Push-forward Measure Research Articles

Nonconvergence of a sum-of-squares hierarchy for global polynomial optimization based on push-forward measures

Nonconvergence of a sum-of-squares hierarchy for global polynomial optimization based on push-forward measures

Splitting for some classes of homeomorphic and coalescing stochastic flows

The splitting scheme (the Kato-Trotter formula) is applied to stochastic flows with common noise of the type introduced by Th.E. Harris. The case of possibly coalescing flows with continuous infinitesimal covariance is considered and the weak convergence of the corresponding finite-dimensional motions is established. As applications, results for the convergence of the associated pushforward measures and dual flows are given. Similarities between splitting and the Euler-Maruyama scheme yield estimates of the speed of the convergence under additional regularity assumptions.

Transversal families of nonlinear projections and generalizations of Favard length

Projections detect information about the size, geometric arrangement, and dimension of sets. To approach this, one can study the energies of measures supported on a set and the energies for the corresponding pushforward measures on the projection side. For orthogonal projections, quantitative estimates rely on a separation condition: most points are well-differentiated by most projections. It turns out that this idea also applies to a broad class of nonlinear projection-type operators satisfying a \textit{transversality condition}. In this work, we establish that several important classes of nonlinear projections are transversal. This leads to quantitative lower bounds for decay rates for nonlinear variants of Favard length, including Favard curve length (as well as a new generalization to higher dimensions, called Favard surface length) and visibility measurements associated to radial projections. As one application, we provide a simplified proof for the decay rate of the Favard curve length of generations of the four corner Cantor set, first established by Cladek, Davey, and Taylor.

Estimates of the Bergman kernel on Teichmüller space

In this paper, a comparison between the Bergman kernel form and the pushforward measure of the Masur-Veech measure on the Teichmüller space of genus g ≥ 2 g\ge 2 is obtained.

Topological Regularization for Representation Learning via Persistent Homology

Generalization is challenging in small-sample-size regimes with over-parameterized deep neural networks, and a better representation is generally beneficial for generalization. In this paper, we present a novel method for controlling the internal representation of deep neural networks from a topological perspective. Leveraging the power of topology data analysis (TDA), we study the push-forward probability measure induced by the feature extractor, and we formulate a notion of “separation” to characterize a property of this measure in terms of persistent homology for the first time. Moreover, we perform a theoretical analysis of this property and prove that enforcing this property leads to better generalization. To impose this property, we propose a novel weight function to extract topological information, and we introduce a new regularizer including three items to guide the representation learning in a topology-aware manner. Experimental results in the point cloud optimization task show that our method is effective and powerful. Furthermore, results in the image classification task show that our method outperforms the previous methods by a significant margin.

Invariant measures in non-conformal fibered systems with singularities

We study a large class of transformations FT which are only piecewise differentiable on countably many pieces and also non-conformal. There exist random countably generated limit sets JT,ω in the fibers of FT. We prove a Global Volume Lemma implying that the push-forward measures of equilibrium measures from a countable shift space, are exact dimensional on the non-compact global basic set JT. A dimension formula of Ledrappier-Young type is obtained for these measures, by using their Lyapunov exponents and marginal entropies. Then, we study the geometric potentials ψT,s on ΣI, and we prove that the dimensions of the push-forward measures νsω in fibers are independent of ω, and they depend real-analytically on the parameter s from an interval F(T). Moreover, we establish a Variational Principle for dimension in fibers. Our results apply in particular to new types of multidimensional continued fractions.

