Estimation of a regression function on a manifold by fully connected deep neural networks

• URL: http://arxiv.org/abs/2107.09532v1
• Date: Tue, 20 Jul 2021 14:43:59 GMT
• Title: Estimation of a regression function on a manifold by fully connected deep neural networks
• Authors: Michael Kohler, Sophie Langer and Ulrich Reif
• Abstract summary: The rate of convergence of least squares estimates based on fully connected spaces of deep neural networks with ReLU activation function is analyzed. It is shown that in case that the distribution of the predictor variable is concentrated on a manifold, these estimates achieve a rate of convergence which depends on the dimension of the manifold and not on the number of components of the predictor variable.
• Score: 6.058868817939519
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Estimation of a regression function from independent and identically distributed data is considered. The $L_2$ error with integration with respect to the distribution of the predictor variable is used as the error criterion. The rate of convergence of least squares estimates based on fully connected spaces of deep neural networks with ReLU activation function is analyzed for smooth regression functions. It is shown that in case that the distribution of the predictor variable is concentrated on a manifold, these estimates achieve a rate of convergence which depends on the dimension of the manifold and not on the number of components of the predictor variable.

Related papers

• Semiparametric conformal prediction [79.6147286161434]
  Risk-sensitive applications require well-calibrated prediction sets over multiple, potentially correlated target variables. We treat the scores as random vectors and aim to construct the prediction set accounting for their joint correlation structure. We report desired coverage and competitive efficiency on a range of real-world regression problems.
  arXiv  Detail & Related papers  (2024-11-04T14:29:02Z)
• Multivariate root-n-consistent smoothing parameter free matching estimators and estimators of inverse density weighted expectations [51.000851088730684]
  We develop novel modifications of nearest-neighbor and matching estimators which converge at the parametric $sqrt n $-rate. We stress that our estimators do not involve nonparametric function estimators and in particular do not rely on sample-size dependent parameters smoothing.
  arXiv  Detail & Related papers  (2024-07-11T13:28:34Z)
• Semi-Supervised Deep Sobolev Regression: Estimation, Variable Selection and Beyond [3.782392436834913]
  We propose SDORE, a semi-supervised deep Sobolev regressor, for the nonparametric estimation of the underlying regression function and its gradient. We conduct a comprehensive analysis of the convergence rates of SDORE and establish a minimax optimal rate for the regression function. We also derive a convergence rate for the associated plug-in gradient estimator, even in the presence of significant domain shift.
  arXiv  Detail & Related papers  (2024-01-09T13:10:30Z)
• Conformal inference for regression on Riemannian Manifolds [49.7719149179179]
  We investigate prediction sets for regression scenarios when the response variable, denoted by $Y$, resides in a manifold, and the covariable, denoted by X, lies in Euclidean space. We prove the almost sure convergence of the empirical version of these regions on the manifold to their population counterparts.
  arXiv  Detail & Related papers  (2023-10-12T10:56:25Z)
• Structured Radial Basis Function Network: Modelling Diversity for Multiple Hypotheses Prediction [51.82628081279621]
  Multi-modal regression is important in forecasting nonstationary processes or with a complex mixture of distributions. A Structured Radial Basis Function Network is presented as an ensemble of multiple hypotheses predictors for regression problems. It is proved that this structured model can efficiently interpolate this tessellation and approximate the multiple hypotheses target distribution.
  arXiv  Detail & Related papers  (2023-09-02T01:27:53Z)
• Kernel-based off-policy estimation without overlap: Instance optimality beyond semiparametric efficiency [53.90687548731265]
  We study optimal procedures for estimating a linear functional based on observational data. For any convex and symmetric function class $mathcalF$, we derive a non-asymptotic local minimax bound on the mean-squared error.
  arXiv  Detail & Related papers  (2023-01-16T02:57:37Z)
• Learning Distributions by Generative Adversarial Networks: Approximation and Generalization [0.6768558752130311]
  We study how well generative adversarial networks learn from finite samples by analyzing the convergence rates of these models. Our analysis is based on a new inequality oracle that decomposes the estimation error of GAN into the discriminator and generator approximation errors. For generator approximation error, we show that neural network can approximately transform a low-dimensional source distribution to a high-dimensional target distribution.
  arXiv  Detail & Related papers  (2022-05-25T09:26:17Z)
• Robust Estimation for Nonparametric Families via Generative Adversarial Networks [92.64483100338724]
  We provide a framework for designing Generative Adversarial Networks (GANs) to solve high dimensional robust statistics problems. Our work extend these to robust mean estimation, second moment estimation, and robust linear regression. In terms of techniques, our proposed GAN losses can be viewed as a smoothed and generalized Kolmogorov-Smirnov distance.
  arXiv  Detail & Related papers  (2022-02-02T20:11:33Z)
• Fluctuations, Bias, Variance & Ensemble of Learners: Exact Asymptotics for Convex Losses in High-Dimension [25.711297863946193]
  We develop a theory for the study of fluctuations in an ensemble of generalised linear models trained on different, but correlated, features. We provide a complete description of the joint distribution of the empirical risk minimiser for generic convex loss and regularisation in the high-dimensional limit.
  arXiv  Detail & Related papers  (2022-01-31T17:44:58Z)
• Equivalence of Convergence Rates of Posterior Distributions and Bayes Estimators for Functions and Nonparametric Functionals [4.375582647111708]
  We study the posterior contraction rates of a Bayesian method with Gaussian process priors in nonparametric regression. For a general class of kernels, we establish convergence rates of the posterior measure of the regression function and its derivatives. Our proof shows that, under certain conditions, to any convergence rate of Bayes estimators there corresponds the same convergence rate of the posterior distributions.
  arXiv  Detail & Related papers  (2020-11-27T19:11:56Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.