Showing papers on "Bounded function published in 2003"

Rademacher and gaussian complexities: risk bounds and structural results

Abstract: We investigate the use of certain data-dependent estimates of the complexity of a function class, called Rademacher and Gaussian complexities. In a decision theoretic setting, we prove general risk bounds in terms of these complexities. We consider function classes that can be expressed as combinations of functions from basis classes and show how the Rademacher and Gaussian complexities of such a function class can be bounded in terms of the complexity of the basis classes. We give examples of the application of these techniques in finding data-dependent risk bounds for decision trees, neural networks and support vector machines.

Revealing information while preserving privacy

Abstract: We examine the tradeoff between privacy and usability of statistical databases. We model a statistical database by an n-bit string d1,..,dn, with a query being a subset q ⊆ [n] to be answered by Σieqdi. Our main result is a polynomial reconstruction algorithm of data from noisy (perturbed) subset sums. Applying this reconstruction algorithm to statistical databases we show that in order to achieve privacy one has to add perturbation of magnitude (Ω√n). That is, smaller perturbation always results in a strong violation of privacy. We show that this result is tight by exemplifying access algorithms for statistical databases that preserve privacy while adding perturbation of magnitude O(√n).For time-T bounded adversaries we demonstrate a privacypreserving access algorithm whose perturbation magnitude is ≈ √T.

Computing and rendering point set surfaces

Abstract: We advocate the use of point sets to represent shapes. We provide a definition of a smooth manifold surface from a set of points close to the original surface. The definition is based on local maps from differential geometry, which are approximated by the method of moving least squares (MLS). The computation of points on the surface is local, which results in an out-of-core technique that can handle any point set. We show that the approximation error is bounded and present tools to increase or decrease the density of the points, thus allowing an adjustment of the spacing among the points to control the error. To display the point set surface, we introduce a novel point rendering technique. The idea is to evaluate the local maps according to the image resolution. This results in high quality shading effects and smooth silhouettes at interactive frame rates.

Introduction to Operator Space Theory

Abstract: The theory of operator spaces is very recent and can be described as a non-commutative Banach space theory. An 'operator space' is simply a Banach space with an embedding into the space B(H) of all bounded operators on a Hilbert space H. The first part of this book is an introduction with emphasis on examples that illustrate various aspects of the theory. The second part is devoted to applications to C*-algebras, with a systematic exposition of tensor products of C*-algebras. The third (and shorter) part of the book describes applications to non self-adjoint operator algebras, and similarity problems. In particular the author's counterexample to the 'Halmos problem' is presented, as well as work on the new concept of 'length' of an operator algebra. Graduate students and professional mathematicians interested in functional analysis, operator algebras and theoretical physics will find that this book has much to offer.

Generators and representability of functors in commutative and noncommutative geometry

Abstract: We give a sufficient condition for an Ext-finite triangulated category to be saturated. Saturatedness means that every contravariant cohomological functor of finite type to vector spaces is representable. The condition consists in the existence of a strong generator. We prove that the bounded derived categories of coherent sheaves on smooth proper commutative and noncommutative varieties have strong generators, and are hence saturated. In contrast, the similar category for a smooth compact analytic surface with no curves is not saturated. 2000 Math. Subj. Class. Primary 18E30.

An Iterative Approach to Quadratic Optimization

Abstract: Assume that C 1, . . . , C N are N closed convex subsets of a real Hilbert space H having a nonempty intersection C. Assume also that each C i is the fixed point set of a nonexpansive mapping T i of H. We devise an iterative algorithm which generates a sequence (x n ) from an arbitrary initial x 0∈H. The sequence (xn) is shown to converge in norm to the unique solution of the quadratic minimization problem min x∈C (1/2)〈Ax, x〉−〈x, u〉, where A is a bounded linear strongly positive operator on H and u is a given point in H. Quadratic–quadratic minimization problems are also discussed.

Image Decomposition and Restoration Using Total Variation Minimization and the H1

Abstract: In this paper, we propose a new model for image restoration and image decomposition into cartoon and texture, based on the total variation minimization of Rudin, Osher, and Fatemi [Phys. D, 60 (1992), pp. 259--268], and on oscillatory functions, which follows results of Meyer [Oscillating Patterns in Image Processing and Nonlinear Evolution Equations, Univ. Lecture Ser. 22, AMS, Providence, RI, 2002]. This paper also continues the ideas introduced by the authors in a previous work on image decomposition models into cartoon and texture [L. Vese and S. Osher, J. Sci. Comput., to appear]. Indeed, by an alternative formulation, an initial image f is decomposed here into a cartoon part u and a texture or noise part v. The u component is modeled by a function of bounded variation, while the v component is modeled by an oscillatory function, bounded in the norm dual to $|\cdot|_{H^1_0}$. After some transformation, the resulting PDE is of fourth order, envolving the Laplacian of the curvature of level lines. Fina...

