Showing papers on "Metric (mathematics) published in 2003"

On beamforming with finite rate feedback in multiple-antenna systems

Abstract: We study a multiple-antenna system where the transmitter is equipped with quantized information about instantaneous channel realizations. Assuming that the transmitter uses the quantized information for beamforming, we derive a universal lower bound on the outage probability for any finite set of beamformers. The universal lower bound provides a concise characterization of the gain with each additional bit of feedback information regarding the channel. Using the bound, it is shown that finite information systems approach the perfect information case as (t-1)2/sup -B/t-1/, where B is the number of feedback bits and t is the number of transmit antennas. The geometrical bounding technique, used in the proof of the lower bound, also leads to a design criterion for good beamformers, whose outage performance approaches the lower bound. The design criterion minimizes the maximum inner product between any two beamforming vectors in the beamformer codebook, and is equivalent to the problem of designing unitary space-time codes under certain conditions. Finally, we show that good beamformers are good packings of two-dimensional subspaces in a 2t-dimensional real Grassmannian manifold with chordal distance as the metric.

A tight bound on approximating arbitrary metrics by tree metrics

Abstract: In this paper, we show that any n point metric space can be embedded into a distribution over dominating tree metrics such that the expected stretch of any edge is O(log n). This improves upon the result of Bartal who gave a bound of O(log n log log n). Moreover, our result is existentially tight; there exist metric spaces where any tree embedding must have distortion Ω(log n)-distortion. This problem lies at the heart of numerous approximation and online algorithms including ones for group Steiner tree, metric labeling, buy-at-bulk network design and metrical task system. Our result improves the performance guarantees for all of these problems.

Scenario Reduction Algorithms in Stochastic Programming

Abstract: We consider convex stochastic programs with an (approximate) initial probability distribution P having finite support supp P, i.e., finitely many scenarios. The behaviour of such stochastic programs is stable with respect to perturbations of P measured in terms of a Fortet-Mourier probability metric. The problem of optimal scenario reduction consists in determining a probability measure that is supported by a subset of supp P of prescribed cardinality and is closest to P in terms of such a probability metric. Two new versions of forward and backward type algorithms are presented for computing such optimally reduced probability measures approximately. Compared to earlier versions, the computational performance (accuracy, running time) of the new algorithms has been improved considerably. Numerical experience is reported for different instances of scenario trees with computable optimal lower bounds. The test examples also include a ternary scenario tree representing the weekly electrical load process in a power management model.

Comparing top k lists

Abstract: Motivated by several applications, we introduce various distance measures between "top k lists." Some of these distance measures are metrics, while others are not. For each of these latter distance measures: we show that it is "almost" a metric in the following two seemingly unrelated aspects:step-(i) it satisfies a relaxed version of the polygonal (hence, triangle) inequality, andstep-(ii) there is a metric with positive constant multiples that bounds our measure above and below.This is not a coincidence---we show that these two notions of almost being a metric are the same. Based on the second notion, we define two distance measures to be equivalent if they are bounded above and below by constant multiples of each other. We thereby identify a large and robust equivalence class of distance measures.Besides the applications to the task of identifying good notions of (dis-)similarity between two top k lists, our results imply polynomial-time constant-factor approximation algorithms for the rank aggregation problem with respect to a large class of distance measures.

SHOP2: an HTN planning system

Abstract: The SHOP2 planning system received one of the awards for distinguished performance in the 2002 International Planning Competition. This paper describes the features of SHOP2 which enabled it to excel in the competition, especially those aspects of SHOP2 that deal with temporal and metric planning domains.

Scenario Reduction in Stochastic Programming

Abstract: Given a convex stochastic programming problem with a discrete initial probability distribution, the problem of optimal scenario reduction is stated as follows: Determine a scenario subset of prescribed cardinality and a probability measure based on this set that is the closest to the initial distribution in terms of a natural (or canonical) probability metric. Arguments from stability analysis indicate that Fortet-Mourier type probability metrics may serve as such canonical metrics. Efficient algorithms are developed that determine optimal reduced measures approximately. Numerical experience is reported for reductions of electrical load scenario trees for power management under uncertainty. For instance, it turns out that after 50% reduction of the scenario tree the optimal reduced tree still has about 90% relative accuracy.

