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

“A CNN is composed of the following main building blocks [32]: (i) a convolution, (ii) a non-linearity, and, optionally, (iii) a down-sampling operation, (iv) a pooling operation, and (v) a normalization.”

“Fully convolutional networks (FCNs) have been introduced by [42] and are mostly used for the semantic segmentation of images, a dense classification task.”

“The batch normalization layer [33] computes Y = BN(X) such as”

“Observables that cover large parts of the sky, such as the Cosmic Microwave Background (CMB) [1–3], neutral hydrogen [4, 5], galaxy clustering [6], gravitational lensing [7, 8], and others, have been used to constrain cosmological and astrophysical models.”

“We use the graph CNN formulation introduced in [26], and a pooling strategy that exploits a hierarchical pixelisation of the sphere to analyse the data at multiple scales.”

“As HEALPix [27] is our target application, we tailor the method to that particular sampling in the subsequent exposition.”

“For this contribution, we chose the formulation of [26] as its root on strong graph signal processing theory makes the concept of convolutions and filters explicit [36].”

“Many formulations of graph neural networks, reviewed by [59] and [60], have been proposed.”

“Second, they did not take 11 Heisenberg’s uncertainty principle states that a filter cannot be arbitrarily concentrated in one domain without being de-concentrated in the other. advantage of a hierarchical pixelization of the sphere and resorted to dynamic pooling [58].”

“We employ the Hierarchical Equal Area isoLatitude Pixelation (HEALPix) [10], which was designed for efficient and incremental discretization of full-sky maps in application to the satellite missions to measure the cosmic microwave background in astrophysics [30].”

“Although we implemented spherical convolution layers as described by Esteves et al. (2018), other architectures also exist and warrant investigation in the context of microstructural parameter estimation.”

“Most graph-based convolutions eliminate orientation ambiguity by utilizing radially isotropic filters [PDKS19, ZXW*19].”

“Frameworks that use spherical harmonics CNNs can be divided into two groups: Point-based SCNN that extract features based on point maps or pairwise relations [9], [8] and the other group that uses spherical harmonics convolution on images casted on the sphere [6], [7].”

“, SphereNet [20], Spherical CNNs [19,22], Spherical CNNs on Unstructured Grids [26], and DeepSphere [44].”

“HEALPix has also been used in other machine learning contexts where spherical data is natural [14, 15].”

“…Smets et al., 2020), 3D rotations (Cohen et al., 2018; Esteves et al., 2018; Worrall & Brostow, 2018; Weiler et al., 2018a; Kondor et al., 2018a; Perraudin et al., 2019), permutations (Zaheer et al., 2017; Preprint Hartford et al., 2018; Kondor et al., 2018b; Maron et al., 2019a; 2020), general…”

“There have been a number of studies proposing CNNs for omnidirectional data [5, 8, 6, 29, 21, 36].”

“Other approaches that do not use 2D planar CNNs, such as generalized FFT-based CNNs on sphere [9, 11], distortion-aware sampling [24, 6], and graph convolution on sampled grid on sphere [18, 27], were also pro-”

“…data, SO 3 -equivariance can be obtained with harmonic methods (Cohen et al., 2018; Kondor et al., 2018; Ocampo et al., 2023) or via isotropic Laplacian convolutions on spherical graphs (Perraudin et al., 2019), the latter of which we extend to ℝ 3 × S 2 dMRI inputs for E 3 × SO 3 equivariance.”

“Other examples include LiDAR data in autonomous driving [5], planetary data in climate science [6], and observations of theuniverseincosmology[7].”

“In recent studies, DeepSphere [17] proposes a spherical CNNs by using graph neural networks for cosmological data analysis.”

“For spherical data, SO ( 3 ) -equivariant convolutions are often achieved via spherical harmonics-parameterized convolutions [16, 21, 42] or isotropic convolutions on spherical graphs [53].”

“Perraudin et al. [153] used Hierarchical Equal Area isoLatitude Pixelisation (HEALPix) sampling of the sphere on DeepSphere for cosmological applications.”