Numerical Analysis of Wavelet Methods

Abstract: Introduction. Notations. 1. Basic examples. 1.1 Introduction. 1.2 The Haar system. 1.3 The Schauder hierarchical basis. 1.4 Multivariate constructions. 1.5 Adaptive approximation. 1.6 Multilevel preconditioning. 1.7 Conclusions. 1.8 Historical notes. 2. Multiresolution approximation. 2.1 Introduction. 2.2 Multiresolution analysis. 2.3 Refinable functions. 2.4 Subdivision schemes. 2.5 Computing with refinable functions. 2.6 Wavelets and multiscale algorithms. 2.7 Smoothness analysis. 2.8 Polynomial exactness. 2.9 Duality, orthonormality and interpolation. 2.10 Interpolatory and orthonormal wavelets. 2.11 Wavelets and splines. 2.12 Bounded domains and boundary conditions. 2.13 Point values, cell averages, finite elements. 2.14 Conclusions. 2.15 Historical notes. 3. Approximation and smoothness. 3.1 Introduction. 3.2 Function spaces. 3.3 Direct estimates. 3.4 Inverse estimates. 3.5 Interpolation and approximation spaces. 3.6 Characterization of smoothness classes. 3.7 Lp-unstable approximation and 0 1. 3.8 Negative smoothness and Lp-spaces. 3.9 Bounded domains. 3.10 Boundary conditions. 3.11 Multilevel preconditioning. 3.12 Conclusions. 3.13 Historical notes. 4. Adaptivity. 4.1 Introduction. 4.2 Nonlinear approximation in Besov spaces. 4.3 Nonlinear wavelet approximation in Lp. 4.4 Adaptive finite element approximation. 4.5 Other types of nonlinear approximations. 4.6 Adaptive approximation of operators. 4.7 Nonlinear approximation and PDE's. 4.8 Adaptive multiscale processing. 4.9 Adaptive space refinement. 4.10 Conclusions. 4.11 Historical notes. References. Index.

Bounded geometries, fractals, and low-distortion embeddings

Abstract: The doubling constant of a metric space (X, d) is the smallest value /spl lambda/ such that every ball in X can be covered by /spl lambda/ balls of half the radius. The doubling dimension of X is then defined as dim (X) = log/sub 2//spl lambda/. A metric (or sequence of metrics) is called doubling precisely when its doubling dimension is bounded. This is a robust class of metric spaces which contains many families of metrics that occur in applied settings. We give tight bounds for embedding doubling metrics into (low-dimensional) normed spaces. We consider both general doubling metrics, as well as more restricted families such as those arising from trees, from graphs excluding a fixed minor, and from snowflaked metrics. Our techniques include decomposition theorems for doubling metrics, and an analysis of a fractal in the plane according to T. J. Laakso (2002). Finally, we discuss some applications and point out a central open question regarding dimensionality reduction in L/sub 2/.

Exponential Hedging and Entropic Penalties

Abstract: We solve the problem of hedging a contingent claim B by maximizing the expected exponential utility of terminal net wealth for a locally bounded semimartingale X. We prove a duality relation between this problem and a dual problem for local martingale measures Q for X where we either minimize relative entropy minus a correction term involving B or maximize the Q-price of B subject to an entropic penalty term. Our result is robust in the sense that it holds for several choices of the space of hedging strategies. Applications include a new characterization of the minimal martingale measure and risk-averse asymptotics.

A convergent and universally bounded interpolation scheme for the treatment of advection

Abstract: A high resolution scheme with improved iterative convergence properties was devised by incorporating total-variation diminishing constraints, appropriate for unsteady problems, into an implicit time-marching method used for steady flow problems. The new scheme, referred to as Convergent and Universally Bounded Interpolation Scheme for the Treatment of Advection (CUBISTA), has similar accuracy to the well-known SMART scheme, both being formally third-order accurate on uniform meshes for smooth flows. Three demonstration problems are considered: (1) advection of three scalar profiles, a step, a sine-squared, and a semi-ellipse; (2) Newtonian flow over a backward-facing step; and (3) viscoelastic flow through a planar contraction and around a cylinder. For the case of the viscoelastic flows, in which the high resolution schemes are also used to represent the advective terms in the constitutive equation, it is shown that only the new scheme is able to provide a converged solution to the prescribed tolerance. Copyright © 2003 John Wiley & Sons, Ltd.

The maximal function on variable spaces.

Abstract: We give continuity conditions on the exponent function p(x) which are su-- cient for the Hardy{Littlewood maximal operator to be bounded on the variable Lebesgue space L p(x) (›) , where › is any open subset of R n . Further, our conditions are necessary on R. Our result extends the recent work of Pick and R"••a (20), Diening (3) and Nekvinda (19). We also show that under much weaker assumptions on p(x) , the maximal operator satisfles a weak-type modular inequality.