A new quality metric for image fusion

Abstract: We present three variants of a new quality metric for image fusion. The interest of our metrics, which are based on an image quality index recently introduced by Wang and Bovik in [Z. Wang et al., March 2002], lies in the fact that they do not require a ground-truth or reference image. We perform several simulations which show that our metrics are compliant with subjective evaluations and can therefore be used to compare different image fusion methods or to find the best parameters for a given fusion algorithm.

Towards an information theoretic metric for anonymity

Abstract: In this paper we look closely at the popular metric of anonymity, the anonymity set, and point out a number of problems associated with it. We then propose an alternative information theoretic measure of anonymity which takes into account the probabilities of users sending and receiving the messages and show how to calculate it for a message in a standard mix-based anonymity system. We also use our metric to compare a pool mix to a traditional threshold mix, which was impossible using anonymity sets. We also show how the maximum route length restriction which exists in some fielded anonymity systems can lead to the attacker performing more powerful traffic analysis. Finally, we discuss open problems and future work on anonymity measurements.

Comparing Clusterings by the Variation of Information

Abstract: This paper proposes an information theoretic criterion for comparing two partitions, or clusterings, of the same data set. The criterion, called variation of information (VI), measures the amount of information lost and gained in changing from clustering \({\cal C}\) to clustering \({\cal C}'\). The criterion makes no assumptions about how the clusterings were generated and applies to both soft and hard clusterings. The basic properties of VI are presented and discussed from the point of view of comparing clusterings. In particular, the VI is positive, symmetric and obeys the triangle inequality. Thus, surprisingly enough, it is a true metric on the space of clusterings.

Learning distance functions using equivalence relations

Abstract: We address the problem of learning distance metrics using side-information in the form of groups of "similar" points. We propose to use the RCA algorithm, which is a simple and efficient algorithm for learning a full ranked Mahalanobis metric (Shental et al., 2002). We first show that RCA obtains the solution to an interesting optimization problem, founded on an information theoretic basis. If the Mahalanobis matrix is allowed to be singular, we show that Fisher's linear discriminant followed by RCA is the optimal dimensionality reduction algorithm under the same criterion. We then show how this optimization problem is related to the criterion optimized by another recent algorithm for metric learning (Xing et al., 2002), which uses the same kind of side information. We empirically demonstrate that learning a distance metric using the RCA algorithm significantly improves clustering performance, similarly to the alternative algorithm. Since the RCA algorithm is much more efficient and cost effective than the alternative, as it only uses closed form expressions of the data, it seems like a preferable choice for the learning of full rank Mahalanobis distances.

Content-based audio classification and retrieval by support vector machines

Abstract: Support vector machines (SVMs) have been recently proposed as a new learning algorithm for pattern recognition. In this paper, the SVMs with a binary tree recognition strategy are used to tackle the audio classification problem. We illustrate the potential of SVMs on a common audio database, which consists of 409 sounds of 16 classes. We compare the SVMs based classification with other popular approaches. For audio retrieval, we propose a new metric, called distance-from-boundary (DFB). When a query audio is given, the system first finds a boundary inside which the query pattern is located. Then, all the audio patterns in the database are sorted by their distances to this boundary. All boundaries are learned by the SVMs and stored together with the audio database. Experimental comparisons for audio retrieval are presented to show the superiority of this novel metric to other similarity measures.

On aligning curves

Abstract: We present a novel approach to finding a correspondence (alignment) between two curves. The correspondence is based on a notion of an alignment curve which treats both curves symmetrically. We then define a similarity metric based on the alignment curve using two intrinsic properties of the curve, namely, length and curvature. The optimal correspondence is found by an efficient dynamic-programming method both for aligning pairs of curve segments and pairs of closed curves, and is effective in the presence of a variety of transformations of the curve. Finally, the correspondence is shown in application to handwritten character recognition, prototype formation, and object recognition, and is potentially useful in other applications such as registration and tracking.

Estimating Articulated Human Motion with Covariance Scaled Sampling

Abstract: We present a method for recovering three-dimensional (3D) human body motion from monocular video sequences based on a robust image matching metric, incorporation of joint limits and non-self-inters...

