Pointwise estimates of pseudo-differential operators

Non-Sibsonian interpolation on arbitrary system of points in Euclidean space and adaptive isolines generation

Two strongly NP-hard problems of clustering a finite set of points in Euclidean space are considered. In the first problem, given an input set, we need to find a cluster (i.e., a subset) of given size that minimizes the sum of the squared distances between the elements of this cluster and its centroid (geometric center). Every point outside this cluster is considered a singleton cluster. In the second problem, we need to partition a finite set into two clusters minimizing the sum, over both clusters, of weighted intracluster sums of the squared distances between the elements of the clusters and their centers. The center of one of the clusters is unknown and is determined as its centroid, while the center of the other cluster is set at some point of space (without loss of generality, at the origin). The weighting factors for both intracluster sums are the given cluster sizes. Parameterized randomized algorithms are presented for both problems. For given upper bounds on the relative error and the failure probability, the parameter value is defined for which both algorithms find approximation solutions in polynomial time. This running time is linear in the space dimension and the size of the input set. The conditions are found under which the algorithms are asymptotically exact and their time complexity is linear in the space dimension and quadratic in the input set size.

Basic program for a significance test for two clusters in euclidean space as measured by their overlap

Low Rank Representation (LRR) based unsupervised clustering methods have achieved great success since these methods could explore low-dimensional subspace structure embedded in original data effectively. The conventional LRR methods generally treat the data as the points in Euclidean space. However, it is no longer suitable for high-dimension data (such as video or imageset). That is because high-dimension data are always considered as non-linear manifold data such as Grassmann manifold. Besides, the typical LRR methods always adopt the traditional single nuclear norm based low rank constraint which can not fully reveal the low rank property of the data representation and often leads to suboptimal solution. In this paper, a new LRR based clustering model is constructed on Grassmann manifold for high-dimension data. In the proposed method, each high-dimension data is formed as a sample on Grassmann manifold with non-linear metric. Meanwhile, a non-convex low rank representation is adopt to reveal the intrinsic property of these high-dimension data and reweighted rank minimization constraint is introduced. The experimental results on several public datasets show that the proposed method outperforms the state-of-the-art clustering methods.

This paper presents an algorithmic method for solving the two-plant simultaneous bounded domain stabilization problem for SISO LTI systems. This problem has no closed form solution. The solution provides robust performance in the presence of sensor or actuator failure, or other major parameter changes. Vidyasagar (1987) studied a similar problem involving partially bounded stability domains. However, stability with respect to partially bounded domains only partially bound performance characteristics, such as control energy and transient response. The current investigation gives necessary conditions for simultaneous bounded domain stability and demonstrates a geometry-based solution algorithm which can be automated. The possible solutions to the problem and the admissible solutions are represented as sets of points in Euclidean space. The solution to the problem is found by using computational geometric techniques to detect points in the intersection of these two sets, if there is one, and deducing the simultaneous stabilizing compensator design from the points found in the intersection.

On prescribing total orders and preorders to pairwise distances of points in Euclidean space

Given a set of points in Euclidean space, a minimum log likelihood ratio threshold, and a statistical significance threshold, Elliptical Hotspot Detection (EHD) finds elliptical hotspot areas where the concentration of activities inside is significantly higher than outside. The EHD problem is important to many fields, such as criminology, transportation engineering, and epidemiology. Related work (e.g., SatScan) enumerates only circular candidates using activities as centers, and may miss many significant ellipses. EHD problem is challenging for two reasons, namely the large enumeration space and the lack of monotonicity of the log likelihood ratio. To overcome the challenges and limitations of the related work, this paper proposes a novel algorithm for EHD. A case study on real crime data shows that our algorithm is able to find hotspots that cannot be detected by the related work. Experimental evaluation shows that the proposed algorithm saves substantial amount of computation compared to the naive approach.

We consider some consimilar problems of searching for disjoint subsets (clusters) in the nite set of points in Euclidean space. In these problems, it is required to maximize the minimum subset size such that the value of each intracluster quadratic variation would not exceed a given fraction (constant) of the total quadratic variation of the points of the input set with respect to its centroid. In the paper, we have proved that all the problems are NP-hard even on a line.

