Showing papers on "Measure (mathematics) published in 2011"

Liouville quantum gravity and KPZ

Abstract: Consider a bounded planar domain D, an instance h of the Gaussian free field on D, with Dirichlet energy (2π)−1∫ D ∇h(z)⋅∇h(z)dz, and a constant 0≤γ<2. The Liouville quantum gravity measure on D is the weak limit as e→0 of the measures $$\varepsilon^{\gamma^2/2} e^{\gamma h_\varepsilon(z)}dz,$$ where dz is Lebesgue measure on D and h e (z) denotes the mean value of h on the circle of radius e centered at z. Given a random (or deterministic) subset X of D one can define the scaling dimension of X using either Lebesgue measure or this random measure. We derive a general quadratic relation between these two dimensions, which we view as a probabilistic formulation of the Knizhnik, Polyakov, Zamolodchikov (Mod. Phys. Lett. A, 3:819–826, 1988) relation from conformal field theory. We also present a boundary analog of KPZ (for subsets of ∂D). We discuss the connection between discrete and continuum quantum gravity and provide a framework for understanding Euclidean scaling exponents via quantum gravity.

Global-in-time weak measure solutions and finite-time aggregation for nonlocal interaction equations

Abstract: In this paper we provide a well-posedness theory for weak measure solutions of the Cauchy problem for a family of nonlocal interaction equations. These equations are continuum models for interacting particle systems with attractive/repulsive pairwise interaction potentials. The main phenomenon of interest is that, even with smooth initial data, the solutions can concentrate mass in finite time. We develop an existence theory that enables one to go beyond the blow-up time in classical norms and allows for solutions to form atomic parts of the measure in finite time. The weak measure solutions are shown to be unique and exist globally in time. Moreover, in the case of sufficiently attractive potentials, we show the finite-time total collapse of the solution onto a single point for compactly supported initial measures. Our approach is based on the theory of gradient flows in the space of probability measures endowed with the Wasserstein metric. In addition to classical tools, we exploit the stability of the flow with respect to the transportation distance to greatly simplify many problems by reducing them to questions about particle approximations.

An Introduction to Measure Theory

Abstract: This is a graduate text introducing the fundamentals of measure theory and integration theory, which is the foundation of modern real analysis. The text focuses first on the concrete setting of Lebesgue measure and the Lebesgue integral (which in turn is motivated by the more classical concepts of Jordan measure and the Riemann integral), before moving on to abstract measure and integration theory, including the standard convergence theorems, Fubini's theorem, and the Caratheodory extension theorem. Classical differentiation theorems, such as the Lebesgue and Radamacher differentiation theorems, are also covered, as are connections with probability theory. The material is intended to cover a quarter or semester's worth of material for a first graduate course in real analysis. There is an emphasis in the text on tying together the abstract and the concrete sides of the subject, using the latter to illustrate and motivate the former. The central role of key principles (such as Littlewood's three principles) as providing guiding intuition to the subject is also emphasised. There are a large number of exercises throughout that develop key aspects of the theory, and are thus an integral component of the text. As a supplementary section, a discussion of general problem-solving strategies in analysis is also given. The last three sections discuss optional topics related to the main matter of the book.

Measure of genuine multipartite entanglement with computable lower bounds

Abstract: We introduce an intuitive measure of genuine multipartite entanglement, which is based on the well-known concurrence. We show how lower bounds on this measure can be derived and also meet important characteristics of an entanglement measure. These lower bounds are experimentally implementable in a feasible way enabling quantification of multipartite entanglement in a broad variety of cases.

Computation of minkowski measures on 2d and 3d binary images

Abstract: Minkowski functionals encompass standard geometric parameters such as volume, area, length and the Euler-Poincare characteristic. Software tools for computing approximations of Minkowski functionals on binary 2D or 3D images are now available based on mathematical methods due to Serra, Lang and Ohser. Minkowski functionals can not be used to describe spatial heterogeneity of structures. This description can be performed by using Minkowski measures, which are local versions of Minkowski functionals. In this paper, we discuss how to extend the computation of Minkowski functionals to the computation of Minkowski measures. Approximations of Minkowski measures are computed using fltering and look-up table transformations. The final result is represented as a grey-level image. Approximation errors are investigated based on numerical examples. Convergence and non convergence of the measure approximations are discussed. The measure of surface area is used to describe spatial heterogeneity of a synthetic structure, and of an image of tomato pericarp.

