scholarly journals Deep Network with Approximation Error Being Reciprocal of Width to Power of Square Root of Depth

2021 ◽  
pp. 1-32
Author(s):  
Zuowei Shen ◽  
Haizhao Yang ◽  
Shijun Zhang
Keyword(s):  
Modulus Of Continuity ◽  
Approximation Error ◽  
Activation Function ◽  
Continuous Functions ◽  
Approximation Rate ◽  
Hölder Continuous ◽  
Arbitrary Continuous Function ◽  
New Class ◽  
Hölder Continuous Functions ◽  
Approximation Power

A new network with super-approximation power is introduced. This network is built with Floor ([Formula: see text]) or ReLU ([Formula: see text]) activation function in each neuron; hence, we call such networks Floor-ReLU networks. For any hyperparameters [Formula: see text] and [Formula: see text], we show that Floor-ReLU networks with width [Formula: see text] and depth [Formula: see text] can uniformly approximate a Hölder function [Formula: see text] on [Formula: see text] with an approximation error [Formula: see text], where [Formula: see text] and [Formula: see text] are the Hölder order and constant, respectively. More generally for an arbitrary continuous function [Formula: see text] on [Formula: see text] with a modulus of continuity [Formula: see text], the constructive approximation rate is [Formula: see text]. As a consequence, this new class of networks overcomes the curse of dimensionality in approximation power when the variation of [Formula: see text] as [Formula: see text] is moderate (e.g., [Formula: see text] for Hölder continuous functions), since the major term to be considered in our approximation rate is essentially [Formula: see text] times a function of [Formula: see text] and [Formula: see text] independent of [Formula: see text] within the modulus of continuity.

On a quantitative theory of limits: Estimating the speed of convergence

2020 ◽  
Vol 23 (4) ◽  
pp. 1013-1024
Author(s):  
Renato Spigler
Keyword(s):  
Modulus Of Continuity ◽  
Fractional Derivatives ◽  
Applied Mathematics ◽  
Continuous Functions ◽  
Speed Of Convergence ◽  
Hölder Continuous ◽  
Lipschitz Continuous ◽  
Quantitative Theory ◽  
Hölder Continuous Functions ◽  
Definition Of

AbstractThe classical “ε-δ” definition of limits is of little use to quantitative purposes, as is needed, for instance, for computational and applied mathematics. Things change whenever a realistic and computable estimate of the function δ(ε) is available. This may be the case for Lipschitz continuous and Hölder continuous functions, or more generally for functions admitting of a modulus of continuity. This, provided that estimates for first derivatives, fractional derivatives, or the modulus of continuity might be obtained. Some examples are given.

scholarly journals Continuous solutions to two iterative functional equations

2021 ◽  
Author(s):  
Karol Baron
Keyword(s):  
Probability Measure ◽  
Functional Equations ◽  
Continuous Functions ◽  
Continuous Solutions ◽  
Hölder Continuous ◽  
Hölder Continuous Functions

AbstractBased on iteration of random-valued functions we study the problem of solvability in the class of continuous and Hölder continuous functions $$\varphi $$ φ of the equations $$\begin{aligned} \varphi (x)=F(x)-\int _{\Omega }\varphi \big (f(x,\omega )\big )P(d\omega ),\\ \varphi (x)=F(x)+\int _{\Omega }\varphi \big (f(x,\omega )\big )P(d\omega ), \end{aligned}$$ φ ( x ) = F ( x ) - ∫ Ω φ ( f ( x , ω ) ) P ( d ω ) , φ ( x ) = F ( x ) + ∫ Ω φ ( f ( x , ω ) ) P ( d ω ) , where P is a probability measure on a $$\sigma $$ σ -algebra of subsets of $$\Omega $$ Ω .

scholarly journals Yang–Mills Measure on the Two-Dimensional Torus as a Random Distribution

2019 ◽  
Vol 372 (3) ◽  
pp. 1027-1058
Author(s):  
Ilya Chevyrev
Keyword(s):  
Random Distribution ◽  
Random Variable ◽  
Continuous Functions ◽  
Small Scale ◽  
Dimensional Torus ◽  
Hölder Continuous ◽  
Line Segments ◽  
Yang Mills ◽  
Hölder Continuous Functions ◽  
On Line

