DeepSphere: Efficient spherical convolutional neural network with HEALPix sampling for cosmological applications

Abstract

Convolutional Neural Networks (CNNs) are a cornerstone of the Deep Learning toolbox and have led to many breakthroughs in Artificial Intelligence. So far, these neural networks (NNs) have mostly been developed for regular Euclidean domains such as those supporting images, audio, or video. Because of their success, CNN-based methods are becoming increasingly popular in Cosmology. Cosmological data often comes as spherical maps, which make the use of the traditional CNNs more complicated. The commonly used pixelization scheme for spherical maps is the Hierarchical Equal Area isoLatitude Pixelisation (HEALPix). We present a spherical CNN for analysis of full and partial HEALPix maps, which we call DeepSphere. The spherical CNN is constructed by representing the sphere as a graph. Graphs are versatile data structures that can represent pairwise relationships between objects or act as a discrete representation of a continuous manifold. Using the graph-based representation, we define many of the standard CNN operations, such as convolution and pooling. With filters restricted to being radial, our convolutions are equivariant to rotation on the sphere, and DeepSphere can be made invariant or equivariant to rotation. This way, DeepSphere is a special case of a graph CNN, tailored to the HEALPix sampling of the sphere. This approach is computationally more efficient than using spherical harmonics to perform convolutions. We demonstrate the method on a classification problem of weak lensing mass maps from two cosmological models and compare its performance with that of three baseline classifiers, two based on the power spectrum and pixel density histogram, and a classical 2D CNN. Our experimental results show that the performance of DeepSphere is always superior or equal to the baselines. For high noise levels and for data covering only a smaller fraction of the sphere, DeepSphere achieves typically 10% better classification accuracy than the baselines. Finally, we show how learned filters can be visualized to introspect the NN.Code and examples are available at https://github.com/SwissDataScienceCenter/DeepSphere.

Predicting the retinotopic organization of human visual cortex from anatomy using geometric deep learning

“…Geometric deep learning, being also well-suited for graph data 6 , offers the field a powerful tool given the growing interest in predicting aspects of human behavior from these functional connectivity graphs. Beyond neuroscience lies a wide range of other potential applications for these geometric deep learning techniques – including application to topological data in earth science 40 , genetic and protein structure data in biological science 41–43 , drug discovery in chemistry science 44 , cosmological data in physical science 45 , and more.…”

HexagDLy—Processing hexagonally sampled data with CNNs in PyTorch

“…Following the growing interest in CNNs, increasing efforts to adapt convolution operations to non-Cartesian data can be observed, as for example for spherical data [19,20] and non-Euclidean manifolds [21]. Besides [12], HexagDLy presents a solution for hexagonally sampled data.…”

“…Starting with hexagonally sampled data, a conversion to a square grid representation therefore implies less efficient data processing. Additionally, re- 1,24,20] Output tensor C C.size() → [1,1,24,20] * Kernel k size = 1 stride = 1 sampling hexagonally sampled data to a square grid can introduce sampling artefacts and often requires an increase in resolution to reduce distortions.…”

Intra-coding of 360-degree images on the sphere

“…More precisely, for each S-block we construct a weighted undirected graph G = (V, E, W), where the set of nodes in the graph V represents the set of pixels, E is the set of edges which represents the connectivity between pixels, and W is the weighted adjacency matrix. We use 8-connected neighbors (except for the boundary pixels of the S-block which have fewer neighbors) to define E. We use the weighted adjacency matrix W suggested in [12]…”

“…Finally, the output of the prediction is further processed to remove within the Sblock redundancies. To do so, a graph transform is used (see Section III), similar to the transform used in [12].…”