Showing papers on "Metric (mathematics) published in 1994"

Statistical distance and the geometry of quantum states

Abstract: By finding measurements that optimally resolve neighboring quantum states, we use statistical distinguishability to define a natural Riemannian metric on the space of quantum-mechanical density operators and to formulate uncertainty principles that are more general and more stringent than standard uncertainty principles.

Partial Metric Topology

Abstract: Metric spaces are inevitably Hausdorff and so cannot, for example, be used to study non-Hausdorff topologies such as those required in the Tarskian approach to programming language semantics. This paper presents a symmetric generalised metric for such topologies, an approach which sheds new light on how metric tools such as Banach's Theorem can be extended to non-Hausdorff topologies.

A motion planner for nonholonomic mobile robots

Abstract: This paper considers the problem of motion planning for a car-like robot (i.e., a mobile robot with a nonholonomic constraint whose turning radius is lower-bounded). We present a fast and exact planner for our mobile robot model, based upon recursive subdivision of a collision-free path generated by a lower-level geometric planner that ignores the motion constraints. The resultant trajectory is optimized to give a path that is of near-minimal length in its homotopy class. Our claims of high speed are supported by experimental results for implementations that assume a robot moving amid polygonal obstacles. The completeness and the complexity of the algorithm are proven using an appropriate metric in the configuration space R/sup 2//spl times/S/sup 1/ of the robot. This metric is defined by using the length of the shortest paths in the absence of obstacles as the distance between two configurations. We prove that the new induced topology and the classical one are the same. Although we concentrate upon the car-like robot, the generalization of these techniques leads to new theoretical issues involving sub-Riemannian geometry and to practical results for nonholonomic motion planning. >

Learning Bayesian networks: the combination of knowledge and statistical data

Abstract: We describe algorithms for learning Bayesian networks from a combination of user knowledge and statistical data. The algorithms have two components: a scoring metric and a search procedure. The scoring metric takes a network structure, statistical data, and a user's prior knowledge, and returns a score proportional to the posterior probability of the network structure given the data. The search procedure generates networks for evaluation by the scoring metric. Our contributions are threefold. First, we identify two important properties of metrics, which we call score equivalence and parameter modularity. These properties have been mostly ignored, but when combined, greatly simplify the encoding of a user's prior knowledge. In particular, a user can express his knowledge—for the most part—as a single prior Bayesian network for the domain. Second, we describe greedy hill-climbing and annealing search algorithms to be used in conjunction with scoring metrics. In the special case where each node has at most one parent, we show that heuristic search can be replaced with a polynomial algorithm to identify the networks with the highest score. Third, we describe a methodology for evaluating Bayesian-network learning algorithms. We apply this approach to a comparison of our metrics and search procedures.

Proximity Matching Using Fixed-Queries Trees

Abstract: We present a new data structure, called the fixed-queries tree, for the problem of finding all elements of a fixed set that are close, under some distance function, to a query element. Fixed-queries trees can be used for any distance function, not necessarily even a metric, as long as it satisfies the triangle inequality. We give an analysis of several performance parameters of fixed-queries trees and experimental results that support the analysis. Fixed-queries trees are particularly efficient for applications in which comparing two elements is expensive.

Equilibrium maps between metric spaces

Abstract: We show the existence of harmonic mappings with values in possibly singular and not necessarily locally compact complete metric length spaces of nonpositive curvature in the sense of Alexandrov. As a technical tool, we show that any bounded sequence in such a space has a subsequence whose mean values converge. We also give a general definition of harmonic maps between metric spaces based on mean value properties andΓ-convergence.

Combinatorial pattern discovery for scientific data: some preliminary results

Abstract: Suppose you are given a set of natural entities (e.g., proteins, organisms, weather patterns, etc.) that possess some important common externally observable properties. You also have a structural description of the entities (e.g., sequence, topological, or geometrical data) and a distance metric. Combinatorial pattern discovery is the activity of finding patterns in the structural data that might explain these common properties based on the metric.This paper presents an example of combinatorial pattern discovery: the discovery of patterns in protein databases. The structural representation we consider are strings and the distance metric is string edit distance permitting variable length don't cares. Our techniques incorporate string matching algorithms and novel heuristics for discovery and optimization, most of which generalize to other combinatorial structures. Experimental results of applying the techniques to both generated data and functionally related protein families obtained from the Cold Spring Harbor Laboratory show the effectiveness of the proposed techniques. When we apply the discovered patterns to perform protein classification, they give information that is complementary to the best protein classifier available today.

