Showing papers on "Measure (mathematics) published in 2022"
01 Jan 2022
TL;DR: An optimization scheme based on the Alternating Direction Method of Multipliers that minimizes not only the training loss of an NN but also its Lipschitz constant resulting in a semidefinite programming based training procedure that promotes robustness.
Abstract: Due to their susceptibility to adversarial perturbations, neural networks (NNs) are hardly used in safety-critical applications. One measure of robustness to such perturbations in the input is the Lipschitz constant of the input-output map defined by an NN. In this letter, we propose a framework to train multi-layer NNs while at the same time encouraging robustness by keeping their Lipschitz constant small, thus addressing the robustness issue. More specifically, we design an optimization scheme based on the Alternating Direction Method of Multipliers that minimizes not only the training loss of an NN but also its Lipschitz constant resulting in a semidefinite programming based training procedure that promotes robustness. We design two versions of this training procedure. The first one includes a regularizer that penalizes an accurate upper bound on the Lipschitz constant. The second one allows to enforce a desired Lipschitz bound on the NN at all times during training. Finally, we provide two examples to show that the proposed framework successfully increases the robustness of NNs.
59 citations
TL;DR: For a given p ∈ [ 2, + ∞ ], the authors in this article proved the smooth convergence of the flow for p = 2 in the Euclidean space R n, in the hyperbolic plane H 2, and in the two-dimensional sphere S 2, which implies that such flow in R n or H 2 remains in a bounded region of the space for any time.
Abstract: For a given p ∈ [ 2 , + ∞ ) , we define the p -elastic energy E of a closed curve γ : S 1 → M immersed in a complete Riemannian manifold ( M , g ) as the sum of the length of the curve and the L p -norm of its curvature (with respect to the length measure). We are interested in the convergence of the ( L p , L p ′ ) -gradient flow of these energies to critical points. By means of parabolic estimates, it is usually possible to prove sub-convergence of the flow, that is, convergence to critical points up to reparametrizations and, more importantly, up to isometry of the ambient. Assuming that the flow sub-converges, we are interested in proving the smooth convergence of the flow, that is, the existence of the full limit of the evolving flow. We first give an overview of the general strategy one can apply for proving such a statement. The crucial step is the application of a Łojasiewicz–Simon gradient inequality, of which we present a versatile version. Then we apply such strategy to the flow of E of curves into manifolds, proving the desired improvement of sub-convergence to full smooth convergence of the flow to critical points. As corollaries, we obtain the smooth convergence of the flow for p = 2 in the Euclidean space R n , in the hyperbolic plane H 2 , and in the two-dimensional sphere S 2 . In particular, the result implies that such flow in R n or H 2 remains in a bounded region of the space for any time.
14 citations
TL;DR: Wang et al. as discussed by the authors proposed a new similarity measure of intuitionistic fuzzy sets (IFS), which takes into account the effect of hesitancy degree on membership degree and non-membership degree.
13 citations
TL;DR: In this article, a geometric generalization of contraction theory called k-contraction is proposed, where a dynamical system is called k -contractive if the dynamics contracts k -parallelotopes at an exponential rate.
10 citations
TL;DR: In this paper, the cumulative residual Tsallis entropy (CRTE) was introduced as an alternative dispersion measure for image quality assessment and its properties were investigated. But the authors considered the problem of estimating the CRTE by means of the empirical CRTE and used two empirical estimators of cumulative distribution function to estimate CRTE.
7 citations
TL;DR: Despite the existence of model uncertainties, it is shown that it is possible for a subset of stabilizing controller gains to be characterized appropriately via convex inner approximation, which then further facilitates the determination of the controller by means of convex optimization.
