Low-dimensional Euclidean Space Research Articles

Analysis of the Sustainable Society Index Worldwide: A Study from the Biplot Perspective

Since the Brundtland Report defined the concept of sustainability in 1987, several different indices and indicators have been developed in this area, which is becoming an ever greater concern in society, since it will affect future generations. The main objective of this research study is to analyze whether there are differences in the scores obtained by a broad sample of countries in the Sustainable Society Index according to the geographical area in which the country is located. We apply the HJ-biplot method (Galindo in Questiio 10(1):13–23, 1986), a statistical technique that provides a joint graphical representation in a low dimensional Euclidean space (usually a plane), of a multivariate data matrix; in our study, this is formed by the countries grouped by geographical areas and variables relating to sets of economic, social and environmental indicators included in the Sustainable Society Index. Our findings stress that the variables related to Human Wellbeing fall mainly within the proximity of the countries located in the geographical areas of Europe, America and, to a lesser extent, Asia. In contrast, other variables associated with Environmental Wellbeing, such as greenhouse gases, renewable energy, and air quality are mainly located closer to Africa, and more residually to Asia. In order to represent the most relevant variables in each geographical area and corroborate the results obtained using the HJ-biplot methodology, an analysis was carried out of the radial graph that represents the values of each variable along the independent axes in the form of radii that have their starting point in the centre of the plot and end in the outer ring such that each radius corresponds to a variable. The results obtained show the characteristics of each geographical area in relation to the Sustainable Society Index, and confirm the results obtained with the HJ-biplot.

Supervised embedding of textual predictors with applications in clinical diagnostics for pediatric cardiology.

Electronic health records possess critical predictive information for machine-learning-based diagnostic aids. However, many traditional machine learning methods fail to simultaneously integrate textual data into the prediction process because of its high dimensionality. In this paper, we present a supervised method using Laplacian Eigenmaps to enable existing machine learning methods to estimate both low-dimensional representations of textual data and accurate predictors based on these low-dimensional representations at the same time. We present a supervised Laplacian Eigenmap method to enhance predictive models by embedding textual predictors into a low-dimensional latent space, which preserves the local similarities among textual data in high-dimensional space. The proposed implementation performs alternating optimization using gradient descent. For the evaluation, we applied our method to over 2000 patient records from a large single-center pediatric cardiology practice to predict if patients were diagnosed with cardiac disease. In our experiments, we consider relatively short textual descriptions because of data availability. We compared our method with latent semantic indexing, latent Dirichlet allocation, and local Fisher discriminant analysis. The results were assessed using four metrics: the area under the receiver operating characteristic curve (AUC), Matthews correlation coefficient (MCC), specificity, and sensitivity. The results indicate that supervised Laplacian Eigenmaps was the highest performing method in our study, achieving 0.782 and 0.374 for AUC and MCC, respectively. Supervised Laplacian Eigenmaps showed an increase of 8.16% in AUC and 20.6% in MCC over the baseline that excluded textual data and a 2.69% and 5.35% increase in AUC and MCC, respectively, over unsupervised Laplacian Eigenmaps. As a solution, we present a supervised Laplacian Eigenmap method to embed textual predictors into a low-dimensional Euclidean space. This method allows many existing machine learning predictors to effectively and efficiently capture the potential of textual predictors, especially those based on short texts.

Band-limited Wavelets and Framelets in Low Dimensions

In this paper, we study the problem of constructing non-separable band-limited wavelet tight frames, Riesz wavelets and orthonormal wavelets in \(\mathbb {R}^{2}\) and \(\mathbb {R}^{3}\). We first construct a class of non-separable band-limited refinable functions in low-dimensional Euclidean spaces by using univariate Meyer’s refinable functions along multiple directions defined by classical box-spline direction matrices. These non-separable band-limited definable functions are then used to construct non-separable band-limited wavelet tight frames via the unitary and oblique extension principles. However, these refinable functions cannot be used for constructing Riesz wavelets and orthonormal wavelets in low dimensions as they are not stable. Another construction scheme is then developed to construct stable refinable functions in low dimensions by using a special class of direction matrices. The resulting stable refinable functions allow us to construct a class of MRA-based non-separable band-limited Riesz wavelets and particularly band-limited orthonormal wavelets in low dimensions with small frequency support.

