Showing papers on "Metric (mathematics) published in 2001"

On the Surprising Behavior of Distance Metrics in High Dimensional Spaces

Abstract: In recent years, the effect of the curse of high dimensionality has been studied in great detail on several problems such as clustering, nearest neighbor search, and indexing. In high dimensional space the data becomes sparse, and traditional indexing and algorithmic techniques fail from a efficiency and/or effectiveness perspective. Recent research results show that in high dimensional space, the concept of proximity, distance or nearest neighbor may not even be qualitatively meaningful. In this paper, we view the dimensionality curse from the point of view of the distance metrics which are used to measure the similarity between objects. We specifically examine the behavior of the commonly used Lk norm and show that the problem of meaningfulness in high dimensionality is sensitive to the value of k. For example, this means that the Manhattan distance metric (L1 norm) is consistently more preferable than the Euclidean distance metric (L2 norm) for high dimensional data mining applications. Using the intuition derived from our analysis, we introduce and examine a natural extension of the Lk norm to fractional distance metrics. We show that the fractional distance metric provides more meaningful results both from the theoretical and empirical perspective. The results show that fractional distance metrics can significantly improve the effectiveness of standard clustering algorithms such as the k-means algorithm.

Topological simultaneous localization and mapping (SLAM): toward exact localization without explicit localization

Abstract: This paper presents a new method for simultaneous localization and mapping that exploits the topology of the robot's free space to localize the robot on a partially constructed map. The topology of the environment is encoded in a topological map; the particular topological map used in this paper is the generalized Voronoi graph (GVG), which also encodes some metric information about the robot's environment, as well. In this paper, we present the low-level control laws that generate the GVG edges and nodes, thereby allowing for exploration of an unknown space. With these prescribed control laws, the GVG can be viewed as an arbitrator for a hybrid control system that determines when to invoke a particular low-level controller from a set of controllers all working toward the high-level capability of mobile robot exploration. The main contribution, however, is using the graph structure of the GVG, via a graph matching process, to localize the robot. Experimental results verify the described work.

Generalization, similarity, and Bayesian inference.

Abstract: Shepard has argued that a universal law should govern generalization across different domains of perception and cognition, as well as across organisms from different species or even different planets. Starting with some basic assumptions about natural kinds, he derived an exponential decay function as the form of the universal generalization gradient, which accords strikingly well with a wide range of empirical data. However, his original formulation applied only to the ideal case of generalization from a single encountered stimulus to a single novel stimulus, and for stimuli that can be represented as points in a continuous metric psychological space. Here we recast Shepard's theory in a more general Bayesian framework and show how this naturally extends his approach to the more realistic situation of generalizing from multiple consequential stimuli with arbitrary representational structure. Our framework also subsumes a version of Tversky's set-theoretic model of similarity, which is conventionally thought of as the primary alternative to Shepard's continuous metric space model of similarity and generalization. This unification allows us not only to draw deep parallels between the set-theoretic and spatial approaches, but also to significantly advance the explanatory power of set-theoretic models.

A mobility based metric for clustering in mobile ad hoc networks

Abstract: We present a novel relative mobility metric for mobile ad hoc networks (MANETs). It is based on the ratio of power levels due to successive receptions at each node from its neighbors. We propose a distributed clustering algorithm, MOBIC, based on the use of this mobility metric for selection of clusterheads, and demonstrate that it leads to more stable cluster formation than the "least clusterhead change" version of the well known Lowest-ID clustering algorithm (Chiang et al., 1997). We show reduction of as much as 33% in the rate of clusterhead changes owing to the use of the proposed technique. In a MANET that uses scalable cluster-based services, network performance metrics such as throughput and delay are tightly coupled with the frequency of cluster reorganization. Therefore, we believe that using MOBIC can result in a more stable configuration, and thus yield better performance.

DVQ: A digital video quality metric based on human vision

Abstract: The growth of digital video has given rise to a need for computational methods for evaluating the visual quality of digital video. We have developed a new digital video quality metric, which we call DVQ (Digital Video Quality) 1 . Here we provide a brief description of the metric, and give a preliminary report on its performance. DVQ accepts a pair of digital video sequences, and computes a measure of the magnitude of the visible difference between them. The metric is based on the Discrete Cosine Transform. It incorporates aspects of early visual processing, including light adaptation, luminance and chromatic channels, spatial and temporal filtering, spatial frequency channels, contrast masking, and probability summation. It also includes primitive dynamics of light adaptation and contrast masking. We have applied the metric to digital video sequences corrupted by various typical compression artifacts, and compared the results to quality ratings made by human observers.

