High-dimensional Euclidean Space Research Articles

Extrinsic Local Regression on Manifold-Valued Data

ABSTRACTWe propose an extrinsic regression framework for modeling data with manifold valued responses and Euclidean predictors. Regression with manifold responses has wide applications in shape analysis, neuroscience, medical imaging, and many other areas. Our approach embeds the manifold where the responses lie onto a higher dimensional Euclidean space, obtains a local regression estimate in that space, and then projects this estimate back onto the image of the manifold. Outside the regression setting both intrinsic and extrinsic approaches have been proposed for modeling iid manifold-valued data. However, to our knowledge our work is the first to take an extrinsic approach to the regression problem. The proposed extrinsic regression framework is general, computationally efficient, and theoretically appealing. Asymptotic distributions and convergence rates of the extrinsic regression estimates are derived and a large class of examples is considered indicating the wide applicability of our approach. Supplementary materials for this article are available online.

Emergent community agglomeration from data set geometry.

In the statistical learning language, samples are snapshots of random vectors drawn from some unknown distribution. Such vectors usually reside in a high-dimensional Euclidean space, and thus the "curse of dimensionality" often undermines the power of learning methods, including community detection and clustering algorithms, that rely on Euclidean geometry. This paper presents the idea of effective dissimilarity transformation (EDT) on empirical dissimilarity hyperspheres and studies its effects using synthetic and gene expression data sets. Iterating the EDT turns a static data distribution into a dynamical process purely driven by the empirical data set geometry and adaptively ameliorates the curse of dimensionality, partly through changing the topology of a Euclidean feature space R^{n} into a compact hypersphere S^{n}. The EDT often improves the performance of hierarchical clustering via the automatic grouping information emerging from global interactions of data points. The EDT is not restricted to hierarchical clustering, and other learning methods based on pairwise dissimilarity should also benefit from the many desirable properties of EDT.

Injectivity of minimal immersions and homeomorphic extensions to space

We study a recent general criterion for the injectivity of the conformal immersion of a Riemannian manifold into higher dimensional Euclidean space, and show how it gives rise to important conditions for Weierstrass–Enneper lifts defined in the unit disk $$\mathbb{D}$$ endowed with a conformal metric. Among the corollaries, we obtain a Becker type condition and a sharp condition depending on the Gaussian curvature and the diameter for an immersed geodesically convex minimal disk in $$\mathbb{R}^3$$ to be embedded. Extremal configurations for the criteria are also determined, and can only occur on a catenoid. For non-extremal configurations, we establish fibrations of space by circles in domain and range that give a geometric analogue of the Ahlfors–Weill extension.

Anisotropic geometry-conforming d-simplicial meshing via isometric embeddings

We develop a dimension-independent, Delaunay-based anisotropic mesh generation algorithm suitable for integration with adaptive numerical solvers. As such, the mesh produced by our algorithm conforms to an anisotropic metric prescribed by the solver as well as the domain geometry, given as a piecewise smooth complex. Motivated by the work of Lévy and Dassi [10-12,20], we use a discrete manifold embedding algorithm to transform the anisotropic problem to a uniform one. This work differs from previous approaches in several ways. First, the embedding algorithm is driven by a Riemannian metric field instead of the Gauss map, lending itself to general anisotropic mesh generation problems. Second we describe our method for computing restricted Voronoi diagrams in a dimension-independent manner which is used to compute constrained centroidal Voronoi tessellations. In particular, we compute restricted Voronoi simplices using exact arithmetic and use data structures based on convex polytope theory. Finally, since adaptive solvers require geometry-conforming meshes, we offer a Steiner vertex insertion algorithm for ensuring the extracted dual Delaunay triangulation is homeomorphic to the input geometries.The two major contributions of this paper are: a method for isometrically embedding arbitrary mesh-metric pairs in higher dimensional Euclidean spaces and a dimension-independent vertex insertion algorithm for producing geometry-conforming Delaunay meshes. The former is demonstrated on a two-dimensional anisotropic problem whereas the latter is demonstrated on both 3d and 4d problems.

Rotational surfaces in higher dimensional Euclidean spaces

In the present study we consider the generalized rotational surfaces in Euclidean m-space \(\mathbb {E}^{m}\). Firstly, we introduce some basic concepts of second fundamental form and curvatures of the surfaces in \( \mathbb {E}^{m}\). Further, we obtained some basic properties of generalized rotational surfaces in \(\mathbb {E}^{m}\) and some results related with their curvatures. Finally, we give some examples of generalized rotational surfaces in Euclidean 5-space \(\mathbb {E}^{5}\).

The Atlas of Chinese World Wide Web Ecosystem Shaped by the Collective Attention Flows.