A study on tensor product surfaces in low-dimensional Euclidean spaces

We consider a special case for curves in two-, three-, and four-dimensional Euclidean spaces and obtain a necessary and sufficient condition for the tensor product surfaces of the planar unit circle centered at the origin and these curves to have a harmonic Gauss map.

Shape Variation in Outline Shapes

A general morphometric method for describing shape variation in a sample consisting of landmarks and multiple outline shapes is developed in this article. A distance metric is developed for such data and is used to embed the data in a low-dimensional Euclidean space. The Euclidean space is used to generate summary statistics such as mean and principal shape variation which are implicitly represented in the original space using elements of the sample. A new distance metric for outline shapes is proposed based on Procrustes distance that does not require the extraction of discrete points along the curve. The outline distance metric can be naturally combined with distances between landmarks. A method for aligning outlines and multiple outlines is developed that minimizes the distance metric. The method is compared with semilandmarks on synthetic data and 2 real data sets. Outline methods produce useful and valid results when suitably constrained by landmarks and are useful visualization aids, but questions remain about their suitability for answering biological questions until appropriate distance metrics can be biologically validated.

Embedding complex networks in a low dimensional Euclidean space based on vertex dissimilarities

We propose a method for representing vertices of a complex network as points in a Euclidean space of an appropriate dimension. To this end, we first adopt two widely used quantities as the measures for the dissimilarity between vertices. The dissimilarity is then transformed into its corresponding distance in a Euclidean space via the non-metric multidimensional scaling. We applied the proposed method to real-world as well as models of complex networks. We empirically found that real-world complex networks were embedded in a Euclidean space of relatively lower dimensions and the configuration of vertices in the space was mostly characterized by the self-similarity of a multifractal. In contrast, by applying the same scheme to the network models, we found that, in general, higher dimensions were needed to embed the networks into a Euclidean space and the embedding results usually did not exhibit the self-similar property. From the analysis, we learn that the proposed method serves a way not only to visualize the complex networks in a Euclidean space but to characterize the complex networks in a different manner from conventional ways.

응집 계층 군집화 기법을 이용한 이종 공간정보의 M:N 대응 클래스 군집 쌍 탐색

본 연구는 두 공간정보의 대응 클래스 군집 쌍 탐색을 중심으로 의미론적 정합과정에서 발생하는 M:N 대응관계를 분석하는 방법을 제안한다. 객체의 공유 관계를 이용하여 클래스의 유사도를 측정하고 높은 유사도를 가지는 클래스들을 군집화함으로써 M:N 대응관계를 탐색하고자 한다. 클래스 사이의 유사도를 그래프 모형으로 표현하고 그래프 임베딩 기법을 적용하여 투영공간에서 클래스 사이의 거리가 클래스 중첩분석에 의한 국지적 유사도에 반비례하도록 개별 클래스들의 투영좌표를 계산하고 군집화를 수행함으로써 계층적 대응 군집 쌍을 탐색할 수 있다. 제안된 방법을 평가하기 위하여 경기도 수원시의 수치지형도와 연속지적도에 적용하여 수치지형도의 면 객체 레이어와 연속지적도의 필지 지목의 대응 군집 쌍을 탐색하였다. 탐색된 대응 클래스 쌍의 F-measure를 측정한 결과 약 0.80에서 0.35 사이의 다양한 값을 얻을 수 있었으며, 클래스 명칭과는 상이한 다양한 대응관계를 얻을 수 있었다. In this paper, we propose a method to analyze M:N corresponding relations in semantic matching, especially focusing on feature class matching. Similarities between any class pairs are measured by spatial objects which coexist in the class pairs, and corresponding classes are obtained by clustering with these pairwise similarities. We applied a graph embedding method, which constructs a global configuration of each class in a low-dimensional Euclidean space while preserving the above pairwise similarities, so that the distances between the embedded classes are proportional to the overall degree of similarity on the edge paths in the graph. Thus, the clustering problem could be solved by employing a general clustering algorithm with the embedded coordinates. We applied the proposed method to polygon object layers in a topographic map and land parcel categories in a cadastral map of Suwon area and evaluated the results. F-measures of the detected class pairs were analyzed to validate the results. And some class pairs which would not detected by analysis on nominal class names were detected by the proposed method.

