ChebNet: Efficient and Stable Constructions of Deep Neural Networks with Rectified Power Units using Chebyshev Approximations

In a recent paper [B. Li, S. Tang and H. Yu, arXiv:1903.05858], it was shown that deep neural networks built with rectified power units (RePU) can give better approximation for sufficient smooth functions than those with rectified linear units, by converting polynomial approximation given in power series into deep neural networks with optimal complexity and no approximation error. However, in practice, power series are not easy to compute. In this paper, we propose a new and more stable way to construct deep RePU neural networks based on Chebyshev polynomial approximations. By using a hierarchical structure of Chebyshev polynomial approximation in frequency domain, we build efficient and stable deep neural network constructions. In theory, ChebNets and the deep RePU nets based on Power series have the same upper error bounds for general function approximations. But numerically, ChebNets are much more stable. Numerical results show that the constructed ChebNets can be further trained and obtain much better results than those obtained by training deep RePU nets constructed basing on power series.Comment: 18 pages, 6 figures, 2 table

Learning how to be robust: Deep polynomial regression

Polynomial regression is a recurrent problem with a large number of applications. In computer vision it often appears in motion analysis. Whatever the application, standard methods for regression of polynomial models tend to deliver biased results when the input data is heavily contaminated by outliers. Moreover, the problem is even harder when outliers have strong structure. Departing from problem-tailored heuristics for robust estimation of parametric models, we explore deep convolutional neural networks. Our work aims to find a generic approach for training deep regression models without the explicit need of supervised annotation. We bypass the need for a tailored loss function on the regression parameters by attaching to our model a differentiable hard-wired decoder corresponding to the polynomial operation at hand. We demonstrate the value of our findings by comparing with standard robust regression methods. Furthermore, we demonstrate how to use such models for a real computer vision problem, i.e., video stabilization. The qualitative and quantitative experiments show that neural networks are able to learn robustness for general polynomial regression, with results that well overpass scores of traditional robust estimation methods.Comment: 18 pages, conferenc