Temporal planning with continuous change

Abstract: We present ZENO, a least commitment planner that handles actions occurring over extended intervals of time. Deadline goals, metric preconditions, metric effects, and continuous change are supported. Simultaneous actions are allowed when their effects do not interfere. Unlike most planners that deal with complex languages, the ZENO planning algorithm is sound and complete. The running code is a complete implementation of the formal algorithm, capable of solving simple problems (i.e., those involving less than a dozen steps).

Metrics of negative Ricci curvature

Abstract: One of the most natural and important topics in Riemannian geometry is the relation between curvature and global structure of the underlying manifold. For instance, complete manifolds of negative sectional curvature are always aspherical and in the compact case their fundamental group can only contain abelian subgroups which are infinite cyclic. Furthermore, it seemed to be a natural principle that a (closed) manifold cannot carry two metrics of different signed curvatures, as it is a basic fact that this is true for sectional curvature. But it turned out to be wrong (much later and from a strongly analytic argument) for the scalar curvature S, since each manifold M', n > 3, admits a complete metric with S _-1 (cf. Aubin [A] and Bland, Kalka [BIK]). Hence the situation for Ricci curvature Ric, lying between sectional and scalar curvature, seemed to be quite delicate. Up to now, the most general results concerning Ric < 0 were proved by Gao, Yau [GY] and Brooks [Br] using Thurston's theory of hyperbolic threemanifolds, viz.: Each closed three-manifold admits a metric with Ric < 0. This is obtained from the fact that these manifolds carry hyperbolic metrics with certain singularities; Gao and Yau (resp. Brooks) smoothed these singularities to get a regular metric with Ric < 0. These methods extend to three-manifolds of finite type and certain hyperbolic orbifolds. In any case, the arguments rely on exploiting some extraordinary metric structures, whose existence is neither obvious nor conceptually related to the Ricci curvature problem. Indeed, the existence depends on the assumption that the manifold is three-dimensional and compact. Moreover this approach does not provide insight into the typical behaviour of metrics with Ric < 0 since one is led to very special metrics. In this article we approach negative Ricci curvature using a completely different and new concept (which will become even more significant in [L2]) as we deliberately produce Ric < 0. Actually we will prove the following results; in these notes Ric(g), resp. r(g), denotes the Ricci tensor, resp. curvature of a smooth metric g:

Golden template comparison using efficient image registration

Abstract: Golden Template Comparison (GTC) is a method that can be applied to flaw and defect detection in images of 2-dimensional scenes. When a test image is compared to a golden template image, the images must be registered, and then subtracted. The resulting difference image is then analyzed for features that indicate flaws or defects. The registration step is a major determinant of the performance of GTC, and the invention performs the registration step of GTC using a highly efficient and accurate registration method. The registration method of the invention provides substantial registration of all of the features common to the test image and the golden template image, even when one of the images to be registered is flawed, using a registration metric based on minimization of the sum of the absolute value of the differences between the images to be registered, and an efficient optimization technique, such as a version of gradient descent, wherein a local minimum in a registration metric space is found to the nearest pixel using less computational resources than needed to compute the entire registration metric space.

A study of distance-based machine learning algorithms

Abstract: Distance-based algorithms are machine learning algorithms that classify queries by computing distances between these queries and a number of internally stored exemplars. Exemplars that are closest to the query have the largest influence on the classification assigned to the query. Two specific distance-based algorithms, the nearest neighbor algorithm and the nearest-hyperrectangle algorithm, are studied in detail. It is shown that the k-nearest neighbor algorithm (kNN) outperforms the first nearest neighbor algorithm only under certain conditions. Data sets must contain moderate amounts of noise. Training examples from the different classes must belong to clusters that allow an increase in the value of k without reaching into clusters of other classes. Methods for choosing the value of k for kNN are investigated. It shown that one-fold cross-validation on a restricted number of values for k suffices for best performance. It is also shown that for best performance the votes of the k-nearest neighbors of a query should be weighted in inverse proportion to their distances from the query. Principal component analysis is shown to reduce the number of relevant dimensions substantially in several domains. Two methods for learning feature weights for a weighted Euclidean distance metric are proposed. These methods improve the performance of kNN and NN in a variety of domains. The nearest-hyperrectangle algorithm (NGE) is found to give predictions that are substantially inferior to those given by kNN in a variety of domains. Experiments performed to understand this inferior performance led to the discovery of several improvements to NGE. Foremost of these is BNGE, a batch algorithm that avoids construction of overlapping hyperrectangles from different classes. Although it is generally superior to NGE, BNGE is still significantly inferior to kNN in a variety of domains. Hence, a hybrid algorithm (KBNGE), that uses BNGE in parts of the input space that can be represented by a single hyperrectangle and kNN otherwise, is introduced. The primary contributions of this dissertation are (a) several improvements to existing distance-based algorithms, (b) several new distance-based algorithms, and (c) an experimentally supported understanding of the conditions under which various distance-based algorithms are likely to give good performance.

