and interclass data samples these are shown to be general cases of the global information setup utilized in the fisher criterion and are employed for the construction of different optimization templates to drive the defc model generation for model identification we use a simple but effective bilevel evolutionary optimization which searches for the optimal model and its best model parameters the effectiveness of defc is demonstrated with experiments using noisy synthetic datasets possessing nonlinear distributions and real world datasets from different application fields

ricci curvature for metric measure spaces via optimal transport we dene a notion of a measured length space x having nonnegative n ricci curvature for n 2 1 1 or having1 ricci curvature bounded below byk fork2 r the denitions are in terms of the displacement convexity of certain functions on the associated wasserstein metric space p2 x of probability measures we show that these properties are preserved under measured gromov hausdor limits we give geometric and analytic consequences this paper has dual goals one goal is to extend results about optimal transport from the setting of smooth riemannian manifolds to the setting of length spaces a second goal is to use optimal transport to give a notion for a measured length space to have ricci curvature bounded below we refer to 11 and 44 for background material on length spaces and optimal transport respectively further bibliographic notes on optimal transport are in appendix f in the present introduction we motivate the questions that we address and we state the main results to start on the geometric side there are various reasons to try to extend notions of curvature from smooth riemannian manifolds to more general spaces a fairly general setting is that of length spaces meaning metric spaces x d in which the distance between two points equals the inmum of the lengths of curves joining the points in the rest of this introduction we assume that x is a compact length space alexandrov gave a good notion of a length space having curvature bounded below by k with k a real number in terms of the geodesic triangles in x in the case of a riemannian manifold m with the induced length structure one recovers the riemannian notion of having sectional curvature bounded below by k length spaces with alexandrov curvature bounded below by k behave nicely with respect to the gromovhausdor topology on compact metric spaces modulo isometries they form