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DeepSphere: Efficient spherical Convolutional Neural Network with HEALPix sampling for cosmological applications

Nathanaël Perraudin, Michaël Defferrard, Tomasz Kacprzak, Raphael Sgier

arXiv:1810.12186v2astro-ph.COastro-ph.IMcs.AIcs.CVcs.LG

TL;DR

DeepSphere addresses the difficulty of applying CNNs to irregular spherical cosmological maps, especially when observations cover only part of the sky. It represents HEALPix maps as graphs, uses graph convolutions and hierarchical pooling, and achieves better classification than the evaluated baselines, with advantages growing to 10% for smaller, noisier data. The paper also notes that direct comparison with alternative spherical CNNs remains future work because those methods target equirectangular full-sphere settings.

  • Problem

    CNNs are mainly designed for regular Euclidean domains, while cosmological data are often irregular spherical maps requiring rotation-aware operations.

  • Method

    DeepSphere represents discretized HEALPix spheres as graphs, applies graph convolutions with hierarchical pooling, and supports full or partial spherical observations.

  • Results

    DeepSphere performs better than three baselines in all considered weak-lensing classification cases, with its advantage growing up to 10% for smaller, noisier data.

  • Takeaways & Limitations

    The graph-based spherical CNN provides an efficient and flexible approach for cosmological map classification and other global or dense prediction tasks.

  • Takeaways & Limitations

    Alternative spherical-CNN comparisons are left for future work because the available methods target equirectangular full-sphere data and are unsuitable for direct partial-HEALPix comparison.

Abstract

from arXiv · show

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.

1. Introduction

DeepSphere addresses the difficulty of applying CNNs to irregular spherical maps by representing the sphere as a graph and using hierarchical pixelization for multiscale processing. The method targets rotation-aware, computationally efficient analysis of full or partial HEALPix data.

  • Cosmological observations often take the form of spherical sky maps used to constrain cosmological and astrophysical models.
  • CNNs are attractive because shared trainable weights can capture nonlinear patterns without relearning features at every spatial location.
  • The main spherical-CNN challenge is defining convolutions for irregular sampling while preserving rotation equivariance and related spherical-domain properties.
  • Existing planar-grid and projected-patch approaches are not rotation equivariant, whereas spherical Fourier methods are computationally expensive and poorly accelerated for partial-sky maps.
  • DeepSphere models HEALPix pixels as a graph for convolution and uses hierarchical pixelization for pooling, supporting rotation equivariance, efficiency, and partial-sky observations.
  • The model handles global or dense classification and regression for spherical maps with single or multiple values per pixel.

2. Method

DeepSphere adapts CNN operations to spherical data by constructing a weighted graph from HEALPix pixels and defining graph Fourier convolutions. Hierarchical HEALPix structure supplies multiscale down-sampling while graph spectral modes approximate spherical harmonics.

  • DeepSphere adapts convolution and down-sampling to spherical domains while retaining point-wise nonlinearities, normalization, and permutation-invariant pooling.
  • The method uses hierarchical spherical pixelization to down-sample data across multiple scales despite irregular sampling.
  • HEALPix provides hierarchical, equal-area, isoLatitude sampling; the graph construction and parent selection are the only sampling-dependent method components.
  • Each HEALPix pixel becomes a graph vertex connected to its 8 or 7 neighboring pixels, with distance-based edge weights accounting for unequal pixel separations.
  • The normalized graph Laplacian supplies an orthonormal eigenvector basis for graph Fourier transforms, with eigenvalues interpreted as squared frequencies.
  • Graph Fourier modes of the HEALPix graph resemble spherical harmonics, indicating that the construction captures spherical properties of the sampling.
  • Graph convolutions use localized polynomial filters whose neighborhoods follow graph paths, enabling weighted combinations of neighboring pixel values.

2.5. Efficient convolutions

DeepSphere replaces computationally expensive graph Fourier filtering with localized polynomial graph convolutions and hierarchical pooling on HEALPix graphs. The resulting operations support multiscale processing while scaling linearly with the number of pixels.

  • Graph Fourier filtering is computationally cumbersome because dense transforms cost O(Npix^2), while Laplacian eigendecomposition adds a one-time O(Npix^3) cost.
  • Polynomial graph filters hθ(λ) of degree K avoid explicit Fourier transforms and use K+1 trainable coefficients.
  • Radial filters combine Laplacian neighborhoods into weighted local sums, making filter values depend on distance rather than direction and yielding rotation-equivariant convolutions.
  • Chebyshev evaluation requires O(K) sparse Laplacian multiplications, reducing convolution cost to O(Npix) because the graph has fewer than 8Npix edges.For Nside = 2048, the method was ten to twenty times faster than libsharp spherical-harmonic filtering for K = 20 and K = 5.
  • Hierarchical coarsening merges equal groups of HEALPix child cells, while pooling summarizes their values with permutation-invariant operations such as maximum, sum, or average.For HEALPix, each parent has a constant number of children, and successive pooling reduces the pixel count.
  • DeepSphere stacks graph convolutions, pooling, batch normalization, fully connected layers, and an optional softmax layer.