Optimistic Optimization of a Deterministic Function without the Knowledge of its Smoothness

Abstract: We consider a global optimization problem of a deterministic function f in a semi-metric space, given a finite budget of n evaluations. The function f is assumed to be locally smooth (around one of its global maxima) with respect to a semi-metric l We describe two algorithms based on optimistic exploration that use a hierarchical partitioning of the space at all scales. A first contribution is an algorithm, DOO, that requires the knowledge of l. We report a finite-sample performance bound in terms of a measure of the quantity of near-optimal states. We then define a second algorithm, SOO, which does not require the knowledge of the semi-metric l under which f is smooth, and whose performance is almost as good as DOO optimally-fitted.

Asymptotic expansion of beta matrix models in the one-cut regime

Abstract: We prove the existence of a 1/N expansion to all orders in beta matrix models with a confining, off-critical potential corresponding to an equilibrium measure with a connected support. Thus, the coefficients of the expansion can be obtained recursively by the "topological recursion" of Chekhov and Eynard. Our method relies on the combination of a priori bounds on the correlators and the study of Schwinger-Dyson equations, thanks to the uses of classical complex analysis techniques. These a priori bounds can be derived following Boutet de Monvel, Pastur and Shcherbina, or for strictly convex potentials by using concentration of measure. Doing so, we extend the strategy of Guionnet and Maurel-Segala, from the hermitian models (beta = 2) and perturbative potentials, to general beta models. The existence of the first correction in 1/N has been considered previously by Johansson and more recently by Kriecherbauer and Shcherbina. Here, by taking similar hypotheses, we extend the result to all orders in 1/N.

BAM: a bounded adjusted measure of efficiency for use with bounded additive models

Abstract: A decade ago the Range Adjusted Measure (RAM) was introduced for use with Additive Models. The empirical experience gained since then recommends developing a new measure with similar characteristics but with more discriminatory power. This task is accomplished in this paper by introducing the Bounded Adjusted Measure (BAM) in connection with a new family of Data Envelopment Analysis (DEA) additive models that incorporate lower bounds for inputs and upper bounds for outputs while accepting any returns to scale imposed on the production technology.

An incremental Hausdorff distance calculation algorithm

Abstract: The Hausdorff distance is commonly used as a similarity measure between two point sets. Using this measure, a set X is considered similar to Y iff every point in X is close to at least one point in Y. Formally, the Hausdorff distance HausDist(X, Y) can be computed as the Max-Min distance from X to Y, i.e., find the maximum of the distance from an element in X to its nearest neighbor (NN) in Y. Although this is similar to the closest pair and farthest pair problems, computing the Hausdorff distance is a more challenging problem since its Max-Min nature involves both maximization and minimization rather than just one or the other. A traditional approach to computing HausDist(X, Y) performs a linear scan over X and utilizes an index to help compute the NN in Y for each x in X. We present a pair of basic solutions that avoid scanning X by applying the concept of aggregate NN search to searching for the element in X that yields the Hausdorff distance. In addition, we propose a novel method which incrementally explores the indexes of the two sets X and Y simultaneously. As an example application of our techniques, we use the Hausdorff distance as a measure of similarity between two trajectories (represented as point sets). We also use this example application to compare the performance of our proposed method with the traditional approach and the basic solutions. Experimental results show that our proposed method outperforms all competitors by one order of magnitude in terms of the tree traversal cost and total response time.