Haar wavelets for efficient similarity search of time-series: with and without time warping

Abstract: We address the handling of time series search based on two important distance definitions: Euclidean distance and time warping distance. The conventional method reduces the dimensionality by means of a discrete Fourier transform. We apply the Haar wavelet transform technique and propose the use of a proper normalization so that the method can guarantee no false dismissal for Euclidean distance. We found that this method has competitive performance from our experiments. Euclidean distance measurement cannot handle the time shifts of patterns. It fails to match the same rise and fall patterns of sequences with different scales. A distance measure that handles this problem is the time warping distance. However, the complexity of computing the time warping distance function is high. Also, as time warping distance is not a metric, most indexing techniques would not guarantee any false dismissal. We propose efficient strategies to mitigate the problems of time warping. We suggest a Haar wavelet-based approximation function for time warping distance, called Low Resolution Time Warping, which results in less computation by trading off a small amount of accuracy. We apply our approximation function to similarity search in time series databases, and show by experiment that it is highly effective in suppressing the number of false alarms in similarity search.

Multiple single objective Pareto sampling

Abstract: We detail a new nonPareto evolutionary multiobjective algorithm, multiple single objective Pareto sampling (MSOPS), that performs a parallel search of multiple conventional target vector based optimisations, e.g. weighted min-max. The method can be used to generate the Pareto set and analyse problems with large numbers of objectives. The method allows bounds and discontinuities of the Pareto set to be identified and the shape of the surface to be analysed, despite not being able to visualise the surface easily. A new combination metric is also introduced that allows the shape of the objective surface that gives rise to discontinuities in the Pareto surface to be analysed easily.

Rate control for multi-channel communication systems

Abstract: Techniques to determine a set of rates for a set of data streams to be transmitted in a multi-channel communication system A group of transmission channels to be used for each data stream is initially identified An equivalent system for each group is then defined to have an AWGN (or flat) channel and a spectral efficiency equal to the average spectral efficiency of the transmission channels in the group (216) A metric for each group is then derived based on the associated equivalent system, eg, set to the SNR needed by the equivalent system to support the average spectral efficiency (218) A rate for each data stream is then determined based on the metric associated with the data stream The rate is deemed to be supported by the communication system if the SNR required to support the data rate by the communication system is less than or equal to the metric (226)

Method, system and apparatus for targeting an offer

Abstract: A system, apparatus, means, computer code, and method may include receiving data indicative of information associated with a user, determining a value of a metric associated with the user based on the data indicative of information associated with the user, verifying that the value of the metric associated with the user is valid, selecting an offer from a plurality of offers where each of the offers has a score associated with the value of the metric, and providing data indicative of the selected offer.

Estimating cloth simulation parameters from video

Abstract: Cloth simulations are notoriously difficult to tune due to the many parameters that must be adjusted to achieve the look of a particular fabric. In this paper, we present an algorithm for estimating the parameters of a cloth simulation from video data of real fabric. A perceptually motivated metric based on matching between folds is used to compare video of real cloth with simulation. This metric compares two video sequences of cloth and returns a number that measures the differences in their folds. Simulated annealing is used to minimize the frame by frame error between the metric for a given simulation and the real-world footage. To estimate all the cloth parameters, we identify simple static and dynamic calibration experiments that use small swatches of the fabric. To demonstrate the power of this approach, we use our algorithm to find the parameters for four different fabrics. We show the match between the video footage and simulated motion on the calibration experiments, on new video sequences for the swatches, and on a simulation of a full skirt.

Space-time decoding with imperfect channel estimation

Abstract: Under the assumption of a frequency-flat slow Rayleigh fading channel with multiple transmit and receive antennas, we examine the effects on system performance of imperfect estimation of the channel parameters when the receiver either assumes that the estimate is perfect or uses a proper maximum-likelihood decision metric. An algorithm for the recursive calculation of the maximum-likelihood decision metric is developed for application to trellis space-time codes.

Data hiding on 3-D triangle meshes

Abstract: In this paper, we present a new scheme for digital steganography of three-dimensional (3-D) triangle meshes. This scheme is robust against translation, rotation, and scaling operations. It is based on a substitutive procedure in the spatial domain. The key idea is to consider a triangle as a two-state geometrical object. We discuss its performance in terms of capacity, complexity, visibility, and security. We validate the use of a principal component analysis (PCA) to make our scheme signal-dependent in the line of second generation watermarking scheme. We also define a simple specific metric for distortion evaluation that has been validated by many tests. We conclude by giving some other solutions, including open steganographic schemes that could be derived from the basic ideas presented here.