Constructions and Properties of Quasi Sigma-Algebra in Topological Measure Space

The topological views of a measure space provide deep insights. In this paper, the sigma-set algebraic structure is extended in a Hausdorff topological space based on the locally compactable neighborhood systems without considering strictly (metrized) Borel variety. The null extension gives rise to a quasi sigma-semiring based on sigma-neighborhoods, which are rectifiable in view of Dieudonné measure in n-space. The concepts of symmetric signed measure, uniformly pushforward measure, and its interval-valued Lebesgue variety within a topological measure space are introduced. The symmetric signed measure preserves the total ordering on the real line; however, the collapse of symmetry admits Dieudonné measure within the topological space. The locally constant measures in compact supports in sigma-neighborhood systems are invariant under topological deformation retraction in a simply connected space where the sequence of deformation retractions induces a strongly convergent sequence of measures. Moreover, the extended sigma-structures in an automorphic and isomorphic topological space preserve the properties of uniformly pushforward measure. The Haar-measurable group algebraic structures equivalent to additive integer groups arise under the locally constant and signed measures as long as the topological space is non-compact and the null-extended sigma-neighborhood system admits compact groups. The comparative analyses of the proposed concepts with respect to existing results are outlined.

Discrete-Time Observations of Brownian Motion on Lie Groups and Homogeneous Spaces: Sampling and Metric Estimation

We present schemes for simulating Brownian bridges on complete and connected Lie groups and homogeneous spaces. We use this to construct an estimation scheme for recovering an unknown left- or right-invariant Riemannian metric on the Lie group from samples. We subsequently show how pushing forward the distributions generated by Brownian motions on the group results in distributions on homogeneous spaces that exhibit a non-trivial covariance structure. The pushforward measure gives rise to new non-parametric families of distributions on commonly occurring spaces such as spheres and symmetric positive tensors. We extend the estimation scheme to fit these distributions to homogeneous space-valued data. We demonstrate both the simulation schemes and estimation procedures on Lie groups and homogenous spaces, including SPD(3)=GL+(3)/SO(3) and S2=SO(3)/SO(2).

Topological Sigma-Semiring Separation and Ordered Measures in Noetherian Hyperconvexes

The interplay between topological hyperconvex spaces and sigma-finite measures in such spaces gives rise to a set of analytical observations. This paper introduces the Noetherian class of k-finite k-hyperconvex topological subspaces (NHCs) admitting countable finite covers. A sigma-finite measure is constructed in a sigma-semiring in a NHC under a topological ordering of NHCs. The topological ordering relation maintains the irreflexive and anti-symmetric algebraic properties while retaining the homeomorphism of NHCs. The monotonic measure sequence in a NHC determines the convexity and compactness of topological subspaces. Interestingly, the topological ordering in NHCs in two isomorphic topological spaces induces the corresponding ordering of measures in sigma-semirings. Moreover, the uniform topological measure spaces of NHCs need not always preserve the pushforward measures, and a NHC semiring is functionally separable by a set of inner-measurable functions.

Bayesian topological signal processing

<p style='text-indent:20px;'>Topological data analysis encompasses a broad set of techniques that investigate the shape of data. One of the predominant tools in topological data analysis is persistent homology, which is used to create topological summaries of data called persistence diagrams. Persistent homology offers a novel method for signal analysis. Herein, we aid interpretation of the sublevel set persistence diagrams of signals by 1) showing the effect of frequency and instantaneous amplitude on the persistence diagrams for a family of deterministic signals, and 2) providing a general equation for the probability density of persistence diagrams of random signals via a pushforward measure. We also provide a topologically-motivated, efficiently computable statistical descriptor analogous to the power spectral density for signals based on a generalized Bayesian framework for persistence diagrams. This Bayesian descriptor is shown to be competitive with power spectral densities and continuous wavelet transforms at distinguishing signals with different dynamics in a classification problem with autoregressive signals.</p>

Ledrappier\u2013Young formulae for a family of nonlinear attractors

We study a natural class of invariant measures supported on the attractors of a family of nonlinear, non-conformal iterated function systems introduced by Falconer, Fraser and Lee. These are pushforward quasi-Bernoulli measures, a class which includes the well-known class of Gibbs measures for Hölder continuous potentials. We show that these measures are exact dimensional and that their exact dimensions satisfy a Ledrappier–Young formula.