Spectral gaps for a Metropolis-Hastings algorithm in infinite dimensions

Abstract: We study the problem of sampling high and infinite dimensional target measures arising in applications such as conditioned diffusions and inverse problems. We focus on those that arise from approximating measures on Hilbert spaces defined via a density with respect to a Gaussian reference measure. We consider the Metropolis-Hastings algorithm that adds an accept-reject mechanism to a Markov chain proposal in order to make the chain reversible with respect to the target measure. We focus on cases where the proposal is either a Gaussian random walk (RWM) with covariance equal to that of the reference measure or an Ornstein-Uhlenbeck proposal (pCN) for which the reference measure is invariant. Previous results in terms of scaling and diffusion limits suggested that the pCN has a convergence rate that is independent of the dimension while the RWM method has undesirable dimension-dependent behaviour. We confirm this claim by exhibiting a dimension-independent Wasserstein spectral gap for pCN algorithm for a large class of target measures. In our setting this Wasserstein spectral gap implies an $L^2$-spectral gap. We use both spectral gaps to show that the ergodic average satisfies a strong law of large numbers, the central limit theorem and nonasymptotic bounds on the mean square error, all dimension independent. In contrast we show that the spectral gap of the RWM algorithm applied to the reference measures degenerates as the dimension tends to infinity.

An Exact Algorithm for F-Measure Maximization

Abstract: The F-measure, originally introduced in information retrieval, is nowadays routinely used as a performance metric for problems such as binary classification, multi-label classification, and structured output prediction. Optimizing this measure remains a statistically and computationally challenging problem, since no closed-form maximizer exists. Current algorithms are approximate and typically rely on additional assumptions regarding the statistical distribution of the binary response variables. In this paper, we present an algorithm which is not only computationally efficient but also exact, regardless of the underlying distribution. The algorithm requires only a quadratic number of parameters of the joint distribution (with respect to the number of binary responses). We illustrate its practical performance by means of experimental results for multi-label classification.

Smooth metric measure spaces with non-negative curvature

Abstract: We study both function theoretic and spectral properties on complete noncompact smooth metric measure space $(M,g,e^{-f}dv)$ with nonnegative Bakry-\'{E}mery Ricci curvature. Among other things, we derive a gradient estimate for positive $f$-harmonic functions and obtain as a consequence the strong Liouville property under the optimal sublinear growth assumption on $f.$ We also establish a sharp upper bound of the bottom spectrum of the $f$-Laplacian in terms of the linear growth rate of $f.$ Moreover, we show that if equality holds and $M$ is not connected at infinity, then $M$ must be a cylinder. As an application, we conclude steady Ricci solitons must be connected at infinity.

The Logic of Explanatory Power

Abstract: This article introduces and defends a probabilistic measure of the explanatory power that a particular explanans has over its explanandum. To this end, we propose several intuitive, formal conditions of adequacy for an account of explanatory power. Then, we show that these conditions are uniquely satisfied by one particular probabilistic function. We proceed to strengthen the case for this measure of explanatory power by proving several theorems, all of which show that this measure neatly corresponds to our explanatory intuitions. Finally, we briefly describe some promising future projects inspired by our account.

k-NN Regression Adapts to Local Intrinsic Dimension

Abstract: Many nonparametric regressors were recently shown to converge at rates that depend only on the intrinsic dimension of data. These regressors thus escape the curse of dimension when high-dimensional data has low intrinsic dimension (e.g. a manifold). We show that k-NN regression is also adaptive to intrinsic dimension. In particular our rates are local to a query x and depend only on the way masses of balls centered at x vary with radius. Furthermore, we show a simple way to choose k = k(x) locally at any x so as to nearly achieve the minimax rate at x in terms of the unknown intrinsic dimension in the vicinity of x. We also establish that the minimax rate does not depend on a particular choice of metric space or distribution, but rather that this minimax rate holds for any metric space and doubling measure.

Rough stochastic PDEs

Abstract: In this article, we show how the theory of rough paths can be used to provide a notion of solution to a class of nonlinear stochastic PDEs of Burgers type that exhibit too-high spatial roughness for classical analytical methods to apply. In fact, the class of SPDEs that we consider is genuinely ill-posed in the sense that different approximations to the nonlinearity may converge to different limits. Using rough path theory, a pathwise notion of solution to these SPDEs is formulated, and we show that this yields a well-posed problem that is stable under a large class of perturbations, including the approximation of the rough-driving noise by a mollified version and the addition of hyperviscosity. We also show that under certain structural assumptions on the coefficients, the SPDEs under consideration generate a reversible Markov semigroup with respect to a diffusion measure that can be given explicitly.