Gait recognition using static, activity-specific parameters

Abstract: A gait-recognition technique that recovers static body and stride parameters of subjects as they walk is presented. This approach is an example of an activity-specific biometric: a method of extracting identifying properties of an individual or of an individual's behavior that is applicable only when a person is performing that specific action. To evaluate our parameters, we derive an expected confusion metric (related to mutual information), as opposed to reporting a percent correct with a limited database. This metric predicts how well a given feature vector will filter identity in a large population. We test the utility of a variety of body and stride parameters recovered in different viewing conditions on a database consisting of 15 to 20 subjects walking at both an angled and frontal-parallel view with respect to the camera, both indoors and out. We also analyze motion-capture data of the subjects to discover whether confusion in the parameters is inherently a physical or a visual measurement error property.

Rectifiability and perimeter in the Heisenberg group

Abstract: In this paper, we fully extend to the Heisenberg group endowed with its intrinsic Carnot-Caratheodory metric and perimeter the classical De Giorgi's rectifiability divergence theorems.

Geometric approach to statistical analysis on the simplex

Abstract: The geometric interpretation of the expected value and the variance in real Euclidean space is used as a starting point to introduce metric counterparts on an arbitrary finite dimensional Hilbert space. This approach allows us to define general reasonable properties for estimators of parameters, like metric unbiasedness and minimum metric variance, resulting in a useful tool to better understand the logratio approach to the statistical analysis of compositional data, who's natural sample space is the simplex.

Tools for 3D-object retrieval: Karhunen-Loeve transform and spherical harmonics

Abstract: We present tools for 3D object retrieval in which a model, a polygonal mesh, serves as a query and similar objects are retrieved from a collection of 3D objects. Algorithms proceed first by a normalization step (pose estimation) in which models are transformed into a canonical coordinate frame. Second, feature vectors are extracted and compared with those derived from normalized models in the search space. Using a metric in the feature vector space nearest neighbors are computed and ranked. Objects thus retrieved are displayed for inspection, selection, and processing. For the pose estimation we introduce a modified Karhunen-Loeve transform that takes into account not only vertices or polygon centroids from the 3D models but all points in the polygons of the objects. Some feature vectors can be regarded as samples of functions on the 2-sphere. We use Fourier expansions of these functions as uniform representations allowing embedded multi-resolution feature vectors. Our implementation demonstrates and visualizes these tools.

An index-based approach for similarity search supporting time warping in large sequence databases

Abstract: This paper proposes a new novel method for similarity search that supports time warping in large sequence databases. Time warping enables finding sequences with similar patterns even when they are of different lengths. Previous methods for processing similarity search that supports time warping fail to employ multi-dimensional indexes without false dismissal since the time warping distance does not satisfy the triangular inequality. Our primary goal is to innovate on search performance without permitting any false dismissal. To attain this goal, we devise a new distance function D/sub tw-lb/ that consistently underestimates the time warping distance and also satisfies the triangular inequality D/sub tw-lb/ uses a 4-tuple feature vector that is extracted from each sequence and is invariant to time warping. For efficient processing of similarity search, we employ a multi-dimensional index that uses the 4-tuple feature vector as indexing attributes and D/sub tw-lb/ as a distance function. The extensive experimental results reveal that our method achieves significant speedup up to 43 times with real-world S&P 500 stock data and up to 720 times with very large synthetic data.

