Related papers: Strong Divergence for System Approximations

The reduction of dynamical systems has a rich history, with many important applications related to stability, control and verification. Reduction of nonlinear systems is typically performed in an exact manner - as is the case with…

Deep learning models lack shift invariance, making them sensitive to input shifts that cause changes in output. While recent techniques seek to address this for images, our findings show that these approaches fail to provide…

We study the fixed design segmented regression problem: Given noisy samples from a piecewise linear function $f$, we want to recover $f$ up to a desired accuracy in mean-squared error. Previous rigorous approaches for this problem rely on…

In the first part we have shown that, for $L_2$-approximation of functions from a separable Hilbert space in the worst-case setting, linear algorithms based on function values are almost as powerful as arbitrary linear algorithms if the…

We prove an abstract form of the strong convergence of the Halpern-type and Tikhonov-type proximal point algorithms in CAT(0) spaces. In addition, we derive uniform and computable rates of metastability (in the sense of Tao) for these…

Non-convex sampling is a key challenge in machine learning, central to non-convex optimization in deep learning as well as to approximate probabilistic inference. Despite its significance, theoretically there remain many important…

Understanding the limiting behavior of eigenvalues of random matrices is the central problem of random matrix theory. Classical limit results are known for many models, and there has been significant recent progress in obtaining more…

Positive definite kernels and their associated Reproducing Kernel Hilbert Spaces provide a mathematically compelling and practically competitive framework for learning from data. In this paper we take the approximation theory point of view…

Random features is a powerful universal function approximator that inherits the theoretical rigor of kernel methods and can scale up to modern learning tasks. This paper views uncertain system models as unknown or uncertain smooth functions…

We consider approximating analytic functions on the interval $[-1,1]$ from their values at a set of $m+1$ equispaced nodes. A result of Platte, Trefethen \& Kuijlaars states that fast and stable approximation from equispaced samples is…

This paper proposes a novel reaction-diffusion system approximation tailored for singular diffusion problems, typified by the fast diffusion equation. While such approximation methods have been successfully applied to degenerate parabolic…

Strong invariance principles describe the error term of a Brownian approximation of the partial sums of a stochastic process. While these strong approximation results have many applications, the results for continuous-time settings have…

The conditional distribution of the next outcome given the infinite past of a stationary process can be inferred from finite but growing segments of the past. Several schemes are known for constructing pointwise consistent estimates, but…

In high-dimensional data, many sparse regression methods have been proposed. However, they may not be robust against outliers. Recently, the use of density power weight has been studied for robust parameter estimation and the corresponding…

Let $H$ be an infinite-dimensional separable Hilbert space and let $(X,d,\mu)$ be a metric measure space satisfying the doubling and upper Alhfors regularity conditions at small scale. We prove that every bounded continuous tight frame…

Paley-Wiener theorem for frames for Hilbert spaces, Banach frames, Schauder frames and atomic decompositions for Banach spaces are known. In this paper, we derive Paley-Wiener theorem for p-approximate Schauder frames for separable Banach…

We provide a comprehensive study of interrelations between different measures of smoothness of functions on various domains and smoothness properties of approximation processes. Two general approaches to this problem have been developed:…

Given an arbitrary finite set of data F= {f_1,..., f_m} in L2(Rd) we prove the existence and show how to construct a "small shift invariant space" that is "closest" to the data F over certain class of closed subspaces of L2(Rd). The…

Suppose that a continuous-time linear infinite-dimensional system with a static state-feedback controller is strongly stable. We address the following question: If we convert the continuous-time controller to a sampled-data controller by…

We consider the problem of computing the maximal invariant set of discrete-time linear systems subject to a class of non-convex constraints that admit quadratic relaxations. These non-convex constraints include semialgebraic sets and other…