Designing a network on 3D surface for non-rigid shape analysis is a challenging task. In this work, we propose a novel spectral transform network on 3D surface to learn shape descriptors. The proposed network architecture consists of four stages: raw descriptor extraction, surface second-order pooling, mixture of power function-based spectral transform, and metric learning. The proposed network is simple and shallow. Quantitative experiments on challenging benchmarks show its effectiveness for non-rigid shape retrieval and classification, e.g., it achieved the highest accuracies on SHREC'14, 15 datasets as well as the "range" subset of SHREC'17 dataset.Keywords: non-rigid shape analysis · spectral transform · shape representation Recently, a promising trend in non-rigid shape representation is the learning-based methods on 3D surface for tasks of non-rigid shape retrieval and classification. Many learning-based methods take low-level shape descriptors as inputs and extract highlevel descriptors by integrating over the entire shape. In the work of [8], they first extract SIHKS and WKS, and then integrate them to form a global descriptor followed by LMNN embedding. Global shape descriptors are learned by Long-Short Term Memory (LSTM) network in [45] based on spectral descriptors. The eigen-shape and Fishershape descriptors are learned by a modified auto-encoder based on spectral descriptors in [12]. These works have shown impressive results in learning global shape descriptors. Though these advances have been achieved, designing learning-based methods on 3D surface is still an emerging and challenging task, including how to design feature aggregation and feature learning on 3D surface for non-rigid shape representation.In this work, we propose a novel learning-based spectral transform network on 3D surface to learn discriminative shape descriptor for non-rigid shape retrieval and classification. First, we define a second-order pooling operation on 3D surface which models the second-order statistics of input raw descriptors on 3D surfaces. Second, considering that the pooled second-order descriptors lie on a manifold of symmetric positive definite matrices (SPDM-manifold), we define a novel manifold transform for feature learning by learning a mixture of power function on the singular values of the SPDM descriptors. Third, by concatenating the stages of raw descriptor extraction, surface second-order pooling, transform on SPDM-manifold and metric learning, we propose a novel network architecture, dubbed as spectral transform network as shown in Fig. 1, which can learn discriminative shape descriptors for non-rigid shape analysis.To the best of our knowledge, this is the first paper that learns second-order poolingbased shape descriptors on 3D surfaces using a network architecture. Our network structure is simple and easily to be trained, and is justified to be able to significantly improve the discriminative ability of input raw descriptors. It is adaptive to various non-rigid shapes such as watertight meshes, parti...