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Pooling and unpooling are two essential operations in constructinghierarchical spherical convolutional neural networks (HS-CNNs) forcomprehensive feature learning in the spherical domain. Most existing modelsemploy downsampling-based pooling, which will inevitably incur information lossand cannot adapt to different spherical signals and tasks. Besides, thepreserved information after pooling cannot be well restored by the subsequentunpooling to characterize the desirable features for a task. In this paper, wepropose a novel framework of HS-CNNs with a lifting structure to learn adaptivespherical wavelets for pooling and unpooling, dubbed LiftHS-CNN, which ensuresa more efficient hierarchical feature learning for both image- and pixel-leveltasks. Specifically, adaptive spherical wavelets are learned with a liftingstructure that consists of trainable lifting operators (i.e., update andpredict operators). With this learnable lifting structure, we can adaptivelypartition a signal into two sub-bands containing low- and high-frequencycomponents, respectively, and thus generate a better down-scaled representationfor pooling by preserving more information in the low-frequency sub-band. Theupdate and predict operators are parameterized with graph-based attention tojointly consider the signal's characteristics and the underlying geometries. Wefurther show that particular properties are promised by the learned wavelets,ensuring the spatial-frequency localization for better exploiting the signal'scorrelation in both spatial and frequency domains. We then propose an unpoolingoperation that is invertible to the lifting-based pooling, where an inversewavelet transform is performed by using the learned lifting operators torestore an up-scaled representation. Extensive empirical evaluations on variousspherical domain tasks validate the superiority of the proposed LiftHS-CNN.

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