An efficient tree-based computation of a metric comparable to a natural diffusion distance

Using diffusion to define distances between points on a manifold (or a sampled data set) has been successfully employed in various applications such as data organization and approximately isometric embedding of high dimensional data in low dimensional Euclidean space. Recently, P. Jones has proposed a diffusion distance which is both intuitively appealing and scales appropriately with increasing time. In the first part of our paper, we present an efficient tree-based approach to computing an approximation to Jonesʼs diffusion distance. We also show our approximation is comparable to Jonesʼs distance. Neither Jonesʼs distance, nor our approximation, satisfies the triangle inequality; in particular, in the case of heat flow on Rn, Jonesʼs separation distance gives a scaled square of the Euclidean distance. In the second part of our paper, we present a general construction to obtain an “almost” metric from a general distance. We also discuss a numerical procedure to implement our construction. Additionally, we show that in the case of heat flow on Rn, we recover (scaled) Euclidean distance from Jonesʼs distance.

Phoenix: A Weight-Based Network Coordinate System Using Matrix Factorization

Network coordinate (NC) systems provide a lightweight and scalable way for predicting the distances, i.e., round-trip latencies among Internet hosts. Most existing NC systems embed hosts into a low dimensional Euclidean space. Unfortunately, the persistent occurrence of Triangle Inequality Violation (TIV) on the Internet largely limits the distance prediction accuracy of those NC systems. Some alternative systems aim at handling the persistent TIV, however, they only achieve comparable prediction accuracy with Euclidean distance based NC systems. In this paper, we propose an NC system, so-called Phoenix, which is based on the matrix factorization model. Phoenix introduces a weight to each reference NC and trusts the NCs with higher weight values more than the others. The weight-based mechanism can substantially reduce the impact of the error propagation. Using the representative aggregate data sets and the newly measured dynamic data set collected from the Internet, our simulations show that Phoenix achieves significantly higher prediction accuracy than other NC systems. We also show that Phoenix quickly converges to steady state, performs well under host churn, handles the drift of the NCs successfully by using regularization, and is robust against measurement anomalies. Phoenix achieves a scalable yet accurate end-to-end distances monitoring. In addition, we study how well an NC system can characterize the TIV property on the Internet by introducing two new quantitative metrics, so-called RE <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">RPL</sub> and AE <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">RPL</sub> . We show that Phoenix is able to characterize TIV better than other existing NC systems.

Some Liouville theorems for Hénon type elliptic equations

We investigate here the nonlinear elliptic equations − Δ u = | x | α e u and − Δ u = | x | α | u | p − 1 u with α > − 2 , p > 1 and N ⩾ 2 . In particular, we prove some Liouville type theorems for weak solutions with finite Morse index in the low dimensional Euclidean spaces or half spaces.

Good quality virtual realization of unit disk graphs

We consider the problem of finding a realization of an n-vertex unit disk graph (UDG) expressed in general form , say, as an adjacency matrix. The problem is to construct an embedding of the graph in low-dimensional Euclidean space so that the ratio of the length of the longest edge under the embedding to the length of the shortest non-edge under the embedding is as small as possible; the measure is known as the quality of the realization. Thus, an optimum quality realization has quality between 1/2 and 1. Kuhn et al. gave a O (log 3.5 n (loglog n ) 1/2 }) quality realization that requires solving a linear program with exponentially many constraints by using the ellipsoid algorithm. In this article, we give a combinatorial algorithm that achieves an O (log 3 n ) quality realization of an n -vertex UDG expressed in general form. Thus, not only is our algorithm an improvement, it also bypasses the standard and costly technique of solving a linear program with exponentially many “spreading constraints.” As a side effect of our construction, we get the first constant-factor approximation to the minimum clique partition problem for UDGs expressed in general form. Such a clique partition also represents our key technical contribution. If the embedding is allowed to reside in higher dimensional space, we obtain improved results: a quality-2 embedding in constant dimensional Euclidean space.

Shifting strategy for geometric graphs without geometry

We give a simple framework which is an alternative to the celebrated and widely used shifting strategy of Hochbaum and Maass (J. ACM 32(1):103---136, 1985) which has yielded efficient algorithms with good approximation bounds for numerous optimization problems in low-dimensional Euclidean space. Our framework does not require the input graph/metric to have a geometric realization--it only requires that the input graph satisfy some weak property referred to as growth boundedness. Growth bounded graphs form an important graph class that has been used to model wireless networks. We show how to apply the framework to obtain a polynomial time approximation scheme (PTAS) for the maximum (weighted) independent set problem on this important graph class; the problem is W[1]-complete. Via a more sophisticated application of our framework, we show how to obtain a PTAS for the maximum (weighted) independent set for intersection graphs of (low-dimensional) fat objects that are expressed without geometry. Erlebach et al. (SIAM J. Comput. 34(6):1302---1323, 2005) and Chan (J. Algorithms 46(2):178---189, 2003) independently gave a PTAS for maximum weighted independent set problem for intersection graphs of fat geometric objects, say ball graphs, which required a geometric representation of the input. Our result gives a positive answer to a question of Erlebach et al. (SIAM J. Comput. 34(6):1302---1323, 2005) who asked if a PTAS for this problem can be obtained without access to a geometric representation.