A notion of total variation depending on a metric with discontinuous coefficients

Abstract: Given a function u : Ω ⊆ _ ℝ n → ℝ , we introduce a notion of total variation of u depending on a possibly discontinuous Finsler metric. We prove some integral representation results for this total variation, and we study the connections with the theory of relaxation.

Finsler structures for the part metric and Hilbert's projective metric and applications to ordinary differential equations

Abstract: A vibration accommodating clamp for securing pipes which carry brake lines and the like to the underframes of railway cars. The clamp is initially mounted by placing a U-shaped bolt over the pipe and through holes in a mounting bracket and permanently securing a collar on a grooved portion of one leg of the bolt. The second leg of the bolt remains unused until it is necessary to remove the clamp. The clamp is removed by cutting off the collar and reapplied by placing a nut and washer on the threaded portion of the unused leg.

Handwritten character classification using nearest neighbor in large databases

Abstract: Shows that systems built on a simple statistical technique and a large training database can be automatically optimized to produce classification accuracies of 99% in the domain of handwritten digits. It is also shown that the performance of these systems scale consistently with the size of the training database, where the error rate is cut by more than half for every tenfold increase in the size of the training set from 10 to 100,000 examples. Three distance metrics for the standard nearest neighbor classification system are investigated: a simple Hamming distance metric, a pixel distance metric, and a metric based on the extraction of penstroke features. Systems employing these metrics were trained and tested on a standard, publicly available, database of nearly 225,000 digits provided by the National Institute of Standards and Technology. Additionally, a confidence metric is both introduced by the authors and also discovered and optimized by the system. The new confidence measure proves to be superior to the commonly used nearest neighbor distance. >

Solving Recursive Domain Equations with Enriched Categories.

Abstract: : Both pre-orders and metric spaces have been used at various times as a foundation for the solution of recursive domain equations in the area of denotational semantics. In both cases the central theorem states that a'converging' sequence of 'complete' domains/spaces with 'continuous'retraction pairs between them has a limit in the category of complete domains/spaces with retraction pairs as morphisms. The pre-order version was discovered first by Scott in 1969, and is referred to as Scott's inverse limit theorem. The metric version was mainly developed by de Bakker and Zucker and refined and generalized by America and Rutten. The theorem in both its versions provides the main tool for solving recursive domain equations. The proofs of the two versions of the theorem look astonishingly similar, but until now the preconditions for the pre-order and the metric versions have seemed to be fundamentally different. In this thesis we establish a more general theory of domains based on the notions of enriched categories, and prove Scott's inverse limit theorem in this theory. The metric and pre-order versions are special cases, obtained just by using different logics as parameter to the general theory.

A self-organizing network for hyperellipsoidal clustering (HEC)

Abstract: We propose a self-organizing network (HEC) for hyper-ellipsoidal clustering. The HEC network performs a partitional clustering using the regularized Mahalanobis distance. This regularized Mahalanobis distance measure is proposed to deal with the problems in estimating the Mahalanobis distance when the number of patterns in a cluster is less than (ill-posed problem) or not considerably larger than (poorly-posed problem) the dimensionality of the feature space in clustering multidimensional data. This regularized distance also achieves a tradeoff between hyperspherical and hyperellipsoidal cluster shapes so as to prevent the HEC network from producing unusually large or unusually small clusters. The significance level of the Kolmogrov-Smirnov test on the distribution of the Mahalanobis distances of patterns in a cluster to the cluster center under the multivariate Gaussian assumption is used as a measure of cluster compactness. The HEC network has been tested on a number of artificial data sets and real data sets. Experiments show that the HEC network gives better clustering results compared to the well-known K-means algorithm with the Euclidean distance metric. >

An adaptive multiuser detector with joint amplitude and delay estimation

Abstract: A multiuser detector is developed in which the delays and amplitudes of the incoming waveforms are estimated recursively. The algorithm is an extension of the symbol-by-symbol detector of Abend and Fritchman, originally derived for intersymbol interference (ISI) channels, to the multiuser application. In order to make the multiuser detector adaptive, the likelihoods in the symbol-by-symbol metric update are approximated using a set of extended Kalman filter (EKF) innovations. The EKF's provide, in addition to the likelihoods, joint estimates of the signal delays and amplitudes. The resulting algorithm is quite complex, due to the large number of possible composite symbols corresponding to the multiple users. However, it is shown that the likelihood computations and EKF updates can be expressed it terms of a set of cross-correlation functions, which need only be computed for a subset of the possible symbols. The cross-correlator outputs are updated at the bit rate, and thus the EKF and metric update computations need be performed only at a function of the spread-spectrum chip rate. A metric pruning technique is proposed that further reduces the number of EKF delay/amplitude estimators required. Finally, an important sampling simulation strategy is used to obtain bit-error rate estimates for the adaptive multiuser detector. >

Metric inference for social networks

Abstract: Using a natural metric on the space of networks, we define a probability measure for network-valued random variables. This measure is indexed by two parameters, which are interpretable as a location parameter and a dispersion parameter. From this structure, one can develop maximum likelihood estimates, hypothesis tests and confidence regions, all in the context of independent and identically distributed networks. The value of this perspective is illustrated through application to portions of the friedship cognitive social structure data gathered by Krackhardt (1987).