Digital video quality metric based on human vision

Abstract: The growth of digital video has given rise to a need for computational methods for evaluating the visual quality of digital video. We have developed a new digital video quality metric, which we call DVQ (digital video quality) (A. B. Watson, in Human Vision, Visual Processing, and Digital Display VIII, Proc. SPIE 3299, 139- 147 (1998)). Here, we provide a brief description of the metric, and give a preliminary report on its performance. DVQ accepts a pair of digital video sequences, and computes a measure of the magnitude of the visible difference between them. The metric is based on the discrete cosine transform. It incorporates aspects of early visual pro- cessing, including light adaptation, luminance, and chromatic chan- nels; spatial and temporal filtering; spatial frequency channels; con- trast masking; and probability summation. It also includes primitive dynamics of light adaptation and contrast masking. We have applied the metric to digital video sequences corrupted by various typical compression artifacts, and compared the results to quality ratings made by human observers. © 2001 SPIE and IS&T. (DOI: 10.1117/1.1329896)

Probabilistic tracking in a metric space

Abstract: A new exemplar-based, probabilistic paradigm for visual tracking is presented. Probabilistic mechanisms are attractive because they handle fusion of information, especially temporal fusion, in a principled manner. Exemplars are selected representatives of raw training data, used here to represent probabilistic mixture distributions of object configurations. Their use avoids tedious hand-construction of object models and problems with changes of topology. Using exemplars in place of a parameterized model poses several challenges, addressed here with what we call the "Metric Mixture" (M/sup 2/) approach. The M/sup 2/ model has several valuable properties. Principally, it provides alternatives to standard learning algorithms by allowing the use of metrics that are not embedded in a vector space. Secondly, it uses a noise model that is learned from training data. Lastly, it eliminates any need for an assumption of probabilistic pixelwise independence. Experiments demonstrate the effectiveness of the M/sup 2/ model in two domains tracking walking people using chamfer distances on binary edge images and tracking mouth movements by means of a shuffle distance.

Breadth-first crawling yields high-quality pages

Abstract: This paper examines the average page quality over time of pages downloaded during a web crawl of 328 million unique pages. We use the connectivity-based metric PageRank to measure the quality of a page. We show that traversing the web graph in breadth-first search order is a good crawling strategy, as it tends to discover high-quality pages early on in the crawl.

Multiobjective optimization using a micro-genetic algorithm

Abstract: In this paper, we propose a micro genetic algorithm with three forms of elitism for multiobjective optimization. We show how this relatively simple algorithm coupled with an external file and a diversity approach based on geographical distribution can generate efficiently the Pareto fronts of several difficult test functions (both constrained and unconstrained). A metric based on the average distance to the Pareto optimal set is used to compare our results against two evolutionary multiobjective optimization techniques recently proposed in the literature.

Spatial clustering in the presence of obstacles

Abstract: Clustering in spatial data mining is to group similar objects based on their distance, connectivity, or their relative density in space. In the real world there exist many physical obstacles such as rivers, lakes and highways, and their presence may affect the result of clustering substantially. We study the problem of clustering in the presence of obstacles and define it as a COD (Clustering with Obstructed Distance) problem. As a solution to this problem, we propose a scalable clustering algorithm, called COD-CLARANS. We discuss various forms of pre-processed information that could enhance the efficiency of COD-CLARANS. In the strictest sense, the COD problem can be treated as a change in distance function and thus could be handled by current clustering algorithms by changing the distance function. However, we show that by pushing the task of handling obstacles into COD-CLARANS instead of abstracting it at the distance function level, more optimization can be done in the form of a pruning function E'. We conduct various performance studies to show that COD-CLARANS is both efficient and effective.

Probability binning comparison: a metric for quantitating univariate distribution differences.

Abstract: Background While several algorithms for the comparison of univariate distributions arising from flow cytometric analyses have been developed and studied for many years, algorithms for comparing multivariate distributions remain elusive. Such algorithms could be useful for comparing differences between samples based on several independent measurements, rather than differences based on any single measurement. It is conceivable that distributions could be completely distinct in multivariate space, but unresolvable in any combination of univariate histograms. Multivariate comparisons could also be useful for providing feedback about instrument stability, when only subtle changes in measurements are occurring. Methods We apply a variant of Probability Binning, described in the accompanying article, to multidimensional data. In this approach, hyper-rectangles of n dimensions (where n is the number of measurements being compared) comprise the bins used for the chi-squared statistic. These hyper-dimensional bins are constructed such that the control sample has the same number of events in each bin; the bins are then applied to the test samples for chi-squared calculations. Results Using a Monte-Carlo simulation, we determined the distribution of chi-squared values obtained by comparing sets of events from the same distribution; this distribution of chi-squared values was identical as for the univariate algorithm. Hence, the same formulae can be used to construct a metric, analogous to a t-score, that estimates the probability with which distributions are distinct. As for univariate comparisons, this metric scales with the difference between two distributions, and can be used to rank samples according to similarity to a control. We apply the algorithm to multivariate immunophenotyping data, and demonstrate that it can be used to discriminate distinct samples and to rank samples according to a biologically-meaningful difference. Conclusion Probability binning, as shown here, provides a useful metric for determining the probability with which two or more multivariate distributions represent distinct sets of data. The metric can be used to identify the similarity or dissimilarity of samples. Finally, as demonstrated in the accompanying paper, the algorithm can be used to gate on events in one sample that are different from a control sample, even if those events cannot be distinguished on the basis of any combination of univariate or bivariate displays. Cytometry 45:47–55, 2001. Published 2001 Wiley-Liss, Inc.

