Rethinking 3D Convolution in $\ell_p$-norm Space

Li Zhang, Yan Zhong, Jianan Wang, Zhe Min, RujingWang, Liu Liu

Keywords: $\ell_p$-norm; 3D Convolution

Abstract:

Convolution is a fundamental operation in the 3D backbone. However, under certain conditions, the feature extraction ability of traditional convolution methods may be weakened. In this paper, we introduce a new convolution method based on $\ell_p$-norm.
For theoretical support, we prove the universal approximation theorem for $\ell_p$-norm based convolution, and analyze the robustness and feasibility of $\ell_p$-norms in 3D point cloud tasks. Concretely, $\ell_{\infty}$-norm based convolution is prone to feature loss. $\ell_2$-norm based convolution is essentially a linear transformation of the traditional convolution. $\ell_1$-norm based convolution is an economical and effective feature extractor. We propose customized optimization strategies to accelerate the training process of $\ell_1$-norm based Nets and enhance the performance. Besides, a theoretical guarantee is given for the convergence by \textit{regret} argument. We apply our methods to classic networks and conduct related experiments. Experimental results indicate that our approach exhibits competitive performance with traditional CNNs, with lower energy consumption and instruction latency.

Primary Area: Learning theory

Submission Number: 1528