Topological degree for a mean field equation on Riemann surfaces

Abstract: We consider the following mean field equations: (0.1) where M is a compact Riemann surface with volume 1, h is a positive continuous function on M, ρ is a real number, (0.2) and where Ω is a bounded smooth domain, h is a C1 positive function on Ω, and ρ ∈ ℝ. Based on our previous analytic work [14], we prove, among other things, that the degree-counting formula for (0.1) is given by () for ρ ∈ (8mπ, 8(m + 1)π). © 2003 Wiley Periodicals, Inc.

A sharp bilinear restriction estimate for paraboloids

Abstract: Recently Wolff [W3] obtained a sharp L 2 bilinear restriction theorem for bounded subsets of the cone in general dimension. Here we adapt the argument of Wolff to also handle subsets of “elliptic surfaces” such as paraboloids. Except for an endpoint, this answers a conjecture of Machedon and Klainerman, and also improves upon the known restriction theory for the paraboloid and sphere.

Wiener's lemma for twisted convolution and Gabor frames

Abstract: As a consequence, Ca is invertible and bounded on all ?p(Zd) for 1 < p < oo simultaneously. In this article we study several non-commutative generalizations of Wiener's Lemma and their application to Gabor theory. The paper is divided into two parts: the first part (Sections 2 and 3) is devoted to abstract harmonic analysis and extends Wiener's Lemma to twisted convolution. The second part (Section 4) is devoted to the theory of Gabor frames, specifically to the design of dual windows with good time-frequency localization. In particular, we solve a conjecture of Janssen, Feichtinger and one of us [17], [18], [9]. These two topics appear to be completely disjoint, but they are not. The solution of the conjectures about Gabor frames is an unexpected application of methods from non-commutative harmonic analysis to application-oriented mathematics. It turns out that the connection between twisted convolution and the Heisenberg group and the theory of symmetric group algebras are precisely the tools needed to treat the problem motivated by signal analysis. To be more concrete, we formulate some of our main results first and will deal with the details and the technical background later.

Automatic abstraction without counterexamples

Abstract: A method of automatic abstraction is presented that uses proofs of unsatisfiability derived from SAT-based bounded model checking as a guide to choosing an abstraction for unbounded model checking. Unlike earlier methods, this approach is not based on analysis of abstract counterexamples. The performance of this approach on benchmarks derived from microprocessor verification indicates that SAT solvers are quite effective in eliminating logic that is not relevant to a given property. Moreover, benchmark results suggest that when bounded model checking successfully terminates, and the problem is unsatisfiable, the number of state variables in the proof of unsatisfiability tends to be small. In almost all cases tested, when bounded model checking succeeded, unbounded model checking of the resulting abstraction also succeeded.

Practical stabilization of driftless systems on Lie groups: the transverse function approach

Abstract: A general control design approach for the stabilization of controllable driftless nonlinear systems on finite dimensional Lie groups is presented. The approach is based on the concept of bounded transverse functions, the existence of which is equivalent to the system's controllability. Its outcome is the practical stabilization of any trajectory, i.e., not necessarily a solution of the control system, in the state-space. The possibility of applying the approach to an arbitrary controllable smooth driftless system follows in turn from the fact that any controllable homogeneous approximation of this system can be lifted (via a dynamic extension) to a system on a Lie group. Illustrative examples are given.

Renormalisation of \phi^4-theory on noncommutative R^2 in the matrix base

Abstract: As a first application of our renormalisation group approach to non-local matrix models [hep-th/0305066], we prove (super-)renormalisability of Euclidean two-dimensional noncommutative \phi^4-theory. It is widely believed that this model is renormalisable in momentum space arguing that there would be logarithmic UV/IR-divergences only. Although momentum space Feynman graphs can indeed be computed to any loop order, the logarithmic UV/IR-divergence appears in the renormalised two-point function -- a hint that the renormalisation is not completed. In particular, it is impossible to define the squared mass as the value of the two-point function at vanishing momentum. In contrast, in our matrix approach the renormalised N-point functions are bounded everywhere and nevertheless rely on adjusting the mass only. We achieve this by introducing into the cut-off model a translation-invariance breaking regulator which is scaled to zero with the removal of the cut-off. The naive treatment without regulator would not lead to a renormalised theory.

