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

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.