2.8. Network architectures

DeepSphere supports global and dense prediction for both classification and regression by combining graph-convolutional representations with task-specific output layers. Rotation-equivariant dense predictions can be averaged to obtain rotation-invariant global outputs.

  • DeepSphere handles global or dense prediction tasks in either classification or regression, with output size determined by the task.Global outputs do not depend on input size, whereas dense tasks produce one prediction per pixel.
  • Fully convolutional networks preserve spatial outputs, while fully connected layers convert representations into global predictions.
  • The receptive field is an isotropic local disk whose radius depends on graph-convolution order K and any pooling down-sampling.
  • Averaging dense predictions produces global outputs that are invariant to rotation on SO(3).
  • Replacing average pooling with a fully connected layer lets CNNs learn where to place attention, sacrificing built-in rotation invariance.
  • Classification uses cross-entropy with a softmax output, whereas regression commonly uses mean squared error.
  • Training minimizes empirical risk using back-propagation and stochastic gradient descent over mini-batches.Mini-batches exploit parallelism in modern computing platforms.

3. Related work

Prior spherical CNNs either use coordinate-dependent 2D convolutions or rotation-equivariant spherical harmonics, but these approaches face equivariance, localization, or computational limitations. DeepSphere instead uses graph convolutions on HEALPix, trading exact equivariance for linear scaling and partial-sky efficiency.

  • Grid-based and tangent-plane methods use 2D CNNs efficiently but are not rotation equivariant because they depend on local coordinate systems and anisotropic filters.
  • Spherical-harmonic convolutions provide mathematically well-defined rotation equivariance but remain expensive, with HEALPix transforms costing O(Npix^3/2).
  • DeepSphere scales as O(Npix), unlike spherical-harmonic methods, but its graph-based equivariance is not exact.
  • Partial-sky data is difficult to accelerate in existing 2D and spherical CNNs because computations may still process unseen or mostly empty regions.
  • The selected graph-CNN formulation makes convolutions and filters explicit through graph signal-processing theory and remains close in spirit to spherical-harmonic methods.

4. Experiments

DeepSphere was evaluated on classifying noisy partial-sky convergence maps from two cosmological models with similar power spectra. It outperformed the baselines across configurations, while performance depended on sample size, noise, and architecture.

  • 4.1. Data: DeepSphere classified convergence maps from two cosmological model classes using simulated gravitational-lensing-like data.The experiment used two parameter sets designed to produce similar angular power spectra.
  • 4.2. Problem formulation: Gaussian noise ranged from zero to twice the noiseless pixel-value standard deviation, with fresh noise generated during training for augmentation.The noise model was used to make discrimination harder and reduce over-fitting.
  • 4.5. Results: DeepSphere outperformed all configurations of the 2D ConvNet and SVM baselines, with the performance gap widening as sample size decreased and noise increased.The SVM baselines used pixel histograms or power spectral density features, while the 2D ConvNet operated on flattened projections.
  • 4.5. Results: Accuracy exceeded 97% for DeepSphere at orders o = 1 and o = 2 across all noise levels, declining to 90% for o = 4 at the highest noise.Histogram features reached 80% for o = 4 at the highest noise level.
  • 4.5. Results: FCN variants outperformed CNN variants because the mass maps were empirically consistent with stationary, locally correlated, rotation-invariant structure.The CNN variant may be preferable when the data lacks this property.
  • 4.5. Results: Chebyshev-polynomial filters made training loss decrease more smoothly than monomial filters, but neither this choice nor ℓ2 regularization significantly improved converged accuracy.The authors recommend centered Gaussian initialization for Chebyshev coefficients.
  • 4.6. Filter visualization: Learned filters from the fifth spherical convolutional layer showed radial profiles and gnomonic projections resembling peaks, consistent with concentrated clumps in convergence maps.Visually similar patterns appeared across the trained networks examined.

5. Conclusion

DeepSphere combines graph convolutions with HEALPix sampling to provide efficient, rotation-aware analysis of full and partial spherical data. It outperforms three benchmark methods on weak-lensing cosmological model discrimination, with the largest gains for smaller, noisier maps.

  • DeepSphere uses graph convolutions on HEALPix maps, achieving O(Npix) complexity and supporting partial-sphere observations.The graph representation also captures spherical structure through graph Fourier modes that approximate spherical harmonics.
  • DeepSphere systematically outperforms a 2D CNN and two SVM classifiers on the evaluated weak-lensing mass-map task.The SVM baselines use spherical-harmonic power spectra or pixel-density histograms.
  • The advantage is small for large, noise-free maps but grows up to 10% for smaller, noisier data.
  • The released Python package is intended to make spherical CNNs easier to use for dense and global classification and regression.
  • The authors identify convergence of the graph Laplacian toward the Laplace–Beltrami operator and broader comparisons with spherical CNNs as future work.
  • The graph formulation relaxes the isolatitude constraint that traditionally supports fast spherical convolutions.This may permit sampling schemes with different trade-offs or direct use of measurement positions.