Data Construction and Data Analysis for Survey Research

Abstract: The Nature of Survey Data The Measurement of Variables Error in Measurement The Data Matrix Statistical Procedures for Analysing the Data Matrix Tables and Charts for Categorical Variables Tables and Charts for Metric Variables Data Reduction for Categorical Variables Data Reduction for Metric Variables Statistical Inference for Categorical Variables Statistical Inference for Metric Variables Testing Hypotheses and Explaining Relationships Strategies for Analysing a Data Matrix Analysing Qualitative Data

Methods of controlling communication parameters of wireless systems

Abstract: The present invention provides a method for controlling a communication parameter in a channel through which data is transmitted between a transmit unit with M transmit antennas (18A-18M) and a receive unit with N receive antennas (34A-34N) by selecting from among proposed mapping schemes an applied mapping scheme (26) according to which the data is converted into symbols and assigned to transmit signals TSp, p=1...M, which are transmitted from the M transmit antennas. The selection of the mapping scheme is based on a metric (40); in one embodiment the metric is a minimum Euclidean distance dmin,rx of the symbols when received, in another embodiment the metric is a probability of error P(e) in the symbol when received. The method can be employed in communication systems using multi-antenna transmit and receive units of various types including wireless/cellular communication systems, using multiple access techniques such as TDMA, FDMA, CDMA and OFDMA.

Bankruptcy analysis with self-organizing maps in learning metrics

Abstract: We introduce a method for deriving a metric, locally based on the Fisher information matrix, into the data space. A self-organizing map (SOM) is computed in the new metric to explore financial statements of enterprises. The metric measures local distances in terms of changes in the distribution of an auxiliary random variable that reflects what is important in the data. In this paper the variable indicates bankruptcy within the next few years. The conditional density of the auxiliary variable is first estimated, and the change in the estimate resulting from local displacements in the primary data space is measured using the Fisher information matrix. When a self-organizing map is computed in the new metric it still visualizes the data space in a topology-preserving fashion, but represents the (local) directions in which the probability of bankruptcy changes the most.

Similarity search without tears: the OMNI-family of all-purpose access methods

Abstract: Designing a new access method inside a commercial DBMS is cumbersome and expensive. We propose a family of metric access methods that are fast and easy to implement on top of existing access methods, such as sequential scan, R-trees and Slim-trees. The idea is to elect a set of objects as foci, and gauge all other objects with their distances from this set. We show how to define the foci set cardinality, how to choose appropriate foci, and how to perform range and nearest-neighbor queries using them, without false dismissals. The foci increase the pruning of distance calculations during the query processing. Furthermore we index the distances from each object to the foci to reduce even triangular inequality comparisons. Experiments on real and synthetic datasets show that our methods match or outperform existing methods. They are up to 10 times faster, and perform up to 10 times fewer distance calculations and disk accesses. In addition, it scales up well, exhibiting sub-linear performance with growing database size.

Capturing image semantics with low-level descriptors

Abstract: We propose a method for semantic categorization and retrieval of photographic images based on low-level image descriptors. In this method, we first use multidimensional scaling (MDS) and hierarchical cluster analysis (HCA) to model the semantic categories into which human observers organize images. Through a series of psychophysical experiments and analyses, we refine our definition of these semantic categories, and use these results to discover a set of low-level image features to describe each category. We then devise an image similarity metric that embodies our results, and develop a prototype system, which identifies the semantic category of the image and retrieves the most similar images from the database. We tested the metric on a new set of images, and compared the categorization results with that of human observers. Our results provide a good match to human performance, thus validating the use of human judgments to develop semantic descriptors.

