DeepSphere: Efficient spherical Convolutional Neural Network with HEALPix sampling for cosmological applications (1810.12186v2)

Abstract: Convolutional Neural Networks (CNNs) are a cornerstone of the Deep Learning toolbox and have led to many breakthroughs in Artificial Intelligence. These networks 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 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 the performance of the CNN with that of two baseline classifiers. The results show that the performance of DeepSphere is always superior or equal to both of these baselines. For high noise levels and for data covering only a smaller fraction of the sphere, DeepSphere achieves typically 10% better classification accuracy than those baselines. Finally, we show how learned filters can be visualized to introspect the neural network.

• The paper introduces a novel graph-based spherical CNN that adapts HEALPix sampling for efficient cosmological data analysis.
• It achieves computational complexity of O(Npix) using Chebyshev polynomials, outperforming traditional spherical methods on noisy and partial data.
• DeepSphere delivers approximately 10% higher classification accuracy in challenging conditions, highlighting its promise for astrophysical applications.

Analysis of DeepSphere: Efficient Spherical Convolutional Neural Network with HEALPix Sampling for Cosmological Applications

The paper presents DeepSphere, a convolutional neural network specifically designed for analyzing cosmological data, which often comes in the form of spherical maps. Central to its approach is the adaptation of traditional convolutional neural networks (CNNs) to operate efficiently on spherical domains—a departure from the more common Euclidean domains like images and audio. The authors propose modeling the sphere as a graph, leveraging HEALPix sampling, which facilitates the analysis of cosmological data through spherical CNNs that retain computational efficiency and rotational equivariance.

Cosmological maps, such as those representing the Cosmic Microwave Background (CMB), gravitational lensing, and galaxy clustering, are essential for understanding the universe's structure. These maps often use the Hierarchical Equal Area isoLatitude Pixelisation (HEALPix) to parameterize the sphere, presenting challenges due to their non-Euclidean geometry. DeepSphere addresses these challenges by translating the sphere into a graph—a versatile data structure allowing for the definition of convolution and pool operations, essential for CNNs, while ensuring that these operations are equivariant to rotation. This contrasts with approaches using spherical harmonics, which can be computationally expensive.

The paper demonstrates DeepSphere through a classification task involving weak lensing mass maps generated from two distinct cosmological models. The performance of DeepSphere is compared against three established classifiers: two utilizing power spectrum and pixel density histogram features, and a traditional 2D CNN. DeepSphere's results reveal superior performance, especially in high-noise scenarios and when dealing with partial sky observations. Notably, DeepSphere achieves approximately 10% higher classification accuracy over these baselines under such conditions.

Key Contributions and Results

• Graph-based Spherical Representation: By representing the sphere via a graph, the authors define convolution operations that replicate the behavior and efficacy of CNNs on spherical domains. This model supports functions like down-sampling and pooling, and its radial filters enforce rotational equivariance efficiently.
• Computational Efficiency: Using Chebyshev polynomials to define convolution kernels, DeepSphere achieves $O(N_{pix})$ complexity, which is significantly more efficient than the $O(N_{pix}^2)$ complexity observed with spherical harmonic-based convolutions. This efficiency is critical when scaling to large datasets or partial sky maps.
• Performance on Noisy and Incomplete Data: In a direct comparison across several noise levels and data sizes, DeepSphere consistently outperforms traditional techniques. For instance, even with high noise levels or small data coverage, DeepSphere maintains its classification advantage.

Implications and Future Directions

From a theoretical perspective, DeepSphere broadens the understanding and application of graph neural networks by juxtaposing them with spherical data analysis. Practically, the employment of DeepSphere could revolutionize data handling in fields reliant on spherical data models, such as astrophysics, meteorology, and even geospatial analysis. The graph-based approach could evolve into a standard for handling spherical datasets, offering potent computational efficiency and result accuracy, provided that unconventional sampling schemes can be accommodated similarly.

Future developments might explore optimizing the graph-to-sphere approximation further, potentially improving convolution precision and matching it closer to the spherical harmonics. Additionally, further analyzing the potential of this method in other applications beyond cosmology, particularly where part-of-sphere data is prevalent and graph structures can save computational costs, could elevate the method's relevance across various domains.

By democratizing the use of spherical models and providing accessible tools, DeepSphere paves the way for advanced AI applications that blend computational efficiency with cosmological exigencies. The method's capability to sidestep the iso-latitude constraint typical in other methods presents innovative pathways in the projection and utilization of spherical datasets, a key task for scientific exploration across multiple disciplines.