Application of Coupling Method to the First Eigenvalue on Manifold

Abstract: By using a coupling technique,this paper presents some lower bounds of the first eigenvalue of an adjoint operator △+Z on compact M.This method is new and the proofs are straightforward.The method not only achieves the optimal bounds but also improves some known estimates Denote by g.d and D the Riemanman metric,dimension and diameter of M,respectively.Suppose that RicM≥-Kg for some real number K For the special case Z=0,the lower bound of A,provided by the paper am be summarized as followsBesides,a method of estimating the bound for general operators is given.Two examples,even on non-compact space,show that the estimates obtained by this method can be sharp.

Metric Methods Three Examples and a Theorem

Abstract: ⁄y The existence of a model for a logic program is generally established by lattice-theoretic arguments. We present three examples to show that metric methods can often be used instead, generally in a direct, straightforward way. One example is a game program, which is not stratifled or locally stratifled, but which has a unique supported model whose existence is easily established using metric methods. The second example is a program without a unique supported model, but having a part that is ‘well-behaved.’ The third example is a program in which one part depends on another, illustrating how modularity might be treated metrically. Finally we use ideas from this third example to prove a general result from [3]. The intention in presenting these examples and the theorem is to stimulate interest in metric techniques, and is not to present a fully developed theory.

Existence and Differentiability of Metric Projections in Hilbert Spaces

Abstract: This paper considers metric projections onto a closed subset S of a Hilbert space. If the set S is convex, then it is well known that the corresponding metric projections always exist, unique and directionally differentiable at boundary points of S. These properties of metric projections are considered for possibly nonconvex sets S. In particular, existence and directional differentiability of metric projections for certain classes of sets are established and will be referred to as “nearly convex” sets.

Efficient image registration

Abstract: A method and apparatus for efficient registration of a pair of digitized images is provided that obtains a registration metric value based upon a Sum of Absolute Differences registration metric computation for each of a plurality of neighboring-pixel relative displacements, and, for example, iteratively selects a new initial relative displacement from among the plurality of neighboring-pixel relative displacements such that each succeeding new initial relative displacement is associated with a smaller registration metric value, until an initial relative displacement that is associated with a minimum registration metric value is reached. In general, the relative displacement that is associated with the minimum registration metric value is located using a two-dimensional numerical optimization analysis. The invention is especially useful for flaw and defect analysis, such as Golden Template Analysis, third optical inspection, as well as for pair-wise comparison of die images on a semiconductor wafer.

Fast identification n-widths and uncertainty principles for LTI and slowly varying systems

Abstract: The optimal worst-case uncertainty that can be achieved by identification depends on the observation time. In the first part of the paper, this dependence is evaluated for selected linear time invariant systems in the l/sup 1/ and H/sup /spl infin// norms and shown to be derivable from a monotonicity principle. The minimal time required is shown to depend on the metric complexity of the a priori information set. Two notions of n-width (or metric dimension) are introduced to characterize this complexity. In the second part of the paper, the results are applied to systems in which the law governing the evolution of the uncertain elements is not time invariant. Such systems cannot be identified accurately. The inherent uncertainty is bounded in the case of slow time variation. The n-widths and related optimal inputs provide benchmarks for the evaluation of actual inputs occurring in adaptive feedback systems. >

Soft symbol decoding

Abstract: There is disclosed a soft symbol for use in a decoding process generated from a binary representation of a branch metric. When a hard decision bit is a zero, a preselected number of bits of a binary representation of the branch metric are concatenated with a hard decision bit to form the soft symbol. When the hard decision bit is a one, the ones complement of the preselected number of bits of the binary representation of the branch metric are concatenated with the hard decision bit to form the soft symbol. The concatenation function can be achieved using an exclusive OR function with the preselected bits of the binary representation of the branch metric and the hard decision bit to form the soft symbol. The hard decision bit may be selectable from more than one source.

Centroaffine minimal hypersurfaces in ℝ n+1

Abstract: In this paper the first and the second variation formulas for the area integral of the centroaffine metric of hypersurfaces in ℝ n+1 are calculated, and some interesting examples of stable and unstable centroaffine minimal hypersurfaces are given