On the rates of convergence for learning with convolutional neural networks

We study approximation and learning capacities of convolutional neural networks (CNNs) with one-side zero-padding and multiple channels. Our first result proves a new approximation bound for CNNs with certain constraint on the weights. Our second result gives new analysis on the covering number of feed-forward neural networks with CNNs as special cases. The analysis carefully takes into account the size of the weights and hence gives better bounds than the existing literature in some situations. Using these two results, we are able to derive rates of convergence for estimators based on CNNs in many learning problems. In particular, we establish minimax optimal convergence rates of the least squares based on CNNs for learning smooth functions in the nonparametric regression setting. For binary classification, we derive convergence rates for CNN classifiers with hinge loss and logistic loss. It is also shown that the obtained rates for classification are minimax optimal in some common settings.

Permutation Equivariant Graph Framelets for Heterophilous Graph Learning

The nature of heterophilous graphs is significantly different with that of homophilous graphs, which causes difficulties in early graph neural network models and suggests aggregations beyond 1-hop neighborhood. In this paper, we develop a new way to implement multi-scale extraction via constructing Haar-type graph framelets with desired properties of permutation equivariance, efficiency, and sparsity, for deep learning tasks on graphs. We further design a graph framelet neural network model PEGFAN (Permutation Equivariant Graph Framelet Augmented Network) based on our constructed graph framelets. The experiments are conducted on a synthetic dataset and 9 benchmark datasets to compare performance with other state-of-the-art models. The result shows that our model can achieve best performance on certain datasets of heterophilous graphs (including the majority of heterophilous datasets with relatively larger sizes and denser connections) and competitive performance on the remaining.

Permutation Equivariant Graph Framelets for Heterophilous Semi-supervised Learning

The nature of heterophilous graphs is significantly different with that of homophilous graphs, which suggests aggregations beyond 1-hop neighborhood and causes difficulties in early graph neural network models. In this paper, we develop a new way to implement multi-scale extraction via constructing Haar-type graph framelets with desired properties of permutation equivariance, efficiency, and sparsity, for deep learning tasks on graphs. We further deisgn a graph framelet neural network model PEGFAN using our constructed graph framelets. The experiments are conducted on a synthetic dataset and 9 benchmark datasets to compare performance with other state-of-the-art models. The result shows that our model can achieve best performance on certain datasets of heterophilous graphs (including the majority of heterophilous datasets with relatively larger sizes and denser connections) and competitive performance on the remaining.

Optimal Estimates for Pairwise Learning with Deep ReLU Networks

Pairwise learning refers to learning tasks where a loss takes a pair of samples into consideration. In this paper, we study pairwise learning with deep ReLU networks and estimate the excess generalization error. For a general loss satisfying some mild conditions, a sharp bound for the estimation error of order $O((V\log(n) /n)^{1/(2-\beta)})$ is established. In particular, with the pairwise least squares loss, we derive a nearly optimal bound of the excess generalization error which achieves the minimax lower bound up to a logrithmic term when the true predictor satisfies some smoothness regularities.

A new activation for neural networks and its approximation

Deep learning with deep neural networks (DNNs) has attracted tremendous attention from various fields of science and technology recently. Activation functions for a DNN define the output of a neuron given an input or set of inputs. They are essential and inevitable in learning non-linear transformations and performing diverse computations among successive neuron layers. Thus, the design of activation functions is still an important topic in deep learning research. Meanwhile, theoretical studies on the approximation ability of DNNs with activation functions have been investigated within the last few years. In this paper, we propose a new activation function, named as "DLU", and investigate its approximation ability for functions with various smoothness and structures. Our theoretical results show that DLU networks can process competitive approximation performance with rational and ReLU networks, and have some advantages. Numerical experiments are conducted comparing DLU with the existing activations-ReLU, Leaky ReLU, and ELU, which illustrate the good practical performance of DLU.

Approximation analysis of CNNs from feature extraction view

Deep learning based on deep neural networks has been very successful in many practical applications, but it lacks enough theoretical understanding due to the network architectures and structures. In this paper, we establish the analysis for linear feature extraction by deep multi-channel convolutional neural networks(CNNs), which demonstrates the power of deep learning over traditional linear transformations, like Fourier, Wavelets, and Redundant dictionary coding methods. Moreover, we give an exact construction presenting how linear features extraction can be conducted efficiently with multi-channel CNNs. It can be applied to lower the essential dimension for approximating a high-dimensional function. Rates of function approximation by such deep networks implemented with channels and followed by fully-connected layers are investigated as well. Harmonic analysis for factorizing linear features into multi-resolution convolutions plays an essential role in our work. Nevertheless, a dedicate vectorization of matrices is constructed, which bridges 1D CNN and 2D CNN and allows us have corresponding 2D analysis.

Spherical Image Inpainting with Frame Transformation and Data-driven Prior Deep Networks

Spherical image processing has been widely applied in many important fields, such as omnidirectional vision for autonomous cars, global climate modelling, and medical imaging. It is non-trivial to extend an algorithm developed for flat images to the spherical ones. In this work, we focus on the challenging task of spherical image inpainting with deep learning-based regularizer. Instead of a naive application of existing models for planar images, we employ a fast directional spherical Haar framelet transform and develop a novel optimization framework based on a sparsity assumption of the framelet transform. Furthermore, by employing progressive encoder-decoder architecture, a new and better-performed deep CNN denoiser is carefully designed and works as an implicit regularizer. Finally, we use a plug-and-play method to handle the proposed optimization model, which can be implemented efficiently by training the CNN denoiser prior. Numerical experiments are conducted and show that the proposed algorithms can greatly recover damaged spherical images and achieve the best performance over purely using deep learning denoiser and plug-and-play model.

Convolutional Neural Networks for Spherical Signal Processing via Spherical Haar Tight Framelets

In this paper, we develop a general theoretical framework for constructing Haar-type tight framelets on any compact set with a hierarchical partition. In particular, we construct a novel area-regular hierarchical partition on the 2-sphere and establish its corresponding spherical Haar tight framelets with directionality. We conclude by evaluating and illustrating the effectiveness of our area-regular spherical Haar tight framelets in several denoising experiments. Furthermore, we propose a convolutional neural network (CNN) model for spherical signal denoising which employs the fast framelet decomposition and reconstruction algorithms. Experiment results show that our proposed CNN model outperforms threshold methods, and processes strong generalization and robustness properties.

Radial Basis Function Approximation with Distributively Stored Data on Spheres

This paper proposes a distributed weighted regularized least squares algorithm (DWRLS) based on spherical radial basis functions and spherical quadrature rules to tackle spherical data that are stored across numerous local servers and cannot be shared with each other. Via developing a novel integral operator approach, we succeed in deriving optimal approximation rates for DWRLS and theoretically demonstrate that DWRLS performs similarly as running a weighted regularized least squares algorithm with the whole data on a large enough machine. This interesting finding implies that distributed learning is capable of sufficiently exploiting potential values of distributively stored spherical data, even though every local server cannot access all the data.