ManifoldNorm: Extending normalizations on Riemannian Manifolds

Abstract

Many measurements in computer vision and machine learning manifest as non-Euclidean data samples. Several researchers recently extended a number of deep neural network architectures for manifold valued data samples. Researchers have proposed models for manifold valued spatial data which are common in medical image processing including processing of diffusion tensor imaging (DTI) where images are fields of $3\times 3$ symmetric positive definite matrices or representation in terms of orientation distribution field (ODF) where the identification is in terms of field on hypersphere. There are other sequential models for manifold valued data that recently researchers have shown to be effective for group difference analysis in study for neuro-degenerative diseases. Although, several of these methods are effective to deal with manifold valued data, the bottleneck includes the instability in optimization for deeper networks. In order to deal with these instabilities, researchers have proposed residual connections for manifold valued data. One of the other remedies to deal with the instabilities including gradient explosion is to use normalization techniques including {\it batch norm} and {\it group norm} etc.. But, so far there is no normalization techniques applicable for manifold valued data. In this work, we propose a general normalization techniques for manifold valued data. We show that our proposed manifold normalization technique have special cases including popular batch norm and group norm techniques. On the experimental side, we focus on two types of manifold valued data including manifold of symmetric positive definite matrices and hypersphere. We show the performance gain in one synthetic experiment for moving MNIST dataset and one real brain image dataset where the representation is in terms of orientation distribution field (ODF).

References in corpus (9)

• Empirical Evaluation of Gated Recurrent Neural Networks on Sequence Modeling
• Geometric deep learning: going beyond Euclidean data
• Unsupervised Learning of Video Representations using LSTMs
• On the Generalization of Equivariance and Convolution in Neural Networks to the Action of Compact Groups
• Gauge Equivariant Convolutional Networks and the Icosahedral CNN
• Riemannian batch normalization for SPD neural networks
• ManifoldNet: A Deep Network Framework for Manifold-valued Data
• A CNN for homogneous Riemannian manifolds with applications to Neuroimaging
• MVC-Net: A Convolutional Neural Network Architecture for Manifold-Valued Images With Applications