The web can be regarded as an ecosystem of digital resources connected and shaped by collective successive behaviors of users. Knowing how people allocate limited attention on different resources is of great importance. To answer this, we embed the most popular Chinese web sites into a high dimensional Euclidean space based on the open flow network model of a large number of Chinese users’ collective attention flows, which both considers the connection topology of hyperlinks between the sites and the collective behaviors of the users. With these tools, we rank the web sites and compare their centralities based on flow distances with other metrics. We also study the patterns of attention flow allocation, and find that a large number of web sites concentrate on the central area of the embedding space, and only a small fraction of web sites disperse in the periphery. The entire embedding space can be separated into 3 regions(core, interim, and periphery). The sites in the core (1%) occupy a majority of the attention flows (40%), and the sites (34%) in the interim attract 40%, whereas other sites (65%) only take 20% flows. What’s more, we clustered the web sites into 4 groups according to their positions in the space, and found that similar web sites in contents and topics are grouped together. In short, by incorporating the open flow network model, we can clearly see how collective attention allocates and flows on different web sites, and how web sites connected each other.

An approximation algorithm for the Euclidean incremental median problem

In the incremental version of the k-median problem, we find a sequence of facility sets F1⊆F2⊆⋯⊆Fn, where each Fk contains at most k facilities. This sequence is said to be δ-competitive if the cost of each Fk is at most δ times the optimum cost of k facilities. The best deterministic (randomized) algorithm available for the metric space has a competitive ratio of 8 (7.656). The best one for the one-dimensional problem finds a 5.828-competitive sequence. We give a 7.076-competitive solution for the high-dimensional Euclidean space.

Approximate furthest neighbor with application to annulus query

Much recent work has been devoted to approximate nearest neighbor queries. Motivated by applications in recommender systems, we consider approximate furthest neighbor (AFN) queries and present a simple, fast, and highly practical data structure for answering AFN queries in high-dimensional Euclidean space. The method builds on the technique of Indyk (SODA 2003), storing random projections to provide sublinear query time for AFN. However, we introduce a different query algorithm, improving on Indyk׳s approximation factor and reducing the running time by a logarithmic factor. We also present a variation based on a query-independent ordering of the database points; while this does not have the provable approximation factor of the query-dependent data structure, it offers significant improvement in time and space complexity. We give a theoretical analysis and experimental results. As an application, the query-dependent approach is used for deriving a data structure for the approximate annulus query problem, which is defined as follows: given an input set S and two parameters r>0 and w≥1, construct a data structure that returns for each query point q a point p∈S such that the distance between p and q is at least r/w and at most wr.

Schwarz Problems for Poly-Hardy Space on the Unit Ball

In this paper we study the Schwarz boundary value problems \({\big(}\) for short BVP\({\big)}\) for the poly-Hardy space defined on the unit ball of higher dimensional Euclidean space \({\mathbb{R}^n}\) . We first discuss the boundary behavior of functions belonging to the poly-Hardy class. Then we construct the Schwarz kernel function, and describe the boundary properties of the Schwarz-type integrable operator. Finally, we study the Schwarz BVPs for the Hardy class and the poly-Hardy class on the unit ball of higher dimensional Euclidean space \({\mathbb{R}^n}\), and obtain the expressions of solutions, explicitly. As an application, the monogenic signals considered for the Hardy spaces defined on the unit sphere are reconstructed when the scalar- and sub-algebra-valued data are given, which is the extension of the analytic signals for the Hardy spaces on the unit circle of the complex plane.

On the Number of Rich Lines in High Dimensional Real Vector Spaces

In this short note we use the polynomial partitioning lemma to strengthen a recent result of Dvir and Gopi about the number of rich lines in high dimensional Euclidean spaces. Our result shows that if there are sufficiently many rich lines incident to a set of points then large fraction of them must be contained in a hyperplane.

The Chaotic Attractor Analysis of DJIA Based on Manifold Embedding and Laplacian Eigenmaps

By using the techniques of Manifold Embedding and Laplacian Eigenmaps, a novel strategy has been proposed in this paper to detect the chaos of Dow Jones Industrial Average. Firstly, the chaotic attractor of financial time series is assumed to lie on a low-dimensional manifold that is embedded into a high-dimensional Euclidean space. Then, an improved phase space reconstruction method and a nonlinear dimensionality reduction method are introduced to help reveal the structure of the chaotic attractor. Next, the empirical study on the financial time series of Dow Jones Industrial Average shows that there exists an attractor which lies on a manifold constructed by the time sequence of Moving average convergence divergence; finally, Determinism Test, Poincaré section, and translation analysis are used as test approaches to prove both whether it is a chaos and how it works.

