Statistical assumption

Statistics, like all mathematical disciplines, does not infer valid conclusions from nothing. Inferring interesting conclusions about real statistical populations almost always requires some background assumptions. Those assumptions must be made carefully, because incorrect assumptions can generate wildly inaccurate conclusions. Here are some examples of statistical assumptions: *Independence of observations from each other (this assumption is an especially common error). *Independence of observational error from potential confounding effects. *Exact or approximate normality of observations (or errors). *Linearity of graded responses to quantitative stimuli, e.g., in linear regression. Classes of assumptions There are two approaches to statistical inference: model-based inference and design-based inference. Both approaches rely on some statistical model to represent the data-generating process. In the model-based approach, the model is taken to be initially unknown, and on

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Normal approximation of the solution to the stochastic heat equation with Levy noise

Carsten Hao Ye Chong

Given a sequence L & x2d9;epsilon of Levy noises, we derive necessary and sufficient conditions in terms of their variances sigma 2(epsilon) such that the solution to the stochastic heat equation with noise sigma(epsilon)-1L & x2d9;epsilon converges in law to the solution to the same equation with Gaussian noise. Our results apply to both equations with additive and multiplicative noise and hence lift the findings of Asmussen and Rosinski (J Appl Probab 38(2):482-493, 2001), Cohen and Rosinski (Bernoulli 13(1):195-210, 2007) for finite-dimensional Levy processes to the infinite-dimensional setting without making distributional assumptions on the solutions such as infinite divisibility. One important ingredient of our proof is to characterize the solution to the limit equation by a sequence of martingale problems. To this end, it is crucial to view the solution processes both as random fields and as cadlag processes with values in a Sobolev space of negative real order.

SPRINGER2020

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Efficiently Learning Fourier Sparse Set Functions

Mikhail Kapralov, Andreas Krause, Amir Zandieh

Learning set functions is a key challenge arising in many domains, ranging from sketching graphs to black-box optimization with discrete parameters. In this paper we consider the problem of efficiently learning set functions that are defined over a ground set of size n and that are sparse (say k-sparse) in the Fourier domain. This is a wide class, that includes graph and hypergraph cut functions, decision trees and more. Our central contribution is the first algorithm that allows learning functions whose Fourier support only contains low degree (say degree d = o(n)) polynomials using O(kd log n) sample complexity and runtime O(kn log(2) k log n log d). This implies that sparse graphs with k edges can, for the first time, be learned from O(k log n) observations of cut values and in linear time in the number of vertices. Our algorithm can also efficiently learn (sums of) decision trees of small depth. The algorithm exploits techniques from the sparse Fourier transform literature and is easily implementable. Lastly, we also develop an efficient robust version of our algorithm and prove l(2)/l(2) approximation guarantees without any statistical assumptions on the noise.

NEURAL INFORMATION PROCESSING SYSTEMS (NIPS)2019

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Structure preserving model reduction of parametric Hamiltonian systems

Jan Sickmann Hesthaven, Babak Maboudi Afkham

While reduced-order models (ROMs) are popular for approximately solving large systems of differential equations, the stability of reduced models over long-time integration remains an open question. We present a greedy approach for ROM generation of parametric Hamiltonian systems which captures the symplectic structure of Hamiltonian systems to ensure stability of the reduced model. Through the greedy selection of basis vectors, two new vectors are added at each iteration to the set of basis vectors to increase the overall accuracy of the reduce basis. We used the error in the Hamiltonian function due to model reduction, as an error indicator to search the parameter space and find the next best basis vectors. We show that the greedy algorithm converges with exponential rate, under natural assumptions on the set of all solutions of the Hamiltonian system under variation of the parameters. Moreover, we demonstrate that combining the greedy basis with the discrete empirical interpolation method also preserves the symplectic structure. This enables the reduction of computational cost for nonlinear Hamiltonian systems. The efficiency, accuracy, and stability of this model reduction technique is illustrated through simulations of the parametric wave equation and the parametric Schroedinger equation.

2017

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