Geometric Wavelet Scattering Networks on Compact Riemannian Manifolds

The Euclidean scattering transform was introduced nearly a decade ago to improve the mathematical understanding of convolutional neural networks. Inspired by recent interest in geometric deep learning, which aims to generalize convolutional neural networks to manifold and graph-structured domains, we define a geometric scattering transform on manifolds. Similar to the Euclidean scattering transform, the geometric scattering transform is based on a cascade of wavelet filters and pointwise nonlinearities. It is invariant to local isometries and stable to certain types of diffeomorphisms. Empirical results demonstrate its utility on several geometric learning tasks. Our results generalize the deformation stability and local translation invariance of Euclidean scattering, and demonstrate the importance of linking the used filter structures to the underlying geometry of the data.Comment: 35 pages; 3 figures; 2 tables; v3: Revisions based on reviewer comment

The Geometry of Neural Nets' Parameter Spaces Under Reparametrization

Model reparametrization -- transforming the parameter space via a bijective differentiable map -- is a popular way to improve the training of neural networks. But reparametrizations have also been problematic since they induce inconsistencies in, e.g., Hessian-based flatness measures, optimization trajectories, and modes of probability density functions. This complicates downstream analyses, e.g. one cannot make a definitive statement about the connection between flatness and generalization. In this work, we study the invariance quantities of neural nets under reparametrization from the perspective of Riemannian geometry. We show that this notion of invariance is an inherent property of any neural net, as long as one acknowledges the assumptions about the metric that is always present, albeit often implicitly, and uses the correct transformation rules under reparametrization. We present discussions on measuring the flatness of minima, in optimization, and in probability-density maximization, along with applications in studying the biases of optimizers and in Bayesian inference

Geometric deep learning: going beyond Euclidean data

Many scientific fields study data with an underlying structure that is a non-Euclidean space. Some examples include social networks in computational social sciences, sensor networks in communications, functional networks in brain imaging, regulatory networks in genetics, and meshed surfaces in computer graphics. In many applications, such geometric data are large and complex (in the case of social networks, on the scale of billions), and are natural targets for machine learning techniques. In particular, we would like to use deep neural networks, which have recently proven to be powerful tools for a broad range of problems from computer vision, natural language processing, and audio analysis. However, these tools have been most successful on data with an underlying Euclidean or grid-like structure, and in cases where the invariances of these structures are built into networks used to model them. Geometric deep learning is an umbrella term for emerging techniques attempting to generalize (structured) deep neural models to non-Euclidean domains such as graphs and manifolds. The purpose of this paper is to overview different examples of geometric deep learning problems and present available solutions, key difficulties, applications, and future research directions in this nascent field

Deep Learning on Lie Groups for Skeleton-based Action Recognition

In recent years, skeleton-based action recognition has become a popular 3D classification problem. State-of-the-art methods typically first represent each motion sequence as a high-dimensional trajectory on a Lie group with an additional dynamic time warping, and then shallowly learn favorable Lie group features. In this paper we incorporate the Lie group structure into a deep network architecture to learn more appropriate Lie group features for 3D action recognition. Within the network structure, we design rotation mapping layers to transform the input Lie group features into desirable ones, which are aligned better in the temporal domain. To reduce the high feature dimensionality, the architecture is equipped with rotation pooling layers for the elements on the Lie group. Furthermore, we propose a logarithm mapping layer to map the resulting manifold data into a tangent space that facilitates the application of regular output layers for the final classification. Evaluations of the proposed network for standard 3D human action recognition datasets clearly demonstrate its superiority over existing shallow Lie group feature learning methods as well as most conventional deep learning methods.Comment: Accepted to CVPR 201

Dynamic Facial Expression Generation on Hilbert Hypersphere with Conditional Wasserstein Generative Adversarial Nets

In this work, we propose a novel approach for generating videos of the six basic facial expressions given a neutral face image. We propose to exploit the face geometry by modeling the facial landmarks motion as curves encoded as points on a hypersphere. By proposing a conditional version of manifold-valued Wasserstein generative adversarial network (GAN) for motion generation on the hypersphere, we learn the distribution of facial expression dynamics of different classes, from which we synthesize new facial expression motions. The resulting motions can be transformed to sequences of landmarks and then to images sequences by editing the texture information using another conditional Generative Adversarial Network. To the best of our knowledge, this is the first work that explores manifold-valued representations with GAN to address the problem of dynamic facial expression generation. We evaluate our proposed approach both quantitatively and qualitatively on two public datasets; Oulu-CASIA and MUG Facial Expression. Our experimental results demonstrate the effectiveness of our approach in generating realistic videos with continuous motion, realistic appearance and identity preservation. We also show the efficiency of our framework for dynamic facial expressions generation, dynamic facial expression transfer and data augmentation for training improved emotion recognition models