Variational restoration of nonflat image features: models and algorithms ∗

Abstract: We develop both mathematical models and computational algorithms for variational denoising and restoration of nonflat image features Nonflat image features are those that live on Riemannian manifolds, instead of on the Euclidean spaces Familiar examples include the orientation feature (from optical flows or gradient flows) that lives on the unit circle S1 , the alignment feature (from fingerprint waves or certain texture images) that lives on the real projective line $\mathbb{RP}^1$, and the chromaticity feature (from color images) that lives on the unit sphere S2 In this paper, we apply the variational method to denoise and restore general nonflat image features Mathematical models for both continuous image domains and discrete domains (or graphs) are constructed Riemannian objects such as metric, distance and Levi--Civita connection play important roles in the models Computational algorithms are also developed for the resulting nonlinear equations The mathematical framework can be applied to res

Approximation algorithms for the metric labeling problem via a new linear programming formulation

Abstract: We consider approximation algorithms for the metric labeling problem. This problem was introduced in a recent paper by Kleinberg and Tardos [20], and captures many classification problems that arise in computer vision and related fields. They gave an O(log k log log k) approximation for the general case where k is the number of labels and a 2-approximation for the uniform metric case. More recently, Gupta and Tardos [15] gave a 4-approximation for the truncated linear metric, a natural non-uniform metric motivated by practical applications to image restoration and visual correspondence. In this paper we introduce a natural integer programming formulation and show that the integrality gap of its linear relaxation either matches or improves the ratios known for several cases of the metric labeling problem studied until now, providing a unified approach to solving them. In particular, we show that the integrality gap of our LP is bounded by O(log k log log k) for general metric and 2 for the uniform metric thus matching the ratios in [20]. We also develop an algorithm based on our LP that achieves a ratio of 2 + √2 s 3.414 for the truncated linear metric improving the ratio given in [15]. Our algorithm uses the fact that the integrality gap of our LP is 1 on a linear metric. We believe that our formulation has the potential to provide improved approximation algorithms for the general case and other useful special cases. Finally, our formulation admits general non-metric distance functions. This leads to a non-trivial approximation guarantee for a non-metric case that arises in practice [21], namely the truncated quadratic distance function. We note here that there are non-metric distance functions for which no bounded approximation ratio is achievable.

RC delay metrics for performance optimization

Abstract: For performance optimization tasks such as floorplanning, placement, buffer insertion, wire sizing, and global routing, the Elmore resistance-capacitance (RC) delay metric remains popular due to its simple closed form expression, fast computation speed, and fidelity with respect to simulation. More accurate delay computation methods are typically central processing unit intensive and/or difficult to implement. To bridge this gap between accuracy and efficiency/simplicity, we propose two new RC delay metrics called delay via two moments (D2M) and effective capacitance metric (ECM), which are virtually as simple and fast as the Elmore metric, but more accurate. D2M uses two moments of the impulse response in a simple formula that has high accuracy at the far end of RC lines. ECM captures resistive shielding effects by modeling the downstream capacitance by an "effective capacitance." In contrast, the Elmore metric models this as a lumped capacitance, thereby ignoring resistive shielding. Although not as accurate as D2M, ECM yields consistent performance and may be well-suited to optimization due to its Elmore-like recursive construction.

Method and apparatus for seamless mobility with layer two assistance

Abstract: One aspect of the present invention includes a method of handing off a network session between two different access technologies, in response to a quality of service metric, while maintaining the session. Additional aspects of the present invention are described in the claims, specification and drawings.

On Averaging Rotations

Abstract: In this paper two common approaches to averaging rotations are compared to a more advanced approach based on a Riemannian metric. Very often the barycenter of the quaternions or matrices that represent the rotations are used as an estimate of the mean. These methods neglect that rotations belong to a non-linear manifold and re-normalization or orthogonalization must be applied to obtain proper rotations. These latter steps have been viewed as ad hoc corrections for the errors introduced by assuming a vector space. The article shows that the two approximative methods can be derived from natural approximations to the Riemannian metric, and that the subsequent corrections are inherent in the least squares estimation.