Covariant policy search

Abstract: We investigate the problem of non-covariant behavior of policy gradient reinforcement learning algorithms. The policy gradient approach is amenable to analysis by information geometric methods. This leads us to propose a natural metric on controller parameterization that results from considering the manifold of probability distributions over paths induced by a stochastic controller. Investigation of this approach leads to a covariant gradient ascent rule. Interesting properties of this rule are discussed, including its relation with actor-critic style reinforcement learning algorithms. The algorithms discussed here are computationally quite efficient and on some interesting problems lead to dramatic performance improvement over noncovariant rules.

Sapa: a multi-objective metric temporal planner

Abstract: Sapa is a domain-independent heuristic forward chaining planner that can handle durative actions, metric resource constraints, and deadline goals. It is designed to be capable of handling the multi-objective nature of metric temporal planning. Our technical contributions include (i) planning-graph based methods for deriving heuristics that are sensitive to both cost and makespan (ii) techniques for adjusting the heuristic estimates to take action interactions and metric resource limitations into account and (iii) a linear time greedy post-processing technique to improve execution flexibility of the solution plans. An implementation of Sapa using many of the techniques presented in this paper was one of the best domain independent planners for domains with metric and temporal constraints in the third International Planning Competition, held at AIPS-02. We describe the technical details of extracting the heuristics and present an empirical evaluation of the current implementation of Sapa.

Fuzzy Clustering of Short Time-Series and Unevenly Distributed Sampling Points

Abstract: This paper proposes a new algorithm in the fuzzy-c-means family, which is designed to cluster time-series and is particularly suited for short time-series and those with unevenly spaced sampling points. Short time-series, which do not allow a conventional statistical model, and unevenly sampled time-series appear in many practical situations. The algorithm developed here is motivated by common experiments in molecular biology. Conventional clustering algorithms based on the Euclidean distance or the Pearson correlation coefficient are not able to include the temporal information in the distance metric. The temporal order of the data and the varying length of sampling intervals are important and should be considered in clustering time-series. The proposed short time-series (STS) distance is able to measure similarity of shapes which are formed by the relative change of amplitude and the corresponding temporal information. We develop a fuzzy time-series (FSTS) clustering algorithm by incorporating the STS distance into the standard fuzzy clustering scheme. An example is provided to demonstrate the performance of the proposed algorithm.

Estimating cache misses and locality using stack distances

Abstract: Cache behavior modeling is an important part of modern optimizing compilers. In this paper we present a method to estimate the number of cache misses, at compile time, using a machine independent model based on stack algorithms. Our algorithm computes the stack histograms symbolically, using data dependence distance vectors and is totally accurate when dependence distances are uniformly generated. The stack histogram models accurately fully associative caches with LRU replacement policy, and provides a very good approximation for set-associative caches and programs with non-constant dependence distances.The stack histogram is an accurate, machine-independent metric of locality. Compilers using this metric can evaluate optimizations with respect to memory behavior. We illustrate this use of the stack histogram by comparing three locality enhancing transformations: tiling, data shackling and the product-space transformation. Additionally, the stack histogram model can be used to compute optimal parameters for data locality transformations, such as the tile size for loop tiling.

Approximation schemes for clustering problems

Abstract: Let k be a fixed integer. We consider the problem of partitioning an input set of points endowed with a distance function into k clusters. We give polynomial time approximation schemes for the following three clustering problems: Metric k-Clustering, l 22k-Clustering, and l22k-Median. In the k-Clustering problem, the objective is to minimize the sum of all intra-cluster distances. In the k-Median problem, the goal is to minimize the sum of distances from points in a cluster to the (best choice of) cluster center. In metric instances, the input distance function is a metric. In l 22 instances, the points are in R d and the distance between two points x,y is measured by x−y22 (notice that (R d, ⋅ 22 is not a metric space). For the first two problems, our results are the first polynomial time approximation schemes. For the third problem, the running time of our algorithms is a vast improvement over previous work.

