Topic
Smoothness
About: Smoothness is a research topic. Over the lifetime, 803 publications have been published within this topic receiving 20537 citations. The topic is also known as: infinitely differentiable function & smooth map.
Papers published on a yearly basis
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Papers
TL;DR: In this article, the authors proposed a smoothness adaptive thresholding procedure, called SureShrink, which is adaptive to the Stein unbiased estimate of risk (sure) for threshold estimates and is near minimax simultaneously over a whole interval of the Besov scale; the size of this interval depends on the choice of mother wavelet.
Abstract: We attempt to recover a function of unknown smoothness from noisy sampled data. We introduce a procedure, SureShrink, that suppresses noise by thresholding the empirical wavelet coefficients. The thresholding is adaptive: A threshold level is assigned to each dyadic resolution level by the principle of minimizing the Stein unbiased estimate of risk (Sure) for threshold estimates. The computational effort of the overall procedure is order N · log(N) as a function of the sample size N. SureShrink is smoothness adaptive: If the unknown function contains jumps, then the reconstruction (essentially) does also; if the unknown function has a smooth piece, then the reconstruction is (essentially) as smooth as the mother wavelet will allow. The procedure is in a sense optimally smoothness adaptive: It is near minimax simultaneously over a whole interval of the Besov scale; the size of this interval depends on the choice of mother wavelet. We know from a previous paper by the authors that traditional smoot...
5,021 citations
TL;DR: In this article, the smoothness of time functions and slicings by Cauchy hypersurfaces has been studied in the context of Lorentzian geometries, and it has been shown that any globally hyperbolic spacetime admits a smooth time function.
Abstract: The folk questions in Lorentzian Geometry which concerns the smoothness of time functions and slicings by Cauchy hypersurfaces, are solved by giving simple proofs of: (a) any globally hyperbolic spacetime (M, g) admits a smooth time function Open image in new window whose levels are spacelike Cauchy hyperfurfaces and, thus, also a smooth global splitting Open image in new window if a spacetime M admits a (continuous) time function t then it admits a smooth (time) function Open image in new window with timelike gradient Open image in new window on all M.
516 citations
TL;DR: In this paper, the authors consider a number of properties of space-time covariance functions and how these relate to the spatial-temporal interactions of the process and obtain a parametric class of spectral densities whose corresponding space time covariance function are infinitely differentiable away from the origin and allow for essentially arbitrary and possibly different degrees of smoothness for the process in space and time.
Abstract: This work considers a number of properties of space–time covariance functions and how these relate to the spatial-temporal interactions of the process. First, it examines how the smoothness away from the origin of a space–time covariance function affects, for example, temporal correlations of spatial differences. Models that are not smoother away from the origin than they are at the origin, such as separable models, have a kind of discontinuity to certain correlations that one might wish to avoid in some circumstances. Smoothness away from the origin of a covariance function is shown to follow from the corresponding spectral density having derivatives with finite moments. These results are used to obtain a parametric class of spectral densities whose corresponding space–time covariance functions are infinitely differentiable away from the origin and that allows for essentially arbitrary and possibly different degrees of smoothness for the process in space and time. Second, this work considers models that ...
514 citations
TL;DR: In this article , a comprehensive overview and survey is presented for activation functions in neural networks for deep learning, including Logistic Sigmoid, Tanh, ReLU, ELU, Swish and Mish.
479 citations
TL;DR: In this article, the authors provide criteria for positive definiteness of radial functions with compact support and derive a series of positive definite and compactly supported radial functions, which will be very useful in applications.
Abstract: We provide criteria for positive definiteness of radial functions with compact support. Based on these criteria we will produce a series of positive definite and compactly supported radial functions, which will be very useful in applications. The simplest ones arecut-off polynomials, which consist of a single polynomial piece on [0, 1] and vanish on [1, ∞). More precisely, for any given dimensionn and prescribedCk smoothness, there is a function inCk(ℝn), which is a positive definite radial function with compact support and is a cut-off polynomial as a function of Euclidean distance. Another example is derived from odd-degreeB-splines.
476 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2026 | 6 |
| 2025 | 388 |
| 2024 | 359 |
| 2023 | 878 |
| 2022 | 1,623 |
| 2021 | 131 |