Symmetries of Monocoronal Tilings

The vertex corona of a vertex of some tiling is the vertex together with the adjacent tiles. A tiling where all vertex coronae are congruent is called monocoronal. We provide a classification of monocoronal tilings in the Euclidean plane and derive a list of all possible symmetry groups of monocoronal tilings. In particular, any monocoronal tiling with respect to direct congruence is crystallographic, whereas any monocoronal tiling with respect to congruence (reflections allowed) is either crystallographic or it has a one-dimensional translation group. Furthermore, bounds on the number of the dimensions of the translation group of monocoronal tilings in higher dimensional Euclidean space are obtained.

A Geometric Representation of Collective Attention Flows.

With the fast development of Internet and WWW, “information overload” has become an overwhelming problem, and collective attention of users will play a more important role nowadays. As a result, knowing how collective attention distributes and flows among different websites is the first step to understand the underlying dynamics of attention on WWW. In this paper, we propose a method to embed a large number of web sites into a high dimensional Euclidean space according to the novel concept of flow distance, which both considers connection topology between sites and collective click behaviors of users. With this geometric representation, we visualize the attention flow in the data set of Indiana university clickstream over one day. It turns out that all the websites can be embedded into a 20 dimensional ball, in which, close sites are always visited by users sequentially. The distributions of websites, attention flows, and dissipations can be divided into three spherical crowns (core, interim, and periphery). 20% popular sites (Google.com, Myspace.com, Facebook.com, etc.) attracting 75% attention flows with only 55% dissipations (log off users) locate in the central layer with the radius 4.1. While 60% sites attracting only about 22% traffics with almost 38% dissipations locate in the middle area with radius between 4.1 and 6.3. Other 20% sites are far from the central area. All the cumulative distributions of variables can be well fitted by “S”-shaped curves. And the patterns are stable across different periods. Thus, the overall distribution and the dynamics of collective attention on websites can be well exhibited by this geometric representation.

Query-aware locality-sensitive hashing for approximate nearest neighbor search

Locality-Sensitive Hashing (LSH) and its variants are the well-known indexing schemes for the c -Approximate Nearest Neighbor ( c -ANN) search problem in high-dimensional Euclidean space. Traditionally, LSH functions are constructed in a query-oblivious manner in the sense that buckets are partitioned before any query arrives. However, objects closer to a query may be partitioned into different buckets, which is undesirable. Due to the use of query-oblivious bucket partition, the state-of-the-art LSH schemes for external memory, namely C2LSH and LSB-Forest, only work with approximation ratio of integer c ≥ 2. In this paper, we introduce a novel concept of query-aware bucket partition which uses a given query as the "anchor" for bucket partition. Accordingly, a query-aware LSH function is a random projection coupled with query-aware bucket partition, which removes random shift required by traditional query-oblivious LSH functions. Notably, query-aware bucket partition can be easily implemented so that query performance is guaranteed. We propose a novel query-aware LSH scheme named QALSH for c -ANN search over external memory. Our theoretical studies show that QALSH enjoys a guarantee on query quality. The use of query-aware LSH function enables QALSH to work with any approximation ratio c > 1. Extensive experiments show that QALSH outperforms C2LSH and LSB-Forest, especially in high-dimensional space. Specifically, by using a ratio c < 2, QALSH can achieve much better query quality.

Extrinsic Local Regression on Manifold-Valued Data

We propose an extrinsic regression framework for modeling data with manifold valued responses and Euclidean predictors. Regression with manifold responses has wide applications in shape analysis, neuroscience, medical imaging and many other areas. Our approach embeds the manifold where the responses lie onto a higher dimensional Euclidean space, obtains a local regression estimate in that space, and then projects this estimate back onto the image of the manifold. Outside the regression setting both intrinsic and extrinsic approaches have been proposed for modeling i.i.d manifold-valued data. However, to our knowledge our work is the first to take an extrinsic approach to the regression problem. The proposed extrinsic regression framework is general, computationally efficient and theoretically appealing. Asymptotic distributions and convergence rates of the extrinsic regression estimates are derived and a large class of examples are considered indicating the wide applicability of our approach.

Embeddings and immersions of tropical curves

We construct immersions of trivalent abstract tropical curves in the Euclidean plane and embeddings of all abstract tropical curves in higher dimensional Euclidean space. Since not all curves have an embedding in the plane, we define the tropical crossing number of an abstract tropical curve to be the minimum number of self-intersections, counted with multiplicity, over all its immersions in the plane. We show that the tropical crossing number is at most quadratic in the number of edges and this bound is sharp. For curves of genus up to two, we systematically compute the crossing number. Finally, we use our immersed tropical curves to construct totally faithful nodal algebraic curves via lifting results of Mikhalkin and Shustin.

