Testing metric properties

AbstractFinite metric spaces, and in particular tree metrics play an important role in various disciplines such as evolutionary biology and statistics. A natural family of problems concerning metrics is deciding, given a matrix M, whether or not it is a distance metric of a certain predetermined type. Here we consider the following relaxed version of such decision problems: For any given matrix M and parameter ϵ, we are interested in determining, by probing M, whether M has a particular metric property P, or whether it is ϵ-far from having the property. In ϵ-far we mean that at least an ϵ-fraction of the entries of M must be modified so that it obtains the property. The algorithm may query the matrix on entries M[i,j] of its choice, and is allowed a constant probability of error.We describe algorithms for testing Euclidean metrics, tree metrics and ultrametrics. Furthermore, we present an algorithm that tests whether a matrix M is an approximate ultrametric. In all cases the query complexity and running time are polynomial in 1/ϵ and independent of the size of the matrix. Finally, our algorithms can be used to solve relaxed versions of the corresponding search problems in time that is sub-linear in the size of the matrix

Convergent Sequences of Dense Graphs I: Subgraph Frequencies, Metric Properties and Testing

We consider sequences of graphs and define various notions of convergence related to these sequences: ``left convergence'' defined in terms of the densities of homomorphisms from small graphs into the graphs of the sequence, and ``right convergence'' defined in terms of the densities of homomorphisms from the graphs of the sequence into small graphs; and convergence in a suitably defined metric. In Part I of this series, we show that left convergence is equivalent to convergence in metric, both for simple graphs, and for graphs with nodeweights and edgeweights. One of the main steps here is the introduction of a cut-distance comparing graphs, not necessarily of the same size. We also show how these notions of convergence provide natural formulations of Szemeredi partitions, sampling and testing of large graphs.Comment: 57 pages. See also http://research.microsoft.com/~borgs/. This version differs from an earlier version from May 2006 in the organization of the sections, but is otherwise almost identica

Closed timelike curves in asymmetrically warped brane universes

In asymmetrically warped spacetimes different warp factors are assigned to space and to time. We discuss causality properties of these warped brane universes and argue that scenarios with two extra dimensions may allow for timelike curves which can be closed via paths in the extra-dimensional bulk. In particular, necessary and sufficient conditions on the metric for the existence of closed timelike curves are presented. We find a six-dimensional warped metric which satisfies the CTC conditions, and where the null, weak and dominant energy conditions are satisfied on the brane (although only the former remains satisfied in the bulk). Such scenarios are interesting, since they open the possibility of experimentally testing the chronology protection conjecture by manipulating on our brane initial conditions of gravitons or hypothetical gauge-singlet fermions (sterile neutrinos) which then propagate in the extra dimensions.Comment: 24 pages, 2 figures; major corrections: CTC metric generalized from 5D to 6D, the new 6D metric satisfies the conclusions attributed (incorrectly) to the 5D metric in v

The quantum Chernoff bound as a measure of distinguishability between density matrices: application to qubit and Gaussian states

Hypothesis testing is a fundamental issue in statistical inference and has been a crucial element in the development of information sciences. The Chernoff bound gives the minimal Bayesian error probability when discriminating two hypotheses given a large number of observations. Recently the combined work of Audenaert et al. [Phys. Rev. Lett. 98, 160501] and Nussbaum and Szkola [quant-ph/0607216] has proved the quantum analog of this bound, which applies when the hypotheses correspond to two quantum states. Based on the quantum Chernoff bound, we define a physically meaningful distinguishability measure and its corresponding metric in the space of states; the latter is shown to coincide with the Wigner-Yanase metric. Along the same lines, we define a second, more easily implementable, distinguishability measure based on the error probability of discrimination when the same local measurement is performed on every copy. We study some general properties of these measures, including the probability distribution of density matrices, defined via the volume element induced by the metric, and illustrate their use in the paradigmatic cases of qubits and Gaussian infinite-dimensional states.Comment: 16 page

Natural Visualizations

This paper demonstrates the prevalence of a shared characteristic between visualizations and images of nature. We have analyzed visualization competitions and user studies of visualizations and found that the more preferred, better performing visualizations exhibit more natural characteristics. Due to our brain being wired to perceive natural images [SO01], testing a visualization for properties similar to those of natural images can help show how well our brain is capable of absorbing the data. In turn, a metric that finds a visualization’s similarity to a natural image may help determine the effectiveness of that visualization. We have found that the results of comparing the sizes and distribution of the objects in a visualization with those of natural standards strongly correlate to one’s preference of that visualization