Approximation in shift-invariant spaces with deep ReLU neural networks

• URL: http://arxiv.org/abs/2005.11949v3
• Date: Sun, 19 Jun 2022 07:13:05 GMT
• Title: Approximation in shift-invariant spaces with deep ReLU neural networks
• Authors: Yunfei Yang, Zhen Li, Yang Wang
• Abstract summary: We study the expressive power of deep ReLU neural networks for approximating functions in dilated shift-invariant spaces. Approximation error bounds are estimated with respect to the width and depth of neural networks.
• Score: 7.7084107194202875
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We study the expressive power of deep ReLU neural networks for approximating functions in dilated shift-invariant spaces, which are widely used in signal processing, image processing, communications and so on. Approximation error bounds are estimated with respect to the width and depth of neural networks. The network construction is based on the bit extraction and data-fitting capacity of deep neural networks. As applications of our main results, the approximation rates of classical function spaces such as Sobolev spaces and Besov spaces are obtained. We also give lower bounds of the $L^p (1\le p \le \infty)$ approximation error for Sobolev spaces, which show that our construction of neural network is asymptotically optimal up to a logarithmic factor.

Related papers

• Addressing caveats of neural persistence with deep graph persistence [54.424983583720675]
  We find that the variance of network weights and spatial concentration of large weights are the main factors that impact neural persistence. We propose an extension of the filtration underlying neural persistence to the whole neural network instead of single layers. This yields our deep graph persistence measure, which implicitly incorporates persistent paths through the network and alleviates variance-related issues.
  arXiv  Detail & Related papers  (2023-07-20T13:34:11Z)
• Globally Optimal Training of Neural Networks with Threshold Activation Functions [63.03759813952481]
  We study weight decay regularized training problems of deep neural networks with threshold activations. We derive a simplified convex optimization formulation when the dataset can be shattered at a certain layer of the network.
  arXiv  Detail & Related papers  (2023-03-06T18:59:13Z)
• Gradient Descent in Neural Networks as Sequential Learning in RKBS [63.011641517977644]
  We construct an exact power-series representation of the neural network in a finite neighborhood of the initial weights. We prove that, regardless of width, the training sequence produced by gradient descent can be exactly replicated by regularized sequential learning.
  arXiv  Detail & Related papers  (2023-02-01T03:18:07Z)
• Optimal Approximation Complexity of High-Dimensional Functions with Neural Networks [3.222802562733787]
  We investigate properties of neural networks that use both ReLU and $x2$ as activation functions. We show how to leverage low local dimensionality in some contexts to overcome the curse of dimensionality, obtaining approximation rates that are optimal for unknown lower-dimensional subspaces.
  arXiv  Detail & Related papers  (2023-01-30T17:29:19Z)
• Bayesian Interpolation with Deep Linear Networks [92.1721532941863]
  Characterizing how neural network depth, width, and dataset size jointly impact model quality is a central problem in deep learning theory. We show that linear networks make provably optimal predictions at infinite depth. We also show that with data-agnostic priors, Bayesian model evidence in wide linear networks is maximized at infinite depth.
  arXiv  Detail & Related papers  (2022-12-29T20:57:46Z)
• Sobolev-type embeddings for neural network approximation spaces [5.863264019032882]
  We consider neural network approximation spaces that classify functions according to the rate at which they can be approximated. We prove embedding theorems between these spaces for different values of $p$. We find that, analogous to the case of classical function spaces, it is possible to trade "smoothness" (i.e., approximation rate) for increased integrability.
  arXiv  Detail & Related papers  (2021-10-28T17:11:38Z)
• Near-Minimax Optimal Estimation With Shallow ReLU Neural Networks [19.216784367141972]
  We study the problem of estimating an unknown function from noisy data using shallow (single-hidden layer) ReLU neural networks. We quantify the performance of these neural network estimators when the data-generating function belongs to the space of functions of second-order bounded variation in the Radon domain.
  arXiv  Detail & Related papers  (2021-09-18T05:56:06Z)
• On the approximation of functions by tanh neural networks [0.0]
  We derive bounds on the error, in high-order Sobolev norms, incurred in the approximation of Sobolev-regular. We show that tanh neural networks with only two hidden layers suffice to approximate functions at comparable or better rates than much deeper ReLU neural networks.
  arXiv  Detail & Related papers  (2021-04-18T19:30:45Z)
• A Convergence Theory Towards Practical Over-parameterized Deep Neural Networks [56.084798078072396]
  We take a step towards closing the gap between theory and practice by significantly improving the known theoretical bounds on both the network width and the convergence time. We show that convergence to a global minimum is guaranteed for networks with quadratic widths in the sample size and linear in their depth at a time logarithmic in both. Our analysis and convergence bounds are derived via the construction of a surrogate network with fixed activation patterns that can be transformed at any time to an equivalent ReLU network of a reasonable size.
  arXiv  Detail & Related papers  (2021-01-12T00:40:45Z)
• Topological obstructions in neural networks learning [67.8848058842671]
  We study global properties of the loss gradient function flow. We use topological data analysis of the loss function and its Morse complex to relate local behavior along gradient trajectories with global properties of the loss surface.
  arXiv  Detail & Related papers  (2020-12-31T18:53:25Z)
• Expressivity of Deep Neural Networks [2.7909470193274593]
  In this review paper, we give a comprehensive overview of the large variety of approximation results for neural networks. While the mainbody of existing results is for general feedforward architectures, we also depict approximation results for convolutional, residual and recurrent neural networks.
  arXiv  Detail & Related papers  (2020-07-09T13:08:01Z)

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.