Abstract We introduce a space of distributional 1-forms $$\Omega ^1_\alpha $$Ωα1 on the torus $$\mathbf {T}^2$$T2 for which holonomies along axis paths are well-defined and induce Hölder continuous functions on line segments. We show that there exists an $$\Omega ^1_\alpha $$Ωα1-valued random variable A for which Wilson loop observables of axis paths coincide in law with the corresponding observables under the Yang–Mills measure in the sense of Lévy (Mem Am Math Soc 166(790), 2003). It holds furthermore that $$\Omega ^1_\alpha $$Ωα1 embeds into the Hölder–Besov space $$\mathcal {C}^{\alpha -1}$$Cα-1 for all $$\alpha \in (0,1)$$α∈(0,1), so that A has the correct small scale regularity expected from perturbation theory. Our method is based on a Landau-type gauge applied to lattice approximations.

scholarly journals Moduli of continuity, functional spaces,\break and elliptic boundary value problems. The full regularity spaces Cα0,λ(Ω̅)

2018 ◽  
Vol 7 (1) ◽  
pp. 15-34 ◽  
Author(s):  
Hugo Beirão da Veiga
Keyword(s):  
Modulus Of Continuity ◽  
Elliptic Boundary ◽  
Continuous Functions ◽  
Second Order ◽  
Hölder Spaces ◽  
Elliptic Boundary Value ◽  
New Class ◽  
Holder Spaces ◽  
Derivatives Of ◽  
Full Regularity

AbstractLet {\boldsymbol{L}} be a second order uniformly elliptic operator, and consider the equation {\boldsymbol{L}u=f} under the boundary condition {u=0}. We assume data f in generical subspaces of continuous functions {D_{\overline{\omega}}} characterized by a given modulus of continuity{\overline{\omega}(r)}, and show that the second order derivatives of the solution u belong to functional spaces {D_{\widehat{\omega}}}, characterized by a modulus of continuity{\widehat{\omega}(r)} expressed in terms of {\overline{\omega}(r)}. Results are optimal. In some cases, as for Hölder spaces, {D_{\widehat{\omega}}=D_{\overline{\omega}}}. In this case we say that full regularity occurs. In particular, full regularity occurs for the new class of functional spaces {C^{0,\lambda}_{\alpha}(\overline{\Omega})} which includes, as a particular case, the classical Hölder spaces {C^{0,\lambda}(\overline{\Omega})=C^{0,\lambda}_{0}(\overline{\Omega})}. Few words, concerning the possibility of generalizations and applications to non-linear problems, are expended at the end of the introduction and also in the last section.

scholarly journals NONCLASSICAL SPECTRAL ASYMPTOTICS AND DIXMIER TRACES: FROM CIRCLES TO CONTACT MANIFOLDS

2017 ◽  
Vol 5 ◽  
Author(s):  
HEIKO GIMPERLEIN ◽  
MAGNUS GOFFENG
Keyword(s):  
Multiplication Operator ◽  
Continuous Functions ◽  
Spectral Asymptotics ◽  
Hölder Continuous ◽  
Contact Manifolds ◽  
Integral Formulas ◽  
Weyl Law ◽  
Wide Range ◽  
Hölder Continuous Functions ◽  
Geometric Origin

We consider the spectral behavior and noncommutative geometry of commutators$[P,f]$, where$P$is an operator of order 0 with geometric origin and$f$a multiplication operator by a function. When$f$is Hölder continuous, the spectral asymptotics is governed by singularities. We study precise spectral asymptotics through the computation of Dixmier traces; such computations have only been considered in less singular settings. Even though a Weyl law fails for these operators, and no pseudodifferential calculus is available, variations of Connes’ residue trace theorem and related integral formulas continue to hold. On the circle, a large class of nonmeasurable Hankel operators is obtained from Hölder continuous functions$f$, displaying a wide range of nonclassical spectral asymptotics beyond the Weyl law. The results extend from Riemannian manifolds to contact manifolds and noncommutative tori.

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