“Following [F22, 39], our fiducial choice of architecture is the TensorFlow [139] implementation of the DeepSphere [140] graph convolutional neural network developed in [112, 113].”

“Therefore, several CNN techniques are developed to perform convolution over spherical data [45–49].”

“Bernreiteret et al. [44] employ spherical CNNs [45], [46], [47] to learn a multi-modal descriptor from spherical and visual images.”

“We use the weighted adjacency matrix W suggested in [12]”

“In contrast, DeepSphere (Perraudin et al. 2019) proposed a deep learning model using Graph Convolutional Networks (GCNs) tailored for HEALPix sky maps, treating each pixel as a node in a graph structure.”

“…et al. 2019; Merten et al. 2019; Peel et al. 2019; Tewes et al. 2019), the cosmic microwave background (Caldeira et al. 2018; Rodriguez et al. 2018; Perraudin et al. 2019; Münchmeyer & Smith 2019; Mishra et al. 2019), LSS for estimating cosmological parameters from the distribution of matter…”

“One of the earliest successes includes architectures like Convolutional Neural Networks [9, 10], and more recently, applications such as medical image analysis [21, 22, 23], cosmology [24, 25], and physics/chemistry [26, 27, 28].”

“2005)) scheme as is done in DeepSphere (Perraudin et al. 2019).”

“One way is to use a spherical CNN approach as presented in [52].”

“Symmetry plays a crucial role in creating efficient neural architectures in various domains (Reiser et al., 2022; Wang et al., 2020; Bogatskiy et al., 2020; Perraudin et al., 2019; Cohen & Welling, 2016; Gordon et al., 2019; Dehmamy et al., 2021).”

“Learning directly on the sphere avoids ERP singularities and allows operators to respect the manifold structure [36, 39, 40].”

“…three residual branches: a dense mixing over all directions with a learned M × M matrix (initialized as the identity) [32], as the sidelobes reach the full field of view, a graph convolution over neighboring directions (Chebyshev filter of order 3 [33], [34]) and a dilated convolution over range.”

“Several studies leveraged the spherical representation of SRP-PHAT maps to develop spherical convolutional recurrent neural networks, enabling efficient DOA estimation with reduced computational complexity [27–29].”

“Spherical-CNN [39] was used for 3D point cloud processing in some of the applica - tions [40, 41 ].”

“SphereNet [41] and its variants [42–44] formulate CNNs on spherical surfaces.”

“Previous work has applied DeepSphere to cosmological mock data with HEALPix [52]; and to KiDS-1000 weak lensing maps for parameter inference [53].”

“…different approaches for spherical convolutions have been proposed such as spectral learning [4,24], 2D processing [28] and graph or multi-graph based representations [12,22,33], it hasn’t been possible to assure these 3 key properties simultaneously to guarantee a proper spherical convolution.”

“The primary objective is to incorporate a sequence of layers to the standard contracting network, with an upsampling operation instead of the pooling operation.”

“Another approach to spherical data is presented in [4, 5, 22] which uses pooling and sampling over a graph created on the surface of a sphere, to learn classification and possibly other tasks.”

“Our method can be seen as an extension of the original ChebNet [Defferrard et al., 2016, Perraudin et al., 2019].”

“Graph-based approaches, notably DeepSphere [23, 73], represent the sphere as a HEALPix graph and apply Chebyshev graph convolutions with controllable equivariance error, achieving performance comparable to spectral methods at lower cost, which suggests that anisotropic filters may be unnecessary…”

“More recently, machine-learning techniques have been applied directly to cosmic-topology classification [20], building on spherical neural-network methods developed for CMB and cosmological data [21, 22].”

“Cohen and Welling (2016) introduced group-equivariant CNNs, followed by a general theoretical treatment in Kondor and Trivedi (2018).”

“…has been shown to be computationally more efficient than harmonic-based methods and yields superior performance in tasks such as cosmological model discrimination, particularly in scenarios involving high noise levels or partial sky coverage, confirming its utility for next-generation surveys [14].”

“Other work has focused on the novelty of pixelization of the sphere, including DeepSphere [27, 79] which employs graph convolutions to address the non-regular grid and varied intra-pixel distances.”