Approximation by fully complex multilayer perceptrons

Abstract: We investigate the approximation ability of a multilayer perceptron (MLP) network when it is extended to the complex domain. The main challenge for processing complex data with neural networks has been the lack of bounded and analytic complex nonlinear activation functions in the complex domain, as stated by Liouville's theorem. To avoid the conflict between the boundedness and the analyticity of a nonlinear complex function in the complex domain, a number of ad hoc MLPs that include using two real-valued MLPs, one processing the real part and the other processing the imaginary part, have been traditionally employed. However, since nonanalytic functions do not meet the Cauchy-Riemann conditions, they render themselves into degenerative backpropagation algorithms that compromise the efficiency of nonlinear approximation and learning in the complex vector field. A number of elementary transcendental functions (ETFs) derivable from the entire exponential function ez that are analytic are defined as fully complex activation functions and are shown to provide a parsimonious structure for processing data in the complex domain and address most of the shortcomings of the traditional approach. The introduction of ETFs, however, raises a new question in the approximation capability of this fully complex MLP. In this letter, three proofs of the approximation capability of the fully complex MLP are provided based on the characteristics of singularity among ETFs. First, the fully complex MLPs with continuous ETFs over a compact set in the complex vector field are shown to be the universal approximator of any continuous complex mappings. The complex universal approximation theorem extends to bounded measurable ETFs possessing a removable singularity. Finally, it is shown that the output of complex MLPs using ETFs with isolated and essential singularities uniformly converges to any nonlinear mapping in the deleted annulus of singularity nearest to the origin.

Bounded Finite State Controllers

Abstract: We describe a new approximation algorithm for solving partially observable MDPs. Our bounded policy iteration approach searches through the space of bounded-size, stochastic finite state controllers, combining several advantages of gradient ascent (efficiency, search through restricted controller space) and policy iteration (less vulnerability to local optima).

A state bounding observer based on zonotopes

Abstract: The proposed observer computes an outer approximation of the set of states which are consistent with a given uncertain state space model and some measurements. The uncertainties are modeled by unknown but bounded inputs. A representation of domains by zonotopes (particular polytopes) is used to reduce the computation of state bounds to rather simple matrix operations and to control the wrapping effect.

Verifying epistemic properties of multi-agent systems via bounded model checking

Abstract: We present a framework for verifying temporal and epistemic properties of multi-agent systems by means of bounded model checking. We use interpreted systems as underlying semantics. We give details of the proposed technique, and show how it can be applied to the "attacking generals problem", a typical example of oordination in multi-agent systems.

Reachability analysis of nonlinear systems using conservative approximation

Abstract: In this paper we present an approach to approximate reachability computation for nonlinear continuous systems. Rather than studying a complex nonlinear system x = g(x), we study an approximating system x = f(x) which is easier to handle. The class of approximating systems we consider in this paper is piecewise linear, obtained by interpolating g over a mesh. In order to be conservative, we add a bounded input in the approximating system to account for the interpolation error. We thus develop a reachability method for systems with input, based on the relation between such systems and the corresponding autonomous systems in terms of reachable sets. This method is then extended to the approximate piecewise linear systems arising in our construction. The final result is a reachability algorithm for nonlinear continuous systems which allows to compute conservative approximations with as great degree of accuracy as desired, and more importantly, it has good convergence rate. If g is a C2 function, our method is of order 2. Furthermore, the method can be straightforwardly extended to hybrid systems.

Repetitive Delone Sets and Quasicrystals

Abstract: This paper studies the problem of characterizing the simplest aperiodic discrete point sets, using invariants based on topological dynamics A Delone set of finite type is a Delone set X such that X - X is locally finite Such sets are characterized by their patch-counting function N(T) of radius T being finite for all T We formulate conjectures relating slow growth of the patch-counting function N(T) to the set X having a nontrivial translation symmetry A Delone set X of finite type is repetitive if there is a function M(T) such that every closed ball of radius M(T) + T contains a complete copy of each kind of patch of radius T that occurs in X This is equivalent to the minimality of an associated topological dynamical system with ℝ-action There is a lower bound for M(T) in terms of N(T), namely M(T) ≥ c(N(T) for some positive constant c depending on the Delone set constants r, R, but there is no general upper bound for M(T) purely in terms of N(T) The complexity of a repetitive Delone set X is measured by the growth rate of its repetitivity function M(T) For example, the function M(T) is bounded if and only if X is a periodic crystal A set X is linearly repetitive if M(T) = O(T) as T → ∞ and is densely repetitive if M(T) = O(N(T)) as T → ∞ We show that linearly repetitive sets and densely repetitive sets have strict uniform patch frequencies, ie the associated topological dynamical system is strictly ergodic It follows that such sets are diffractive, in the sense of having a well-defined diffraction measure In the reverse direction, we construct a repetitive Delone set X in ℝ which has M(T) = O (T (log T) (log log log T)), but does not have uniform patch frequencies Aperiodic linearly repetitive sets have many claims to be the simplest class of aperiodic sets and we propose considering them as a notion of 'perfectly ordered quasicrystals'