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].”

“Such global summarization as the AV layer is commonly used along graph convolutions to classify graphs of varying sizes (as it is invariant to vertex permutation) [45, 46].”

“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.”

“Such global summarization as the AV layer is commonly used along graph convolutions to classify graphs of varying sizes (as it is invariant to vertex permutation) [45, 46].”

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

“This idea has been generalized to arbitrary 2D manifolds for shape alignment and retrieval [50–52].”

“Localization is desired for the transformation to be stable to local deformation [57], and most interactions in the data are local anyway.”

“By combining graph convolutional layers and recurrent layers [66], they can, for example, model structured time series such as tra ffi c on road networks [67], or recursively complete matrices for recommendation [68].”

“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.”

“…of modeling the data domain with a graph allows one to easily model [17], with galactic plane masked, (middle) map of galaxy number counts (number of galaxies per arcmin 2 ) in SDSS DR14 [18], and (right) simulated weak lensing convergence map (dimensionless) simulated with DES DR1 mask [19].”

“This idea has been generalized to arbitrary 2D manifolds for shape alignment and retrieval [50–52].”

“See [56] for a rigorous treatment of convolution and equivariance in NNs.”

“By combining graph convolutional layers and recurrent layers [66], they can, for example, model structured time series such as tra ffi c on road networks [67], or recursively complete matrices for recommendation [68].”

“This idea has been generalized to arbitrary 2D manifolds for shape alignment and retrieval [50–52].”

“Thanks to their versatility, graph neural networks have been used in a variety of tasks, such as identifying diseases from brain connectivity networks [61] or population graphs [62], designing drugs using molecular graphs [63], segmenting 3D point clouds [64], optimizing shapes to be aerodynamic [65], and many more.”

“A first approach, explored by [20] for molecular modeling and [21, 22] for omnidirectional imaging, is to use a 2D CNN on a discretisation of the sphere that is a grid, such as the equirectangular projection (Figure 6, left panel), or a set of grids, such as the cubed-sphere defined by [47] (Figure…”

“This approach has been extensively used for cosmological maps [9, 11, 12, 23] and omnidirectional imaging [48, 49].”

“…log N pix , plus N pix matrix multiplications of size N pix for a total cost of O N 3 / 2 pix operations. leveraged to avoid SHTs between layers (it does a non-linear transformation in the spectral domain), the transform itself costs O ( N 3 / 2 pix ), for no reduction of overall complexity [55].”

“…to their versatility, graph neural networks have been used in a variety of tasks, such as identifying diseases from brain connectivity networks [61] or population graphs [62], designing drugs using molecular graphs [63], segmenting 3D point clouds [64], optimizing shapes to be aerodynamic…”

“Left: a grid using equiang l r spacing in a standard spherical-pol r c ordinate syst m, and Right: An equiang lar c bed-sphere rep sentatio , as describ d in Ronchi et al. (1996).”

“Indeed, [34] showed that the graph Laplacian converges to the Laplace-Beltrami when the number of pixels goes to in-finity providing uniform sampling of the manifold and a fully connected graph built with exponentially decaying weights.”

“…graph neural networks have been used in a variety of tasks, such as identifying diseases from brain connectivity networks [61] or population graphs [62], designing drugs using molecular graphs [63], segmenting 3D point clouds [64], optimizing shapes to be aerodynamic [65], and many more.”

“We refer the reader to [37, Sec 2.2] for a detailed discussion of the connection between translation T i and localization T i .”

“This approach has been extensively used for cosmological maps [9, 11, 12, 23] and omnidirectional imaging [48, 49].”

“Accelerated schemes however exist for some sampling sets [see 39, 53, 54, for examples].”

“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.”

“Accelerated schemes however exist for some sampling sets [see 39, 53, 54, for examples].”

“L-PICOLA [72] is used for fast and approximate N-body simulations.”

“A typical global regression task is to infer the parameters of a cosmological model from maps [11, 23].”

“One approach is to apply a standard 2D CNN to a grid discretisation of the sphere [20–22].”

“Ideally, we would compare DeepSphere to alternative spherical CNNs like [24] and [25].”

“Thanks to their versatility, graph neural networks have been used in a variety of tasks, such as identifying diseases from brain connectivity networks [61] or population graphs [62], designing drugs using molecular graphs [63], segmenting 3D point clouds [64], optimizing shapes to be aerodynamic [65], and many more.”

“Convolutional Neural Networks (CNNs) have been proposed as an alternative analysis tool in cosmology thanks to their ability to automatically design relevant statistics to maximise the precision 1 of the parameter estimation [9–16], while maintaining robustness to noise.”

“A typical global regression task is to infer the parameters of a cosmological model from maps [11, 23].”

“For example, [35] uses the inverse of the distance instead.”

“On a single core, a naive implementation of our method is ten to twenty times faster for N side = 2048, with K = 20 and K = 5, respectively, than using the spherical harmonic transform at (cid:96) max = 3 N side implemented by the highly optimized libsharp [39] library used by HEALPix.”

“As no efficient and general fast Fourier transform (FFT) exists for graphs [38], the execution of the Fourier transform by multiplication of the signal f by the dense matrix U costs O(N2 pix) operations.”

“Segmentation and feature detection [41] are examples of dense classification tasks.”

“Convolutional Neural Networks (CNNs) have been proposed as an alternative analysis tool in cosmology thanks to their ability to automatically design relevant statistics to maximise the precision1 of the parameter estimation [9–16], while maintaining robustness to noise.”

“…graph neural networks have been used in a variety of tasks, such as identifying diseases from brain connectivity networks [61] or population graphs [62], designing drugs using molecular graphs [63], segmenting 3D point clouds [64], optimizing shapes to be aerodynamic [65], and many more.”

“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.”

“This approach has been extensively used for cosmological maps [9, 11, 12, 23] and omnidirectional imaging [48, 49].”

“tures that are (i) the power spectral densities (PSD) of maps, and (ii) the histogram of pixels in the maps [72].”

“The simulations are created using the fast lightcone method UFalcon described in [71].”

“Some dense regression tasks are denoising, interpolation of missing values, or inpainting parts of a map [40].”

“These kind of maps can be created using the gravitational lensing technique [see 29, for review].”

“Another trend, parallel to the modeling of structured data, is the use of graph neural networks for relational reasoning [69].”

“spherical CNNs [19, 5, 20, 21, 22, 23] or more general models of geometric deep learning [24, 25, 26, 27].”

“A possible solution in the horizontal could be to use spherical convolutions (Cohen et al., 2018; Perraudin et al., 2019; Jiang et al., 2019) or to feed in latitude information to the network.”

“Spherical CNNs Besides the general theoretical framework of gauge equivariant convolution, we present in this paper a specific model (the Icosahedral CNN), which can be viewed as a fast and simple alternative to Spherical CNNs (Cohen et al., 2018b; Esteves et al., 2018; Boomsma & Frellsen, 2017; Su & Grauman, 2017; Per…”