Alternative non-Markovianity measure by divisibility of dynamical maps

Abstract: By identifying non-Markovianity with nondivisibility, we propose a measure of non-Markovianity for quantum processes. Three examples are presented, and the measure of non-Markovianity is calculated and discussed for these examples. Comparisons with other measures of non-Markovianity are made. The present non-Markovianity measure has the merit that no optimization procedure is required and it is finite for any quantum process, which greatly enhances the practical relevance of the proposed measure.

Calculation of quantum discord for qubit-qudit or N qubits

Abstract: Quantum discord, a kind of quantum correlation, is defined as the difference between quantum mutual information and classical correlation in a bipartite system. It has been discussed so far for small systems with only a few independent parameters. We extend here to a much broader class of states when the second party is of arbitrary dimension d, so long as the first, measured, party is a qubit. We present two formulae to calculate quantum discord, the first relating to the original entropic definition and the second to a recently proposed geometric distance measure which leads to an analytical formulation. The tracing over the qubit in the entropic calculation is reduced to a very simple prescription. And, when the d-dimensional system is a so-called X state, the density matrix having non-zero elements only along the diagonal and anti-diagonal so as to appear visually like the letter X, the entropic calculation can be carried out analytically. Such states of the full bipartite qubit-qudit system may be named "extended X states", whose density matrix is built of four block matrices, each visually appearing as an X. The optimization involved in the entropic calculation is generally over two parameters, reducing to one for many cases, and avoided altogether for an overwhelmingly large set of density matrices as our numerical investigations demonstrate. Our results also apply to states of a N-qubit system, where "extended X states" consist of (2^(N+2) - 1) states, larger in number than the (2^(N+1) - 1) of X states of N qubits. While these are still smaller than the total number (2^(2N) - 1) of states of N qubits, the number of parameters involved is nevertheless large. In the case of N = 2, they encompass the entire 15-dimensional parameter space, that is, the extended X states for N = 2 represent the full qubit-qubit system.

Extracting rate coefficients from single-molecule photon trajectories and FRET efficiency histograms for a fast-folding protein.

Abstract: Recently developed statistical methods by Gopich and Szabo were used to extract folding and unfolding rate coefficients from single-molecule Forster resonance energy transfer (FRET) data for proteins with kinetics too fast to measure waiting time distributions. Two types of experiments and two different analyses were performed. In one experiment bursts of photons were collected from donor and acceptor fluorophores attached to a 73-residue protein, α(3)D, freely diffusing through the illuminated volume of a confocal microscope system. In the second, the protein was immobilized by linkage to a surface, and photons were collected until one of the fluorophores bleached. Folding and unfolding rate coefficients and mean FRET efficiencies for the folded and unfolded subpopulations were obtained from a photon by photon analysis of the trajectories using a maximum likelihood method. The ability of the method to describe the data in terms of a two-state model was checked by recoloring the photon trajectories with the extracted parameters and comparing the calculated FRET efficiency histograms with the measured histograms. The sum of the rate coefficients for the two-state model agreed to within 30% with the relaxation rate obtained from the decay of the donor-acceptor cross-correlation function, confirming the high accuracy of the method. Interestingly, apparently reliable rate coefficients could be extracted using the maximum likelihood method, even at low (<10%) population of the minor component where the cross-correlation function was too noisy to obtain any useful information. The rate coefficients and mean FRET efficiencies were also obtained in an approximate procedure by simply fitting the FRET efficiency histograms, calculated by binning the donor and acceptor photons, with a sum of three-Gaussian functions. The kinetics are exposed in these histograms by the growth of a FRET efficiency peak at values intermediate between the folded and unfolded peaks as the bin size increases, a phenomenon with similarities to NMR exchange broadening. When comparable populations of folded and unfolded molecules are present, this method yields rate coefficients in very good agreement with those obtained with the maximum likelihood method. As a first step toward characterizing transition paths, the Viterbi algorithm was used to locate the most probable transition points in the photon trajectories.