Abstract: This article presents a convex inner approximation approach for mixed ${H}_2$ / ${H}_\infty$ control of a flexure-based nanopositioning system. Generally for such positioning systems, the inevitable existence of model mismatch renders it often times difficult-to-achieve satisfying system performance. Additionally, it is essential to also note that the high-order resonances typically presented are prone to be activated if the controller is not designed appropriately, especially in the case when the control input variation arising from the design is unnecessarily drastic. Therefore, to circumvent the above undesirable possibilities, this work aims to improve the tracking performance with a suitable controller design that effectively suppresses the control input variation. Furthermore, despite the existence of model uncertainties, it is shown that it is possible for a subset of stabilizing controller gains to be characterized appropriately via convex inner approximation, which then further facilitates the determination of the controller by means of convex optimization. Rather importantly, this approach provides a performance guarantee with an optimized limiting bound to the $H_2$ -norm level (which assures optimal behavior for the system), and also concurrently limits the $H_\infty$ -norm level within a prescribed attenuation level (which satisfies a prescribed robustness measure). Finally, numerical optimization and comparative experiments are carried out for demonstrative purposes.
7 citations
07 Mar 2022
TL;DR: Unlike the real line, the real space, in dimension d ≥ 2, is not canonically ordered as discussed by the authors, and extending to a multivariate context fundamental univariate statistical tools such as quantile...
Abstract: Unlike the real line, the real space, in dimension d ≥ 2, is not canonically ordered. As a consequence, extending to a multivariate context fundamental univariate statistical tools such as quantile...
6 citations
6 citations
TL;DR: In this paper, the authors studied the theory of rectifiability in arbitrary Carnot groups, and in particular with the study of the notion of $$\mathscr {P}$$ -rectifiable measure.
Abstract: This paper deals with the theory of rectifiability in arbitrary Carnot groups, and in particular with the study of the notion of $$\mathscr {P}$$
-rectifiable measure. First, we show that in arbitrary Carnot groups the natural infinitesimal definition of rectifiabile measure, i.e., the definition given in terms of the existence of flat tangent measures, is equivalent to the global definition given in terms of coverings with intrinsically differentiable graphs, i.e., graphs with flat Hausdorff tangents. In general we do not have the latter equivalence if we ask the covering to be made of intrinsically Lipschitz graphs. Second, we show a geometric area formula for the centered Hausdorff measure restricted to intrinsically differentiable graphs in arbitrary Carnot groups. The latter formula extends and strengthens other area formulae obtained in the literature in the context of Carnot groups. As an application, our analysis allows us to prove the intrinsic $$C^1$$
-rectifiability of almost all the preimages of a large class of Lipschitz functions between Carnot groups. In particular, from the latter result, we obtain that any geodesic sphere in a Carnot group equipped with an arbitrary left-invariant homogeneous distance is intrinsic $$C^1$$
-rectifiable.
6 citations
TL;DR: In this article, the authors give some fixed point results based on the technique of measure of noncompactness which extend the classical Darbo-type theorem, and obtain the existence of solution of implicit fractional integral equations in C ( I, l p α ) (collection of all continuous functions from I = [ 0, a ] (a > 0 ) to lp α ), where lpα is a tempered sequence space.
Abstract: The aim of this work is to give some fixed point results based on the technique of measure of noncompactness which extend the classical Darbo’s theorem. With the help of our Darbo-type theorem, we obtain the existence of solution of implicit fractional integral equations in C ( I , l p α ) (collection of all continuous functions from I = [ 0 , a ] ( a > 0 ) to l p α ), where l p α is a tempered sequence space. Finally, we present a numerical example to see the validity and practicability of our existence result.
6 citations
TL;DR: In this paper, an attempt is made to develop a novel measure of similarity between fuzzy numbers, in which the existing measures do not comply with the basic properties of a similarity measure.
TL;DR: The use of integrating sphere has been known as a method to measure hemispherical reflectance of a material sample as discussed by the authors, and mathematical expressions of such reflectance are available in literature.
Abstract: The use of integrating sphere has been known as a method to measure hemispherical reflectance of a material sample. Mathematical expressions of such reflectance are available in literature, but mos...
TL;DR: This work proposes the dependence measure (DM) based on the RDC and the phase space reconstruction theory, aiming to capture linear and nonlinear dynamical features from various kinds of complex signals with higher accuracy, and combines the DM and the CSE to construct the DM-CSE plane.
TL;DR: In this article, the authors characterize surjective isometries of the total variation distance between various subsets of Borel probability measures, i.e., compositum of a measure with a bijection.