Classifying transformation-variant attributed point patterns

This paper presents a classification approach, where a sample is represented by a set of feature vectors called an attributed point pattern. Some attributes of a point are transformational-variant, such as spatial location, while others convey some descriptive feature, such as intensity. The proposed algorithm determines a distance between point patterns by minimizing a Hausdorff-based distance over a set of transformations using a particle swarm optimization. When multiple training samples are available for each class, we implement multidimensional scaling to represent the point patterns in a low-dimensional Euclidean space for visualization and analysis. Results are demonstrated for latent fingerprints from tenprint data and civilian vehicles from circular synthetic aperture radar imagery.

Ultra-low-dimensional embeddings for doubling metrics

We consider the problem of embedding a metric into low-dimensional Euclidean space. The classical theorems of Bourgain, and of Johnson and Lindenstrauss say that any metric on n points embeds into an O (log n )-dimensional Euclidean space with O (log n ) distortion. Moreover, a simple “volume” argument shows that this bound is nearly tight: a uniform metric on n points requires nearly logarithmic number of dimensions to embed with logarithmic distortion. It is natural to ask whether such a volume restriction is the only hurdle to low-dimensional embeddings. In other words, do doubling metrics, that do not have large uniform submetrics, and thus no volume hurdles to low dimensional embeddings, embed in low dimensional Euclidean spaces with small distortion? In this article, we give a positive answer to this question. We show how to embed any doubling metrics into O (log log n ) dimensions with O (log n ) distortion. This is the first embedding for doubling metrics into fewer than logarithmic number of dimensions, even allowing for logarithmic distortion. This result is one extreme point of our general trade-off between distortion and dimension: given an n -point metric (V,d) with doubling dimension dim D , and any target dimension T in the range Ω(dim D log log n ) ≤ T ≤ O (log n ), we show that the metric embeds into Euclidean space R T with O (log n √ dim D / T ) distortion.

Manifold embedding for shape analysis

Shape analysis played important role in computer vision based tasks. The importance of shape information relies that it usually contains perceptual information, and thus can be used for high level visual information analysis. Currently, there are many ways that shapes can be represented as a structural manner using graphs. Hence shapes can be analyzed by using graph methods. This paper describes how graph-spectral methods can be used to transform the node correspondence problem into one of point-sets alignment. We commence by using the ISOMAP algorithm to embed the nodes of a graph in a low-dimensional Euclidean space. With the nodes in the graph transformed to points in a metric space, we can recast the problem of graph-matching into that of aligning the point-sets. Here we use semidefinite programming to develop a robust point-sets correspondences algorithm. Variations in graph structure using the covariance matrix for corresponding embedded point-positions is captured. We construct a statistical point distribution model for the embedded node positions using the eigenvalues and eigenvectors of the covariance matrix. We show how to use this model to project individual graph, i.e. shape into the eigenspace of the point position covariance matrix. We illustrate the utility of the resulting method for shape analysis and recognition on COIL and MPEG-7 databases.

An acceleration method for Ten Berge et al.’s algorithm for orthogonal INDSCAL

INDSCAL (INdividual Differences SCALing) is a useful technique for investigating both common and unique aspects of K similarity data matrices. The model postulates a common stimulus configuration in a low-dimensional Euclidean space, while representing differences among the K data matrices by differential weighting of dimensions by different data sources. Since Carroll and Chang proposed their algorithm for INDSCAL, several issues have been raised: non-symmetric solutions, negative saliency weights, and the degeneracy problem. Orthogonal INDSCAL (O-INDSCAL) which imposes orthogonality constraints on the matrix of stimulus configuration has been proposed to overcome some of these difficulties. Two algorithms have been proposed for O-INDSCAL, one by Ten Berge, Knol, and Kiers, and the other by Trendafilov. In this paper, an acceleration technique called minimal polynomial extrapolation is incorporated in Ten Berge et al.’s algorithm. Simulation studies are conducted to compare the performance of the three algorithms (Ten Berge et al.’s original algorithm, the accelerated algorithm, and Trendafilov’s). Possible extensions of the accelerated algorithm to similar situations are also suggested.