Three chapters of measure theory in Isabelle/HOL

Abstract: Currently published HOL formalizations of measure theory concentrate on the Lebesgue integral and they are restricted to realvalued measures. We lift this restriction by introducing the extended real numbers. We define the Borel s-algebra for an arbitrary type forming a topological space. Then, we introduce measure spaces with extended real numbers as measure values. After defining the Lebesgue integral and verifying its linearity and monotone convergence property, we prove the Radon-Nikodým theorem (which shows the maturity of our framework). Moreover, we formalize product measures and prove Fubini's theorem. We define the Lebesgue measure using the gauge integral available in Isabelle's multivariate analysis. Finally, we relate both integrals and equate the integral on Euclidean spaces with iterated integrals. This work covers most of the first three chapters of Bauer's measure theory textbook.

Optimal Scaling and Diffusion Limits for the Langevin Algorithm in High Dimensions

Abstract: The Metropolis-adjusted Langevin (MALA) algorithm is a sampling algorithm which makes local moves by incorporating information about the gradient of the logarithm of the target density In this paper we study the efficiency of MALA on a natural class of target measures supported on an infinite dimensional Hilbert space These natural measures have density with respect to a Gaussian random field measure and arise in many applications such as Bayesian nonparametric statistics and the theory of conditioned diffusions We prove that, started in stationarity, a suitably interpolated and scaled version of the Markov chain corresponding to MALA converges to an infinite dimensional diffusion process Our results imply that, in stationarity, the MALA algorithm applied to an N-dimensional approximation of the target will take $\mathcal{O}(N^{1/3})$ steps to explore the invariant measure, comparing favorably with the Random Walk Metropolis which was recently shown to require $\mathcal{O}(N)$ steps when applied to the same class of problems

Information-theoretic treatment of tripartite systems and quantum channels

Abstract: A Holevo measure is used to discuss how much information about a given positive operator valued measure (POVM) on system $a$ is present in another system $b$, and how this influences the presence or absence of information about a different POVM on $a$ in a third system $c$. The main goal is to extend information theorems for mutually unbiased bases or general bases to arbitrary POVMs, and especially to generalize ``all-or-nothing'' theorems about information located in tripartite systems to the case of partial information, in the form of quantitative inequalities. Some of the inequalities can be viewed as entropic uncertainty relations that apply in the presence of quantum side information, as in recent work by Berta et al. [Nature Physics 6, 659 (2010)]. All of the results also apply to quantum channels: For example, if $\mathcal{E}$ accurately transmits certain POVMs, the complementary channel $\mathcal{F}$ will necessarily be noisy for certain other POVMs. While the inequalities are valid for mixed states of tripartite systems, restricting to pure states leads to the basis invariance of the difference between the information about $a$ contained in $b$ and $c$.

Rigidity for von Neumann algebras and their invariants

Abstract: We give a survey of recent classification results for von Neumann algebras L∞(X) o Γ arising from measure preserving group actions on probability spaces. This includes II1 factors with uncountable fundamental groups and the construction of W ∗superrigid actions where L∞(X) o Γ entirely remembers the initial group action Γ y X. Mathematics Subject Classification (2000). Primary 46L36; Secondary 46L40,

The conjugacy problem in ergodic theory

Abstract: All common probability preserving transformations can be represented as elements of MPT, the group of measure preserving transformations of the unit interval with Lebesgue measure. This group has a natural Polish topology and the induced topology on the set of ergodic transformations is also Polish. Our main result is that the set of ergodic elements T in MPT that are isomorphic to their inverse is a complete analytic set. This has as a consequence the fact that the isomorphism relation is also a complete analytic set and in particular is not Borel. This is in stark contrast to the situation of unitary operators where the spectral theorem can be used to show that conjugacy relation in the unitary group is Borel. This result explains, perhaps, why the problem of determining whether ergodic transformations are isomorphic or not has proven to be so intractable. The construction that we use is general enough to show that the set of ergodic T ’s with nontrivial centralizer is also complete analytic. On the positive side we show that the isomorphism relation is Borel when restricted to the rank one transformations, which form a generic subset of MPT. It remains an open problem to nd a good explicit method of checking when two rank one transformations are isomorphic. In Memoriam: Prior to the