This paper studies the question of lower bounds on number neurons and examples necessary to program a given task into feedforward neural networks. We introduce notion information complexity network complement that complexity. Neural deals with for resources (numbers neurons) needed by perform within tolerance. Information measures (i.e. examples) about desired input–output function. study interaction two complexities, so building then programming feed-forward nets tasks. show something...

Consider approximating functions based on a finite number of their samples. We show that adaptive algorithms are much more powerful than nonadaptive ones when dealing with piecewise smooth functions. More specifically, let $F_r^1$ be the class scalar $f:[0,T]\to \mathbb {R}$ whose derivatives order up to $r$ continuous at any point except for one unknown singular point. provide an algorithm $\mathcal {A}_n^\textrm {ad}$ uses most $n$ samples $f$ and worst case $L^p$ error ($1\le p<\infty$)...

We present a novel theoretical approach to the analysis of adaptive quadratures and Simpson in particular which leads construction new algorithm for automatic integration. For given function [Formula: see text] with possible endpoint singularities produces an approximation within asymptotically as text]. Moreover, it is optimal among all quadratures, i.e., needs minimal number evaluations obtain text]-approximation runs time proportional

We study the uniform (Chebyshev) approximation of continuous and piecewise r-smooth ($r\ge2$) functions $f:[0,T]\to\mathbb{R}$ with a finite number singular points. The algorithms use only n function values at adaptively or nonadaptively chosen construct nonadaptive algorithm $\mathcal{A}_{r,n}^{\rm non}$ that, for most one point, enjoys best possible convergence rate $n^{-r}$. This is in sharp contrast to results concerning discontinuous functions. For $r\ge3$, this optimal holds asymptotic...

We find the minimal information cost me(∊) of obtaining an ∊–approximation a linear problem, assuming that available consists noisy (perturbed) values I functionals: Perturbations can be absolute or relative, and are assumed to bounded, each bound dependent on consecutive number functional. determine optimal (up constant) functionals, precisions with which they should obtained, as well best algorithm. The results applied problem recovering functions in s variables r continuous derivatives,...

We study the $\omega$-weighted $L^p$ approximation ($1\le p\le\infty$) of piecewise $r$-smooth functions $f:\mathbb{R}\to\mathbb{R}$. Approximations $\mathcal{A}_nf$ are based on $n$ values $f$ at points that can be chosen adaptively. Assuming weight $\omega$ is Riemann integrable any compact interval and asymptotically decreasing, a necessary condition for error to order $n^{-r}$ $\|\omega\|_{L^{1/\gamma}}<\infty$, where $\gamma=r+1/p$. For class $W_r$ globally functions, this also...

Using the Multivariate Decomposition Method (MDM), we develop an efficient algorithm for approximating ∞-variate integral $$\mathcal{I}_{\infty}(f) = \lim\limits_{d\rightarrow \infty} \int\limits_{\mathcal{R}_{+}^{d}}f(x_{1},\ldots,x_{d},0,0,\ldots)\cdot \exp\left(-\sum\limits_{j=1}^{d} x_{j}\right) \mathrm{d} \mathbf{x} $$ a class of functions f that are once differentiable with respect to each variable. MDM requires algorithms d-variate versions problem. Such provided by Smolyak's...