Curve Based Approximation of Measures on Manifolds by Discrepancy Minimization

The approximation of probability measures on compact metric spaces and in particular on Riemannian manifolds by atomic or empirical ones is a classical task in approximation and complexity theory with a wide range of applications. Instead of point measures we are concerned with the approximation by measures supported on Lipschitz curves. Special attention is paid to push-forward measures of Lebesgue measures on the unit interval by such curves. Using the discrepancy as distance between measures, we prove optimal approximation rates in terms of the curve’s length and Lipschitz constant. Having established the theoretical convergence rates, we are interested in the numerical minimization of the discrepancy between a given probability measure and the set of push-forward measures of Lebesgue measures on the unit interval by Lipschitz curves. We present numerical examples for measures on the 2- and 3-dimensional torus, the 2-sphere, the rotation group on mathbb R^3 and the Grassmannian of all 2-dimensional linear subspaces of {mathbb {R}}^4. Our algorithm of choice is a conjugate gradient method on these manifolds, which incorporates second-order information. For efficient gradient and Hessian evaluations within the algorithm, we approximate the given measures by truncated Fourier series and use fast Fourier transform techniques on these manifolds.

Minimizing Rational Functions: A Hierarchy of Approximations via Pushforward Measures

This paper is concerned with minimizing a sum of rational functions over a compact set of high dimension. Our approach relies on the second Lasserre hierarchy (also known as the upper bounds hierar...

Solving Stochastic Inverse Problems for Property–Structure Linkages Using Data-Consistent Inversion and Machine Learning

Determining process–structure–property linkages is one of the key objectives in material science, and uncertainty quantification plays a critical role in understanding both process–structure and structure–property linkages. In this work, we seek to learn a distribution of microstructure parameters that are consistent in the sense that the forward propagation of this distribution through a crystal plasticity finite element model matches a target distribution on materials properties. This stochastic inversion formulation infers a distribution of acceptable/consistent microstructures, as opposed to a deterministic solution, which expands the range of feasible designs in a probabilistic manner. To solve this stochastic inverse problem, we employ a recently developed uncertainty quantification framework based on push-forward probability measures, which combines techniques from measure theory and Bayes’ rule to define a unique and numerically stable solution. This approach requires making an initial prediction using an initial guess for the distribution on model inputs and solving a stochastic forward problem. To reduce the computational burden in solving both stochastic forward and stochastic inverse problems, we combine this approach with a machine learning Bayesian regression model based on Gaussian processes and demonstrate the proposed methodology on two representative case studies in structure–property linkages.

Near-optimal analysis of Lasserre\u2019s univariate measure-based bounds for multivariate polynomial optimization

We consider a hierarchy of upper approximations for the minimization of a polynomial f over a compact set K subseteq mathbb {R}^n proposed recently by Lasserre (arXiv:1907.097784, 2019). This hierarchy relies on using the push-forward measure of the Lebesgue measure on K by the polynomial f and involves univariate sums of squares of polynomials with growing degrees 2r. Hence it is weaker, but cheaper to compute, than an earlier hierarchy by Lasserre (SIAM Journal on Optimization 21(3), 864–885, 2011), which uses multivariate sums of squares. We show that this new hierarchy converges to the global minimum of f at a rate in O(log ^2 r / r^2) whenever K satisfies a mild geometric condition, which holds, eg., for convex bodies and for compact semialgebraic sets with dense interior. As an application this rate of convergence also applies to the stronger hierarchy based on multivariate sums of squares, which improves and extends earlier convergence results to a wider class of compact sets. Furthermore, we show that our analysis is near-optimal by proving a lower bound on the convergence rate in varOmega (1/r^2) for a class of polynomials on K=[-1,1], obtained by exploiting a connection to orthogonal polynomials.

Data-consistent inversion for stochastic input-to-output maps**The views expressed in the article do not necessarily represent the views of the U.S. Department of Energy or the United States Government. Sandia National Laboratories is a multimission laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC., a wholly owned subsidiary of Honeywell International, Inc., for the U.S. Department of