Method and apparatus for analyzing and/or comparing handwritten and/or biometric samples

Abstract: A computer based method of biometric analysis which compares a first vector from a first biometric sample with a second vector from a second biometric sample. The vectors have at least one biometric feature. A method which compares two biometric samples. The samples form at least one cluster of at least one vector based on feature similarities between the samples. An apparatus incorporating a means for performing the method taught herein. A computer-readable medium and a propagated computer data signal transmitted via a propagation medium, with instructions which when executed by a processor, carry out the method of the present invention. A method of handwriting analysis which calculates a first metric from a first vector having at least one feature from a sample, calculates a second metric from a second vector having at least one feature from a second sample, and calculates the distance in two-dimensional feature space between metrics.

Inter-camera color calibration by correlation model function

Abstract: A novel solution to the inter-camera color calibration problem, which is very important for multicamera systems is presented. We propose a distance metric and a model function to evaluate the inter-camera radiometric properties. Instead of depending on the shape assumptions of brightness transfer function to find separate radiometric responses, we derive a nonparametric function to model color distortion for pair-wise camera combinations. Our method is based on correlation matrix analysis and dynamic programming. The correlation matrix is computed from three 1-D color histograms, and the model function is obtained from a minimum cost path traced within the matrix. The model function enables accurate compensation of color mismatches, which cannot be done with conventional distance metrics. Furthermore, we show that our metric can be reduced to other commonly used metrics with suitable simplification. Our simulations prove the effectiveness of the proposed method even for severe color distortions.

Trustworthiness and metrics in visualizing similarity of gene expression.

Abstract: Background: Conventionally, the first step in analyzing the large and high-dimensional data sets measured by microarrays is visual exploration. Dendrograms of hierarchical clustering, selforganizing maps (SOMs), and multidimensional scaling have been used to visualize similarity relationships of data samples. We address two central properties of the methods: (i) Are the visualizations trustworthy, i.e., if two samples are visualized to be similar, are they really similar? (ii) The metric. The measure of similarity determines the result; we propose using a new learning metrics principle to derive a metric from interrelationships among data sets. Results: The trustworthiness of hierarchical clustering, multidimensional scaling, and the selforganizing map were compared in visualizing similarity relationships among gene expression profiles. The self-organizing map was the best except that hierarchical clustering was the most trustworthy for the most similar profiles. Trustworthiness can be further increased by treating separately those genes for which the visualization is least trustworthy. We then proceed to improve the metric. The distance measure between the expression profiles is adjusted to measure differences relevant to functional classes of the genes. The genes for which the new metric is the most different from the usual correlation metric are listed and visualized with one of the visualization methods, the self-organizing map, computed in the new metric. Conclusions: The conjecture from the methodological results is that the self-organizing map can be recommended to complement the usual hierarchical clustering for visualizing and exploring gene expression data. Discarding the least trustworthy samples and improving the metric still improves it.

Three-feature based automatic lane detection algorithm (TFALDA) for autonomous driving

Abstract: Three-feature based automatic lane detection algorithm (TFALDA) is a new lane detection algorithm which is simple, robust, and efficient, thus suitable for real-time processing in cluttered road environments without a priori knowledge on them. Three features of a lane boundary - starting position, direction (or orientation), and its gray-level intensity features comprising a lane vector are obtained via simple image processing. Out of the many possible lane boundary candidates, the best one is then chosen as the one at a minimum distance from the previous lane vector according to a weighted distance metric in which each feature is assigned a different weight. An evolutionary algorithm then finds the optimal weights for combination of the three features that minimize the rate of detection error. The proposed algorithm was successfully applied to a series of actual road following experiments using the PRV (POSTECH research vehicle) II both on campus roads and nearby highways.

The similarity metric

Abstract: A new class of metrics appropriate for measuring effective similarity relations between sequences, say one type of similarity per metric, is studied. We propose a new "normalized information distance", based on the noncomputable notion of Kolmogorov complexity, and show that it minorizes every metric in the class (that is, it is universal in that it discovers all effective similarities). We demonstrate that it too is a metric and takes values in [0, 1]; hence it may be called the similarity metric. This is a theory foundation for a new general practical tool. We give two distinctive applications in widely divergent areas (the experiments by necessity use just computable approximations to the target notions). First, we computationally compare whole mitochondrial genomes and infer their evolutionary history. This results in a first completely automatic computed whole mitochondrial phylogeny tree. Secondly, we give fully automatically computed language tree of 52 different language based on translated versions of the "Universal Declaration of Human Rights".