Direct Sum Simplex Signal Space Codes for DNA Nucleotides and Amino Acid Codons

A genomic symbol to signal conversion concept is introduced and demonstrated on nucleotide base pairs and amino acid codons. Binary encodings of the presence or absence of bioinformatic attributes (e.g., physical, chemical, functional, structural, etc.) onto Euclidean signal space points of $+1$ or $-1$ is performed for individual base pairs or blocks of DNA symbols such as amino acid codons. The symbol to Euclidean space mapping is further encoded into an expanded digital representation based on a direct sum $M$ -ary tree structure called a $\sigma$ -tree. For bioinformatic query sequences, the $\sigma$ -tree provides a rapid search mechanism to find near neighbor similarity sets within warehouses of heterogeneous bioinformatic data. Bioinformatic utility will be achieved in data mining systems based on the proposed symbol-to-signal mapping method through trend analysis of bioinformatic attribute data within similarity set aggregates. Query specific feature/attribute similarity sets are revealed by high dimensional Euclidean signal space clusters. The proposed binary encoded signal vectors provide an equal energy Euclidean space simplex code in the sense of channel coding signal constellation design. Use of an equal energy simplex code prevents biases from being introduced among different attributes during the symbol sequence to signal sequence mapping. The inappropriate use of a multienergy, low dimension, signal point mapping is replaced with a same-energy, high dimension, signal point mapping. The $M$ -ary digital encoding and associated near neighbor search method is extensible to nucleotide sequence blocks of any length, producing digitally expressed signal space simplex representations on a hypersphere of high dimension. Moreover, the direct sum structure empowers computation and memory efficiencies that grow only linearly with increasing sequence length and signal set dimensionality. The bioinformatic attribute encoding method and efficient $\sigma$ -tree expanded digital representation introduced here will provide a foundation for constructing novel genomic driven data mining systems.

Confidence analysis of standard deviational ellipse and its extension into higher dimensional euclidean space.

Standard deviational ellipse (SDE) has long served as a versatile GIS tool for delineating the geographic distribution of concerned features. This paper firstly summarizes two existing models of calculating SDE, and then proposes a novel approach to constructing the same SDE based on spectral decomposition of the sample covariance, by which the SDE concept is naturally generalized into higher dimensional Euclidean space, named standard deviational hyper-ellipsoid (SDHE). Then, rigorous recursion formulas are derived for calculating the confidence levels of scaled SDHE with arbitrary magnification ratios in any dimensional space. Besides, an inexact-newton method based iterative algorithm is also proposed for solving the corresponding magnification ratio of a scaled SDHE when the confidence probability and space dimensionality are pre-specified. These results provide an efficient manner to supersede the traditional table lookup of tabulated chi-square distribution. Finally, synthetic data is employed to generate the 1-3 multiple SDEs and SDHEs. And exploratory analysis by means of SDEs and SDHEs are also conducted for measuring the spread concentrations of Hong Kong’s H1N1 in 2009.

A geometric viewpoint of manifold learning

In many data analysis tasks, one is often confronted with very high dimensional data. The manifold assumption, which states that the data is sampled from a submanifold embedded in much higher dimensional Euclidean space, has been widely adopted by many researchers. In the last 15 years, a large number of manifold learning algorithms have been proposed. Many of them rely on the evaluation of the geometrical and topological of the data manifold. In this paper, we present a review of these methods on a novel geometric perspective. We categorize these methods by three main groups: Laplacian-based, Hessian-based, and parallel field-based methods. We show the connection and difference between these three groups on their continuous and discrete counterparts. The discussion is focused on the problem of dimensionality reduction and semi-supervised learning.

ASKIT: Approximate Skeletonization Kernel-Independent Treecode in High Dimensions

We present a fast algorithm for kernel summation problems in high dimensions. Such problems appear in computational physics, numerical approximation, nonparametric statistics, and machine learning. In our context, the sums depend on a kernel function that is a pair potential defined on a dataset of points in a high-dimensional Euclidean space. A direct evaluation of the sum scales quadratically with the number of points. Fast kernel summation methods can reduce this cost to linear complexity, but the constants involved do not scale well with the dimensionality of the dataset. The main algorithmic components of fast kernel summation algorithms are the separation of the kernel sum between near and far field (which is the basis for pruning) and the efficient and accurate approximation of the far field. We introduce novel methods for pruning and for approximating the far field. Our far field approximation requires only kernel evaluations and does not use analytic expansions. Pruning is not done using bounding boxes but rather combinatorially using a sparsified nearest-neighbor graph of the input distribution. The time complexity of our algorithm depends linearly on the ambient dimension. The error in the algorithm depends on the low-rank approximability of the far field, which in turn depends on the kernel function and on the intrinsic dimensionality of the distribution of the points. The error of the far field approximation does not depend on the ambient dimension. We present the new algorithm along with experimental results that demonstrate its performance. As a highlight, we report results for Gaussian kernel sums for 100 million points in 64 dimensions, for one million points in 1000 dimensions, and for problems in which the Gaussian kernel has a variable bandwidth. To the best of our knowledge, all of these experiments are prohibitively expensive with existing fast kernel summation methods.