TL;DR: In this paper, the authors prove large deviation principles for the weighted spectral measure of unitarily invariant random matrices in two general situations: (1) when the equilibrium measure is not necessarily supported by a single interval and (2) if the potential is a nonnegative polynomial.
TL;DR: The notion of cw-expansive measures linking the continuum theory [16] to measurable dynamics was introduced in this article, where it was shown that for equicontinuous homeomorphisms on compact metric spaces as those supported in the non-locally connected points.
TL;DR: In this article, lower bounds for the first eigenvalue for Bi-drifted Laplacian operator on compact manifolds with boundary and m-Bakry-Emery Ricci curvature bounded below were obtained.
Abstract: In this paper we obtain lower bounds for the first eigenvalue to some kinds of the eigenvalue problems for Bi-drifted Laplacian operator on compact manifolds (also called a smooth metric measure space) with boundary and m-Bakry-Emery Ricci curvature or Bakry-Emery Ricci curvature bounded below. We also address the eigenvalue problem with Wentzell-type boundary condition for drifted Laplacian on smooth metric measure space.
TL;DR: In this paper, the boundedness of difference of weighted composition operators between weak and strong vector-valued Bergman spaces is characterized in three terms: one is a function theoretic characterization of Julia-Caratheodory type, the second is a power type characterization and the other is a measure theoretic characterisation of Carleson type.
TL;DR: In this article, the authors derived further cryptographic properties of the multiplicative inverse function, especially the ones related to higher order differentials, and provided slightly improved bounds over the work of Carlet [IEEE-IT, 2008].
TL;DR: In this article, a self-contained proof of this theorem generalized into metric measure spaces supporting a doubling measure was given, and estimates for Muckenhoupt weights on Whitney chains in the metric setting were given.
Abstract: A theorem by Wolff states that weights defined on a measurable subset of R n and satisfying a Muckenhoupt-type condition can be extended into the whole space as Muckenhoupt weights of the same class. We give a complete and self-contained proof of this theorem generalized into metric measure spaces supporting a doubling measure. Related to the extension problem, we also show estimates for Muckenhoupt weights on Whitney chains in the metric setting.
TL;DR: In this article, the converse Lyapunov theorems for generalized ODEs and measure functional differential equations were obtained for functional dynamic equations on time scales, and necessary and sufficient conditions were established for a system of non-homogeneous nonlinear GDEs defined in a Banach space to be asymptotically controllable.
TL;DR: The important application of using fuzzy measure to provide information about an uncertain variable V, a collection of uncertain ordinal values where the uncertainty is modeled by a measure, is discussed.
TL;DR: In this article, a lower bound for the measure of B (x, r ) ∩ Ω under the assumption that M 1, A ( Ω ) ↪ L A ˆ (Ω ) is a Young function that increases more rapidly than A near infinity is established.
TL;DR: In this paper, the authors present a temporal extension of the slow motion prior model to generate predictions regarding the temporal evolution of the contrast induced speed bias, and further test these predictions using a novel experimental paradigm that allows them to measure the dynamic perceptual difference between stimuli through a series of manual pursuit open loop tasks.
TL;DR: In this paper, the authors study the Besov type function spaces for maps which are defined on abstract metric-measure spaces and extend some of the embedding theorems of the classical Besov spaces to the setting of abstract spaces.
TL;DR: In this article, a Gaussian kernel is used to obtain the consistency of the local correlation dimension for multivariate fractional Brownian motion, which can be used for estimating the Hausdorff dimension of the image of multidimensional stochastic processes.
TL;DR: In this article, the authors considered the continuous parabolic Anderson model with the Gaussian fields under the measure-valued initial conditions, the covariances of which are homogeneous or nonhomogeneous in time and fractional rough in space.
TL;DR: In this article, it was shown that this set is a G δ σ subset of the space of Borel probability measures of the underlying space and that the cw-expansive measures with invariant support are weak* dense on it.
TL;DR: In this article, the authors extended Bochkariev's theorem for some class of variable exponent Lebesgue spaces, such that the Fourier series of it with respect to the system Φ diverges on the set of positive measure.