Piecewise

About: Piecewise is a research topic. Over the lifetime, 21064 publications have been published within this topic receiving 432096 citations. The topic is also known as: piecewise-defined function & hybrid function.

Exponential ergodicity for Markov processes with random switching

Abstract: We study a Markov process with two components: the first component evolves according to one of finitely many underlying Markovian dynamics, with a choice of dynamics that changes at the jump times of the second component. The second component is discrete and its jump rates may depend on the position of the whole process. Under regularity assumptions on the jump rates and Wasserstein contraction conditions for the underlying dynamics, we provide a concrete criterion for the convergence to equilibrium in terms of Wasserstein distance. The proof is based on a coupling argument and a weak form of the Harris theorem. In particular, we obtain exponential ergodicity in situations which do not verify any hypoellipticity assumption, but are not uniformly contracting either. We also obtain a bound in total variation distance under a suitable regularising assumption. Some examples are given to illustrate our result, including a class of piecewise deterministic Markov processes.

Nonlinear Approximation and (Deep) ReLU Networks.

Abstract: This article is concerned with the approximation and expressive powers of deep neural networks. This is an active research area currently producing many interesting papers. The results most commonly found in the literature prove that neural networks approximate functions with classical smoothness to the same accuracy as classical linear methods of approximation, e.g. approximation by polynomials or by piecewise polynomials on prescribed partitions. However, approximation by neural networks depending on n parameters is a form of nonlinear approximation and as such should be compared with other nonlinear methods such as variable knot splines or n-term approximation from dictionaries. The performance of neural networks in targeted applications such as machine learning indicate that they actually possess even greater approximation power than these traditional methods of nonlinear approximation. The main results of this article prove that this is indeed the case. This is done by exhibiting large classes of functions which can be efficiently captured by neural networks where classical nonlinear methods fall short of the task. The present article purposefully limits itself to studying the approximation of univariate functions by ReLU networks. Many generalizations to functions of several variables and other activation functions can be envisioned. However, even in this simplest of settings considered here, a theory that completely quantifies the approximation power of neural networks is still lacking.

L-Splines

Abstract: In this paper, we study the problem of unique interpolation and approximation by a class of spline functions,L-splines, containing as special cases the deficient and generalized spline functions ofAhlberg, Nilson, andWalsh [3, 5, 6], the Chebyshevian spline functions ofKarlin andZiegler [27], and the piecewise Hermite polynomial functions, as considered in [17]. We first give sufficient conditions for unique interpolation byL-spline functions in Section 2. Then, we obtain newL ? andL 2 error estimates for interpolation byL-splines in Section 4, and show that these error estimates are, in a certain sense, sharp. In addition, we make a similar study for theg-splines ofSchoenberg, cf. [44, 3], in Section 5. In Section 6, an application of these new error estimates is made to the analysis of the error made in the use of finite dimensional subspaces ofL-splines andg-splines. in the Rayleigh-Ritz procedure for the class of nonlinear two-point boundary value problems studied in [17]. Because of the rapid growth of the number of papers devoted to or connected with the topic of splines, we believe that a compilation of papers on splines for the reader's use is desirable, and such a list is found in the References at the end of this paper.