Related papers: Covering numbers for bounded variation functions
Lower bounds for variances are often needed to derive central limit theorems. In this paper, we establish a lower bound for the variance of Poisson functionals that uses the difference operator of Malliavin calculus. Poisson functionals,…
Given a bounded domain, we deal with the problem of estimating the distance function from the internal points of the domain to the boundary of the domain. Convolutional and differential distance estimation schemes are considered and, for…
In this note we prove optimal inequalities for bounded functions in terms of their deviation from their mean. These results extend and generalize some known inequalities due to Thong (2011) and Perfetti (2011)
We present and study approximate notions of dimensional and margin complexity, which correspond to the minimal dimension or norm of an embedding required to approximate, rather then exactly represent, a given hypothesis class. We show that…
Avikainen showed that, for any $p,q \in [1,\infty)$, and any function $f$ of bounded variation in $\mathbb{R}$, it holds that $\mathbb{E}[|f(X)-f(\widehat{X})|^{q}] \leq C(p,q) \mathbb{E}[|X-\widehat{X}|^{p}]^{\frac{1}{p+1}}$, where $X$ is…
We suggest an upper bound on binomial coefficients that holds over the entire parameter range and whose form repeats the form of the de Moivre-Laplace approximation of the symmetric binomial distribution. Using the bound, we estimate the…
The first-order Euler-Maclaurin formula relates the sum of the values of a smooth function on an interval of integers with its integral on the same interval on $\mathbb R$. We formulate here the analogue for functions that are just of…
We provide some new estimates for distances in harmonic function spaces of several variables related to mixed norm spaces.Some of them extend previously known assertions in this direction in the unit ball and upperhalfspace.
We obtain bounds for probabilities of deviations of the truncated variation functional of fractional Brownian motions (fBm) of any Hurst index $H \in (0,1)$ from their expected values. Obtained bounds are optimal for large values of…
We introduce the concept of functions of locally bounded variation on abstract Wiener spaces and study their properties. Some nontrivial examples and applications to stochastic analysis are also discussed.
We construct a Lipschitz truncation which approximates functions of bounded variation in the area-strict metric. The Lipschitz truncation changes the original function only on a small set similar to Lusin's theorem. Previous results could…
We consider the question of what functions can be captured by ReLU networks with an unbounded number of units (infinite width), but where the overall network Euclidean norm (sum of squares of all weights in the system, except for an…
In this paper, we define Euclidean minima for function fields and give some bound for this invariant. We furthermore show that the results are analogous to those obtained in the number field case.
In the setting of a metric space that is equipped with a doubling measure and supports a Poincar\'e inequality, we show that the total variation of functions of bounded variation is lower semicontinuous with respect to $L^1$-convergence in…
Computing explicitly the {\epsilon}-subdifferential of a proper function amounts to computing the level set of a convex function namely the conjugate minus a linear function. The resulting theoretical algorithm is applied to the the class…
Extending the notion of bounded variation, a function $u \in L_c^1(\mathbb R^n)$ is of bounded fractional variation with respect to some exponent $\alpha$ if there is a finite constant $C \geq 0$ such that the estimate \[ \biggl|\int u(x)…
We study the random sampling of band-limited functions of several variables. If a bandlimited function with bandwidth one has its essential support on a cube of volume $R^d$, then $\cO (R^d \log R^d)$ random samples suffice to approximate…
In this paper we show that the lengths of the approximating processes in epsilon substitution method are calculable by ordinal recursions in an optimal way.
We study the properties of the set where a generalized function of bounded variation has infinite approximate limit, highlighting in this way the main geometric difference with functions of bounded variation. To this aim we prove a new…
Estimates are constructed for the deviation of the concentration functions of sums of independent random variables with finite variances from the folded normal distribution function without any assumptions concerning the existence of the…