“Perraudin et al. [13] presented a spherical CNN that constructed by representing the sphere as a graph, and utilized the graph-based representation to define the standard CNN operations.”

“Our framework employs the graph-based spherical convolution developed in DeepSphere [15] as the basis of the spherical U-Net architecture [16].”

“Originally developed for cosmic microwave background analysis [54], its remarkable properties have started to attract attention for depth estimation [39] and image compression/rendering [37], [38].”

“Although we implemented spherical convolution layers as described by Esteves et al. (2018), other architectures also exist and warrant investigation in the context of microstructural parameter estimation.”

“It has been adopted for CNNs on the sphere [56] and for learned compression.”

“HEALPix provides an alternative equal-area representation of the sphere [13], which has been adopted by deep learning methods such as DeepSphere, HEALPix CNNs, OSLO, SCE-Depth, and HEALPix-guided diffusion models [14–18], and weather forecasting models including DLWP-HPX, HEAL-ViT, ESyM [19–21].”

“OSLO (Bidgoli et al., 2022) instead learns directly on the sphere, building on HEALPix-based spherical networks such as DeepSphere (Perraudin et al., 2019).”

“…learning models are built to reconstruct the properties of atmospheric neutrinos (direction, energy and PID), including a spherical graph-convolution neural network (GCNN), Deepsphere [3], and two planar convolution neural network (CNN)/Transformer models, EfficientNetV2-S [4] and CoAtNet [5].”

“Examples include full-sky cosmological observations such as cosmic microwave background and large-scale structure maps [6, 24], global climate and Earth observation fields [7, 32], molecular surface representations in computational chemistry [21, 33], and omnidirectional visual data captured by 360…”

“While some methods transform spherical maps to other domains using schemes such as pixelization [29], [30] or the icosahedral polyhedron representation [31], [32], incorporating corresponding CNN feature extraction strategies, a recent study [33] focuses on the sphere surface.”

“More recent graph-based [32–36] and Transformer-based [37–43] spherical models enhance spatial adaptability and nonlinear modeling, yet often embed the sphere as an unstructured graph or sequence, compromising spherical geometric consistency and incurring substantial computational cost [27, 44, 45].”

“The hyperparameters of the LSACAN2Nets model are optimized in this approach, using the Geyser Inspired Algorithm (GIA) [15].”

“The PKO optimizer, as described in [16], simulates the approach pied kingfishers use to find their meals.”

“…developed SCNN was implemented within the PyTorch environment, with spherical data processing partially built upon publicly available DeepSphere 27,28 and HEALPix 30 function was set to mean squared error, with the Adam optimizer set at an initial learning rate of 0.001, which was reduced by a…”

“Meanwhile, other works dedicate to spherical convolution kernels based on spherical harmonics and spectral analysis [14–16], thus enabling direct computation on the spherical domain.”

“A model based on DeepSphere [10] is designed to process the spherical images formed by PMT waveform features.”

“To accommodate large regions of interest (ROI) for which the sky curvature becomes relevant, we build our NNs onto the DeepSphere architecture ([14, 39], see [30, 31, 36] for gamma-ray applications), which models the sky as a graph based on the Healpix tessellation of the sphere [20] and provides…”

“The most recent research efforts have been in attaining more uniform spherical samplings, and designing rotationally equivariant convolutions (Perraudin et al. 2019; Defferrard et al. 2020; Shen et al. 2022).”

“Our CNN architecture is built on the graph-based DeepSphere framework [47,48].”

“The inclusion of nonlinearities together with multiple layers dons Spherical CNNs with an enhanced representation power, capable of capturing nonlinear relationships, and exhibiting superior performance when processing spherical signals [16], [17], [21].”

“We use the weighted adjacency matrix W suggested in [60]”

“Because the HEALPix partitioning scheme is commonly employed for data analysis in cosmology and astronomy, we expect the proposed developments to be of significant importance for the above-mentioned signal processing problems in the fields of cosmology [38], astronomy [39], astrophysics [40] and geoscience [41].”