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Nonparametric regression using deep neural networks with ReLU activation function
- J. Schmidt-Hieber
- Computer ScienceAnnals of Statistics
- 22 August 2017
The theory suggests that for nonparametric regression, scaling the network depth with the sample size is natural and the analysis gives some insights into why multilayer feedforward neural networks perform well in practice.
BAYESIAN LINEAR REGRESSION WITH SPARSE PRIORS
Under compatibility conditions on the design matrix, the posterior distribution is shown to contract at the optimal rate for recovery of the unknown sparse vector, and to give optimal prediction of the response vector.
On adaptive posterior concentration rates
tion of Holder balls and that moreover achieve our lower bound. We analyse the consequences in terms of asymptotic behaviour of poste- rior credible balls as well as frequentist minimax adaptive…
A comparison of deep networks with ReLU activation function and linear spline-type methods
Deep ReLU network approximation of functions on a manifold
- J. Schmidt-Hieber
- Computer Science, MathematicsArXiv
- 2 August 2019
This work studies a regression problem with inputs on a $d^*$-dimensional manifold that is embedded into a space with potentially much larger ambient dimension, and derives statistical convergence rates for the estimator minimizing the empirical risk over all possible choices of bounded network parameters.
Lower bounds for volatility estimation in microstructure noise models
In this paper we derive lower bounds in minimax sense for estimation of the instantaneous volatility if the diffusion type part cannot be observed directly but under some additional Gaussian noise.…
Conditions for Posterior Contraction in the Sparse Normal Means Problem
The first Bayesian results for the sparse normal means problem were proven for spike-and-slab priors. However, these priors are less convenient from a computational point of view. In the meanwhile, a…
Multiscale methods for shape constraints in deconvolution: Confidence statements for qualitative features.
We derive multiscale statistics for deconvolution in order to detect qualitative features of the unknown density. An important example covered within this framework is to test for local monotonicity…
Adaptive wavelet estimation of the diffusion coefficient under additive error measurements
We study nonparametric estimation of the diffusion coefficient from discrete data, when the observations are blurred by additional noise. Such issues have been developed over the last 10 years in…
Nonparametric Estimation of the Volatility Function in a High-Frequency Model corrupted by Noise
We consider the models Yi;n = R i=n 0 (s)dWs + (i=n) i;n, and ~ Yi;n = (i=n)Wi=n + (i=n) i;n, i = 1;:::;n, where (Wt) t2[0;1] denotes a standard Brownian motion and i;n are centered i.i.d. random…