Convolutional neural networks on the HEALPix sphere: a pixel-based algorithm and its application to CMB data analysis
Convolutional neural networks on the HEALPix sphere: a pixel-based algorithm and its application to CMB data analysis
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DOI：
10.1051/0004-6361/201935211
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发表时间：
2019-08-20
影响因子：
6.5
通讯作者：
Tomasi, M.
中科院分区：
文献类型：
--
作者：
Krachmalnicoff, N.;Tomasi, M.
我们描述了一种使用分层等面积纬度像素化方案 (HEALPix) 将卷积神经网络 (CNN) 应用于球体上定义的场的新颖方法。具体来说，我们开发了一种基于像素的方法来在球面上实现卷积层和池化层，类似于应用于欧几里德空间的 CNN 的常见做法。我们算法的主要优点是可以与现有的高度优化的神经网络库（例如 PyTorch、TensorFlow 等）完全集成。我们提出了我们的方法的两个应用：（i）识别投影在球体上的手写数字； (ii) 根据宇宙微波背景（CMB）的模拟图估计宇宙学参数。后者代表了这项探索性工作的主要目标，其目标是展示我们的 CNN 在 CMB 参数估计中的适用性。我们构建了一个简单的神经网络架构，由四个卷积层和池化层组成，并将其用于本文探讨的所有应用程序。对于手写数字的识别，我们的 CNN 达到了接近 95% 的准确率，与其他现有的球形 CNN 相当，而且无论图像在球体上的位置和方向如何，都是如此。对于 CMB 相关的应用，我们在模拟宇宙学参数的估计上测试了 CNN，定义了高斯场的功率谱投影在球体峰值上的角度尺度。在几种情况下，我们直接从模拟地图估计此参数的值：温度和偏振图、白噪声的存在和部分覆盖的地图。对于温度图，神经网络的性能与基于标准频谱的贝叶斯方法的性能相当。对于极化，CNN 的性能比标准算法差大约四倍。尽管如此，我们的结果首次证明，CNN 能够从全天空和掩模图中的偏振场中提取信息，并能够区分像素空间中的 E 模式和 B 模式。最后，我们将 CNN 应用于从模拟 CMB 图估计再电离 (tau) 时的汤姆森散射光学深度。即使没有对神经网络架构进行任何特定的优化，我们也能达到与标准贝叶斯方法相当的精度。这项工作代表了在 CMB 参数估计中利用神经网络的第一步，并证明了我们方法的可行性。
We describe a novel method for the application of convolutional neural networks (CNNs) to fields defined on the sphere, using the Hierarchical Equal Area Latitude Pixelization scheme (HEALPix). Specifically, we have developed a pixel-based approach to implement convolutional and pooling layers on the spherical surface, similarly to what is commonly done for CNNs applied to Euclidean space. The main advantage of our algorithm is to be fully integrable with existing, highly optimized libraries for NNs (e.g., PyTorch, TensorFlow, etc.). We present two applications of our method: (i) recognition of handwritten digits projected on the sphere; (ii) estimation of cosmological parameter from simulated maps of the cosmic microwave background (CMB). The latter represents the main target of this exploratory work, whose goal is to show the applicability of our CNN to CMB parameter estimation. We have built a simple NN architecture, consisting of four convolutional and pooling layers, and we have used it for all the applications explored herein. Concerning the recognition of handwritten digits, our CNN reaches an accuracy of similar to 95%, comparable with other existing spherical CNNs, and this is true regardless of the position and orientation of the image on the sphere. For CMB-related applications, we tested the CNN on the estimation of a mock cosmological parameter, defining the angular scale at which the power spectrum of a Gaussian field projected on the sphere peaks. We estimated the value of this parameter directly from simulated maps, in several cases: temperature and polarization maps, presence of white noise, and partially covered maps. For temperature maps, the NN performances are comparable with those from standard spectrum-based Bayesian methods. For polarization, CNNs perform about a factor four worse than standard algorithms. Nonetheless, our results demonstrate, for the first time, that CNNs are able to extract information from polarization fields, both in full-sky and masked maps, and to distinguish between E and B-modes in pixel space. Lastly, we have applied our CNN to the estimation of the Thomson scattering optical depth at reionization (tau) from simulated CMB maps. Even without any specific optimization of the NN architecture, we reach an accuracy comparable with standard Bayesian methods. This work represents a first step towards the exploitation of NNs in CMB parameter estimation and demonstrates the feasibility of our approach.