We consider a problem of 2-partitioning a finite sequence of points in Euclidean space into two clusters of the given sizes with some additional constraints. The solution criterion is the minimum of the sum (over both clusters) of weighted intracluster sums of squared distances between the elements of each cluster and its center. The weights of the intracluster sums are equal to the cardinalities of the desired clusters. The center of one cluster is given as input, while the center of the other one is unknown and is determined as a geometric center, i.e. as a point of space equal to the mean of the cluster elements. The following constraints hold: the difference between the indices of two subsequent points included in the first cluster is bounded from above and below by given some constants. It is shown that the considered problem is the strongly NP-hard one. An exact algorithm is proposed for the case of integer-valued input of the problem. This algorithm has a pseudopolynomial running time if the space dimension is fixed.

When finite, Shannon entropies of all sub vectors of a random vector are considered for the coordinates of an entropic point in Euclidean space. A linear combination of the coordinates gives rise to an unconstrained information inequality if it is nonnegative for all entropic points. With at least four variables no finite set of linear combinations generates all such inequalities. This is proved by constructing explicitly an infinite sequence of new linear information inequalities and a curve in a special geometric position to the halfspaces defined by the inequalities. The inequalities are constructed recurrently by adhesive pasting of restrictions of polymatroids and the curve ranges in the closure of a set of the entropic points.

We present a new algorithm for clustering longitudinal data. Data of this type can be conceptualized as consisting of individuals and, for each such individual, observations of a time-dependent variable made at various times. Generically, the specific way in which this variable evolves with time is different from one individual to the next. However, there may also be commonalities; specific characteristic features of the time evolution shared by many individuals. The purpose of the method we put forward is to find clusters of individual whose underlying time-dependent variables share such characteristic features. This is done in two steps. The first step identifies each individual to a point in Euclidean space whose coordinates are determined by specific mathematical formulae meant to capture a variety of characteristic features. The second step finds the clusters by applying the Spectral Clustering algorithm to the resulting point cloud.

We consider a strongly NP-hard problem of clustering a finite set of points in Euclidean space into two clusters. In this problem, we find a partition of the input set minimizing the sum over both clusters of the weighted intracluster sums of the squared distances between the elements of the clusters and their centers. The weight factors are the cardinalities of the corresponding clusters and the centers are defined as follows. The center of the first cluster is unknown and determined as the centroid, while the center of the other one is given as input (is the origin without loss of generality). In this paper, we present a polynomial-time exact algorithm for the one-dimensional case of the problem.

The nearest neighbor graph (NNG), defined for a set of points in Euclidean space, has found many uses in computational geometry and clustering analysis. Yet there seems to be surprisingly little knowledge about some basic properties of this graph. In this talk, we ask some natural questions that are motivated by geometric embedding problems. For example, in the simulation of many-body systems, it is desirable to map a set of particles to a regular data array so as to preserve neighborhood relations, i.e., to minimize the dilation of NNG. We will derive bounds on the dilation by studying the diameter of NNG. Other properties and applications of NNG will also be discussed. (Joint work with Mike Paterson)

A set of $n$ points in Euclidean space is partitioned into the $k$ groups that minimize the within groups sum of squares. Under the assumption that the $n$ points come from independent sampling on a fixed distribution, conditions are found to assure asymptotic normality of the vector of means of the $k$ groups. The method of proof makes novel application of a functional central limit theorem for empirical processes--a generalization of Donsker's theorem due to Dudley.

In this paper we show that dimensionality reduction (i.e., Johnson--Lindenstrauss lemma) preserves not only the distances between static points, but also between moving points, and more generally between low-dimensional flats, polynomial curves, curves with low winding degree, and polynomial surfaces. We also show that surfaces with bounded doubling dimension can be embedded into low dimension with small additive error. Finally, we show that for points with polynomial motion, the radius of the smallest enclosing ball can be preserved under dimensionality reduction.