Sensitivity Analysis and Robust Approach in Multidimensional Scaling: An Evaluation of Customer Satisfaction

Multidimensional scaling (MDS) methods are devoted par excellence to representing a set of objects in a low-dimensional Euclidean space by preserving, as far as possible, the proximities between the objects themselves. Like other data analysis techniques, MDS methods treat a set of data as the entire population of interest. It is therefore important to determine the stability and robustness of the results against possible perturbations or errors present in the data. This issue is particularly important in customer satisfaction analysis, where satisfaction indicators and customer ranking can be heavily influenced by such anomalies. These kinds of problems are specifically addressed in this work. Starting from commonly applied methods, a combination of sensitivity and robust analysis will be proposed. A case study will then be considered.

Nonlinear dimensionality reduction for efficient and effective audio similarity searching

In this paper, we address the issue of nonlinear dimensionality reduction to efficiently index spectral audio similarity measures. We propose the embedding of the spectral similarity space to a low-dimensional Euclidean space. This guarantees the triangular inequality and allows the adoption of several indexing schemes. We enlighten the advantages of the proposed indexable method against recently proposed spectral similarity measures that are also indexable. Moreover, our method compares favorably to linear dimensionality reduction methods, like multidimensional scaling (MDS). The proposed method significantly reduces the computation time during the construction process compared to any audio measure and, simultaneously, minimizes the searching cost for similar songs. To the best of our knowledge, the important issue of audio similarity measures' scalability is addressed for the first time.

FINE: Fisher Information Nonparametric Embedding

We consider the problems of clustering, classification, and visualization of high-dimensional data when no straightforward euclidean representation exists. In this paper, we propose using the properties of information geometry and statistical manifolds in order to define similarities between data sets using the Fisher information distance. We will show that this metric can be approximated using entirely nonparametric methods, as the parameterization and geometry of the manifold is generally unknown. Furthermore, by using multidimensional scaling methods, we are able to reconstruct the statistical manifold in a low-dimensional euclidean space; enabling effective learning on the data. As a whole, we refer to our framework as Fisher Information Nonparametric Embedding (FINE) and illustrate its uses on practical problems, including a biomedical application and document classification.

Triangulation and embedding using small sets of beacons

Concurrent with recent theoretical interest in the problem of metric embedding, a growing body of research in the networking community has studied the distance matrix defined by node-to-node latencies in the Internet, resulting in a number of recent approaches that approximately embed this distance matrix into low-dimensional Euclidean space. There is a fundamental distinction, however, between the theoretical approaches to the embedding problem and this recent Internet-related work: in addition to computational limitations, Internet measurement algorithms operate under the constraint that it is only feasible to measure distances for a linear (or near-linear) number of node pairs, and typically in a highly structured way. Indeed, the most common framework for Internet measurements of this type is a beacon-based approach one chooses uniformly at random a constant number of nodes (“beacons”) in the network, each node measures its distance to all beacons, and one then has access to only these measurements for the remainder of the algorithm. Moreover, beacon-based algorithms are often designed not for embedding but for the more basic problem of triangulation , in which one uses the triangle inequality to infer the distances that have not been measured. Here we give algorithms with provable performance guarantees for beacon-based triangulation and embedding. We show that in addition to multiplicative error in the distances, performance guarantees for beacon-based algorithms typically must include a notion of slack —a certain fraction of all distances may be arbitrarily distorted. For metric spaces of bounded doubling dimension (which have been proposed as a reasonable abstraction of Internet latencies), we show that triangulation-based distance reconstruction with a constant number of beacons can achieve multiplicative error 1 + δ on a 1 − ϵ fraction of distances, for arbitrarily small constants δ and ϵ. For this same class of metric spaces, we give a beacon-based embedding algorithm that achieves constant distortion on a 1 − ϵ fraction of distances; this provides some theoretical justification for the success of the recent Global Network Positioning algorithm of Ng and Zhang [2002], and it forms an interesting contrast with lower bounds showing that it is not possible to embed all distances in a doubling metric space with constant distortion. We also give results for other classes of metric spaces, as well as distributed algorithms that require only a sparse set of distances but do not place too much measurement load on any one node.