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In this work, we further develop multigoal-oriented a posteriori error estimation with two objectives in mind. First, we formulate goal-oriented mesh adaptivity for multiple functionals of interest for nonlinear problems in which both the…
This work is concerned with the study of the adaptivity properties of nonparametric regression estimators over the $d$-dimensional sphere within the global thresholding framework. The estimators are constructed by means of a form of…
This paper investigates achievable information rates and error exponents of mismatched decoding when the channel belongs to the class of channels that are close to the decoding metric in terms of relative entropy. For both discrete- and…
This work considers the problem of estimating the parameters of negative mixture models, i.e. mixture models that possibly involve negative weights. The contributions of this paper are as follows. (i) We show that every rational probability…
Reference metrics are used to define the differential structure on multicube representations of manifolds, i.e., they provide a simple and practical way to define what it means globally for tensor fields and their derivatives to be…
Richardson extrapolation is applied to a simple first-order upwind difference scheme for the approximation of solutions of singularly perturbed convection-diffusion problems in one dimension. Robust a posteriori error bounds are derived for…
We consider conformal defects joining two conformal field theories along a line. We define two new quantities associated to such defects in terms of expectation values of the stress tensors and we propose them as measures of the…
Short pitch deformed helix ferroelectric liquid crystals have numerous applications as active materials in displays, optical telemetry and biomedical devices. In this paper, we derive convenient analytical formulas to calculate the…
Given a space it is easy to obtain the system of geodesic equations on it. In this paper the inverse problem of reconstructing the space from the geodesic equations is addressed. A procedure is developed for obtaining the metric tensor from…
We consider the problem of decomposing higher-order moment tensors, i.e., the sum of symmetric outer products of data vectors. Such a decomposition can be used to estimate the means in a Gaussian mixture model and for other applications in…
From Crofton's formula for Minkowski tensors we derive stereological estimators of translation invariant surface tensors of convex bodies in the n-dimensional Euclidean space. The estimators are based on one-dimensional linear sections. In…
In this paper, based on the combination of tensor neural network and a posteriori error estimator, a novel type of machine learning method is proposed to solve high-dimensional boundary value problems with homogeneous and non-homogeneous…
It is shown in this article that if the Einstein Equivalence Principle is valid on a particular metric theory of gravitation in a spherically symmetric space-time, then the time metric component is not equal to the negative of the inverse…
In this paper, we propose a coupled tensor norm regularization that could enable the model output feature and the data input to lie in a low-dimensional manifold, which helps us to reduce overfitting. We show this regularization term is…
We establish the rectifiability of measures satisfying a linear PDE constraint. The obtained rectifiability dimensions are optimal for many usual PDE operators, including all first-order systems and all second-order scalar operators. In…
In [1] it is shown that recurrent neural networks (RNNs) can learn - in a metric entropy optimal manner - discrete time, linear time-invariant (LTI) systems. This is effected by comparing the number of bits needed to encode the…
We study the problem of testing whether two tensors in $\mathbb{R}^\ell\otimes \mathbb{R}^m\otimes \mathbb{R}^n$ are isomorphic under the natural action of orthogonal groups $\textbf{O}(\ell, \mathbb{R})\times\textbf{O}(m,…
Tensor completion and tensor decomposition are important problems in many domains. In this work, we leverage the connection between these problems to learn a distance metric that improves both decomposition and completion. We show that the…
Dimensionality reduction is an effective method for learning high-dimensional data, which can provide better understanding of decision boundaries in human-readable low-dimensional subspace. Linear methods, such as principal component…
This work explores the behaviour of demagnetizing tensors for general ellipsoids under arbitrary rotations in homogeneous magnetic fields. The work is motivated by the concerns in magnetic resonance imaging safety and their practical…