Hierarchical Spherical CNNs with Lifting-Based Adaptive Wavelets for Pooling and Unpooling

Pooling and unpooling are indispensable in constructing hierarchical spherical convolutional neural networks (HS-CNNs). Most existing models employ simple downsampling-based pooling, which ignores the sampling theorem and cannot adapt to different spherical signals (with different spectra) and tasks (dependent on different frequency components), thus suffering a significant information loss. Besides, signals reconstructed by the widely-adopted padding-based unpooling may also change unwantedly the spectra of original signals. To address these, we propose a novel framework of HS-CNNs with lifting structures to learn adaptive spherical wavelets for pooling and unpooling, named LiftHS-CNNs. Specifically, we learn spherical wavelets with a lifting structure to adaptively partition the input signal into low- and high-frequency sub-bands, with the down-scaled representations for pooling generated to preserve more information in the low-frequency sub-band. The lifting structure consists of learnable update and predict operators parameterized with graph attention to jointly consider the signal's characteristics and underlying geometries. We then propose an unpooling operation invertible to the lifting-based pooling for restoring the up-scaled representations, which can well preserve spectral characteristics of the original signal. Particular properties (i.e., spatial locality, vanishing moments, and stability) of the learned wavelets and the information preserving ability of the proposed pooling and unpooling are further studied. Experiments on benchmark spherical datasets for a wide range of tasks verify the superiority of our LiftHS-CNNs.