text: Symmetric Positive Definite (SPD) matrix learning methods have become popular in many image and video processing tasks, thanks to their ability to learn appropriate statistical representations while respecting Riemannian geometry of underlying SPD manifolds. In this paper we build a Riemannian network architecture to open up a new direction of SPD matrix non-linear learning in a deep model. In particular, we devise bilinear mapping layers to transform input SPD matrices to more desirable SPD matrices, exploit eigenvalue rectification layers to apply a non-linear activation function to the new SPD matrices, and design an eigenvalue logarithm layer to perform Riemannian computing on the resulting SPD matrices for regular output layers. For training the proposed deep network, we exploit a new backpropagation with a variant of stochastic gradient descent on Stiefel manifolds to update the structured connection weights and the involved SPD matrix data. We show through experiments that the proposed SPD matrix network can be simply trained and outperform existing SPD matrix learning and state-of-the-art methods in three typical visual classification tasks.
  @cite_31: Convolutional Neural Networks are extremely efficient architectures in image and audio recognition tasks, thanks to their ability to exploit the local translational invariance of signal classes over their domain. In this paper we consider possible generalizations of CNNs to signals defined on more general domains without the action of a translation group. In particular, we propose two constructions, one based upon a hierarchical clustering of the domain, and another based on the spectrum of the graph Laplacian. We show through experiments that for low-dimensional graphs it is possible to learn convolutional layers with a number of parameters independent of the input size, resulting in efficient deep architectures.
  @cite_3: Feature descriptors play a crucial role in a wide range of geometry analysis and processing applications, including shape correspondence, retrieval, and segmentation. In this paper, we introduce Geodesic Convolutional Neural Networks (GCNN), a generalization of the convolutional networks (CNN) paradigm to non-Euclidean manifolds. Our construction is based on a local geodesic system of polar coordinates to extract "patches", which are then passed through a cascade of filters and linear and non-linear operators. The coefficients of the filters and linear combination weights are optimization variables that are learned to minimize a task-specific cost function. We use GCNN to learn invariant shape features, allowing to achieve state-of-the-art performance in problems such as shape description, retrieval, and correspondence.
  @cite_2: In this paper, we propose a generalization of convolutional neural networks (CNN) to non-Euclidean domains for the analysis of deformable shapes. Our construction is based on localized frequency analysis (a generalization of the windowed Fourier transform to manifolds) that is used to extract the local behavior of some dense intrinsic descriptor, roughly acting as an analogy to patches in images. The resulting local frequency representations are then passed through a bank of filters whose coefficient are determined by a learning procedure minimizing a task-specific cost. Our approach generalizes several previous methods such as HKS, WKS, spectral CNN, and GPS embeddings. Experimental results show that the proposed approach allows learning class-specific shape descriptors significantly outperforming recent state-of-the-art methods on standard benchmarks.
summary: Deep neural networks have exhibited their great powers when the processed data own a Euclidean data structure. In many contexts, however, one may be faced with data defined in non-Euclidean domains. To tackle graph-structured data, @cite_31 presented a spectral formulation of convolutional networks by exploiting a notion of non shift-invariant convolution that depends on the analogy between the classical Fourier transform and the Laplace-Beltrami eigenbasis. Following @cite_31 , a localized spectral network was proposed in @cite_2 to non-Euclidean domains by generalizing the windowed Fourier transform to manifolds to extract the local behavior of some dense intrinsic descriptor. Similarly, @cite_3 proposed a geodesic convolution' on non-Euclidean local geodesic system of coordinates to extract local patches' on shape manifolds. The convolutions in this approach were performed by sliding a window over the shape manifolds.