Data-consistent inversion is a recently developed measure-theoretic framework for solving a stochastic inverse problem involving models of physical systems. The goal is to construct a probability measure on model inputs (i.e., parameters of interest) whose associated push-forward measure matches (i.e., is consistent with) a probability measure on the observable outputs of the model (i.e., quantities of interest). Previous implementations required the map from parameters of interest to quantities of interest to be deterministic. This work generalizes this framework for maps that are stochastic, i.e., contain uncertainties and variation not explainable by variations in uncertain parameters of interest. Generalizations of previous theorems of existence, uniqueness, and stability of the data-consistent solution are provided while new theoretical results address the stability of marginals on parameters of interest. A notable aspect of the algorithmic generalization is the ability to query the solution to generate independent identically distributed samples of the parameters of interest without requiring knowledge of the so-called stochastic parameters. This work therefore extends the applicability of the data-consistent inversion framework to a much wider class of problems. This includes those based on purely experimental and field data where only a subset of conditions are either controllable or can be documented between experiments while the underlying physics, measurement errors, and any additional covariates are either uncertain or not accounted for by the researcher. Numerical examples demonstrate application of this approach to systems with stochastic sources of uncertainties embedded within the modeling of a system and a numerical diagnostic is summarized that is useful for determining if a key assumption is verified among competing choices of stochastic maps.

What do we hear from a drum? A data-consistent approach to quantifying irreducible uncertainty on model inputs by extracting information from correlated model output data

In manufacturing processes, controlling system responses with uncertain system inputs, e.g., due to variations in material parameters of critical system sub-components, is a crucial task for performing reliable quality control and verification & validation (V&V) of system design. As a model for a manufacturing process, we consider the production of drums, that is, thin elastic membranes, whose properties are modeled via Dirichlet Laplacian eigenproblems with uncertain diffusion coefficients. Both a quality control and V&V problem are formulated within a data-consistent framework utilizing push-forward and pullback measures. In both problems, the uncertain diffusion coefficients are parameterized for every instance and the corresponding eigen-information defines correlated data streams. Subsequently, the quantities of interest required in the solution to the data-consistent inverse problems are determined by an a posteriori analysis of these data streams using feature extraction techniques. While the methodology proposed here is quite general, the specific efficacy of the proposed methodology is comprehensively explored in the numerical results for both the quality control and V&V problems associated with the manufacturing of drums.

Optimal experimental design for prediction based on push-forward probability measures

Incorporating experimental data is essential for increasing the credibility of simulation-aided decision making and design. This paper presents a method which uses a computational model to guide the optimal acquisition of experimental data to produce data-informed predictions of quantities of interest (QoI). Many strategies for optimal experimental design (OED) select data that maximize some utility that measures the reduction in uncertainty of uncertain model parameters, for example the expected information gain between prior and posterior distributions of these parameters. In this paper, we seek to maximize the expected information gained from the push-forward of an initial (prior) density to the push-forward of the updated (posterior) density through the parameter-to-prediction map. The formulation presented is based upon the solution of a specific class of stochastic inverse problems which seeks a probability density that is consistent with the model and the data in the sense that the push-forward of this density through the parameter-to-observable map matches a target density on the observable data. While this stochastic inverse problem forms the mathematical basis for our approach, we develop a one-step algorithm, focused on push-forward probability measures, that leverages inference-for-prediction to bypass constructing the solution to the stochastic inverse problem. A number of numerical results are presented to demonstrate the utility of this optimal experimental design for prediction and facilitate comparison of our approach with traditional OED.

Matching strings in encoded sequences

We investigate the length of the longest common substring for encoded sequences and its asymptotic behaviour. The main result is a strong law of large numbers for a re-scaled version of this quantity, which presents an explicit relation with the Rényi entropy of the source. We apply this result to the zero-inflated contamination model and the stochastic scrabble. In the case of dynamical systems, this problem is equivalent to the shortest distance between two observed orbits and its limiting relationship with the correlation dimension of the pushforward measure. An extension to the shortest distance between orbits for random dynamical systems is also provided.

About the rate function in concentration inequalities for suprema of bounded empirical processes

We provide new deviation inequalities in the large deviations bandwidth for suprema of empirical processes indexed by classes of uniformly bounded functions associated with independent and identically distributed random variables. The improvements we get concern the rate function which is, as expected, the Legendre transform of the suprema of the log-Laplace transform of the pushforward measure by the functions of the considered class (up to an additional corrective term). Our approach is based on a decomposition in martingale together with some comparison inequalities.