H-CNNs: Convolutional Neural Networks for Riemannian Homogeneous Spaces
H-CNNs: Convolutional Neural Networks for Riemannian Homogeneous Spaces
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H-CNN：黎曼齐次空间的卷积神经网络
DOI：
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发表时间：
2018
期刊：
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通讯作者：
B. Vemuri
中科院分区：
文献类型：
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作者：
Rudrasis Chakraborty;Monami Banerjee;B. Vemuri
卷积神经网络在机器学习应用中无处不在，用于解决各种问题。然而，当数据自然地驻留在常见的流形上时，它们不能被使用，例如球面、特殊正交群、格拉斯曼流形、对称正定矩阵流形等。最近，一些研究小组报告了CNN对驻留在球体上的数据的推广，这些研究小组有几个名称，但我们将其称为球形CNN（SCNN）。SCNN与标准CNN不同的关键属性是它们表现出旋转等方差属性。在本文中，我们从理论上将SCNN推广到黎曼齐次流形，其中包括许多常见的流形，包括上述示例流形。涉及在$（3）的流形上生成的合成数据的概念验证实验 imes 3）$对称正定矩阵与$mathbf{R}^+的乘积流形 矩阵{S}^2$。这些歧管通常在扩散磁共振成像（一种非侵入性医学成像模式）中遇到。
Convolutional neural networks are ubiquitous in Machine Learning applications for solving a variety of problems. They however can not be used as is when data naturally reside on commonly encountered manifolds such as the sphere, the special orthogonal group, the Grassmanian, the manifold of symmetric positive definite matrices and others. Most recently, generalization of CNNs to data residing on a sphere has been reported by some research groups, which go by several names but we will refer to them as spherical CNNs (SCNNs). The key property of SCNNs distinct from the standard CNNs is that they exhibit the rotational equivariance property. In this paper, we theoretically generalize the SCNNs to Riemannian homogeneous manifolds, that include many commonly encountered manifolds including the aforementioned example manifolds. Proof of concept experiments involving synthetic data generated on the manifold of $(3 imes 3)$ symmetric positive definite matrices and the product manifold of $mathbf{R}^+ imes mathbf{S}^2$ respectively, are presented. These manifolds are commonly encountered in diffusion magnetic resonance imaging, a non-invasive medical imaging modality.