Appendix A. Graph Fourier modes and spherical harmonics

The HEALPix graph Fourier basis reproduces key frequency structure of spherical harmonics, especially at low frequencies. Sampling irregularities cause leakage and prevent exact rotational equivariance, leaving convergence as an open theoretical question.

  • Graph Laplacian eigenvalues form groups of 2ℓ + 1, matching the number of spherical-harmonic orders at each degree ℓ.This grouping indicates that graph eigenvectors approximate spherical harmonics.
  • At low frequencies, graph Fourier modes span the same subspaces as spherical harmonics.The correspondence is assessed by comparing graph-eigenvector power spectral densities with spherical-harmonic subspaces.
  • Irregular HEALPix sampling causes energy leakage across frequency bands and affects graph Fourier modes.
  • It remains uncertain whether graph Laplacian eigenvectors converge to spherical harmonics as Nside and Npix increase.
  • These discrepancies make convolution only approximately equivariant to rotation, although experiments suggest they do not hinder neural-network performance.

Appendix B. Example: heat diffusion

The heat-diffusion example interprets graph convolution as spectral filtering by a heat kernel. Graph smoothing resembles spherical Gaussian smoothing, while kernel visualizations connect spectral behavior to spatial localization.

  • The heat-diffusion formulation defines convolution by applying a graph spectral kernel to the initial signal.The solution evolves the signal through the graph Laplacian and is identified as convolution with Kt.
  • The kernel Kt(λ) = e^−τtλ acts on graph eigenvalues interpreted as squared frequencies and generalizes Gaussian smoothing on the sphere.
  • Graph convolution produces results similar to spherical symmetric Gaussian smoothing, with relative differences of 10.4%, 4.8%, and 3.8%.
  • Graph convolution may substitute for spherical-harmonic convolution in applications such as denoising when exact convolution accuracy is unnecessary.The authors also suggest that learned neural-network kernels can compensate for imperfect rotational equivariance.
  • The kernel is visualized in the spectral domain, as a gnomonic projection on the sphere, and through an equatorial cross-section.Localization on the sphere is obtained by convolving the spectral kernel with a Kronecker delta.

Appendix C. Border effects

Reflective border conditions are irrelevant on the complete sphere but make graph filters near partial-sphere boundaries anisotropic. Zero padding can mitigate these effects when samples cover the same region.

  • Reflective border conditions affect convolution only when the graph represents part of the sphere rather than the complete sphere.
  • Zero-padding a small area around the region of interest makes these border effects similar to those in classical CNNs.
  • The authors do not expect border effects to cause problems when data samples cover the same area.
  • A filter localized near a boundary is no longer isotropic under the graph representation.The effect is illustrated for a filter localized with h(L)δi.

Appendix D. Projection onto a 2D plane

The authors avoid computationally expensive radial projections when comparing DeepSphere with a 2D CNN by exploiting HEALPix geometry to construct another projection.

  • Radial projection was considered too computationally expensive, especially with data augmentation, relative to neural-network training.The projection cost was reported to exceed the actual training cost.
  • The alternative projection leverages HEALPix's recursive subdivision of 12 base pixels into groups of four.

Appendix E. Augmentation of the dataset for the SVM classifier

The SVM comparison uses noise augmentation and training-set sizes chosen to match DeepSphere's effective data access. Training and validation errors are monitored until added samples no longer improve performance.

  • Random Gaussian noise is added to every sample before neural-network input to improve noise robustness and reduce overfitting.Because each presentation is noisily perturbed, the network does not see the same sample twice.
  • SVM training-set sizes are increased experimentally until additional training data no longer improves performance, matching DeepSphere's potentially unlimited sample access.
  • Figure E.16 shows converged training and validation errors for histogram features and a linear SVM at order o = 2 and relative noise level 1.5.The validation error may be slightly below the training error because validation performance selected the hyper-parameters.

Appendix F. Convergence Mass Maps

The weak-lensing data are full-sphere mass maps generated from dark-matter simulations and organized through lightcone construction across redshift shells. The analysis uses fast approximate L-PICOLA simulations for two cosmological models.

  • Data and map construction: UFalcon produces two-dimensional full-sphere mass maps representing dimensionless, weighted, projected mass along the line of sight.These projected maps are typically measured using weak gravitational lensing.
  • Data and map construction: Convergence-map generation begins with cosmological N-body simulations that evolve dark-matter particles under gravity and output particle positions at redshift-linked time steps.
  • Lightcone construction: A lightcone arranges particles from different redshifts into concentric shells around an observer, with redshift zero representing the present and higher redshifts earlier times.
  • Map dependence: The convergence map depends on cosmological parameters and on the particle count per pixel within each shell.
  • Simulation setup: The analysis uses fast approximate L-PICOLA simulations with Vsim = (4200 h−1 Mpc)3, mesh = 20483, and 30 realizations per cosmological model.The simulations cover z = 0.1 to z = 0.8 with shell thickness ∆z = 0.01.
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