[论文解读] A CNN for homogneous Riemannian manifolds with applications to Neuroimaging

Rudrasis Chakraborty, Monami Banerjee+1 more|arXiv (Cornell University)|May 14, 2018

Advanced Neuroimaging Techniques and Applications参考文献 10被引用 8

一句话总结

本文提出了一种同构卷积神经网络（HCNN），通过证明线性群等变系统在这些流形上完全由相关性表征，将CNN推广至黎曼齐性流形（如球面、格拉斯曼流形和对称正定矩阵）——实现了对群等变相关性的理论基础构建，并通过基于dMRI扫描的帕金森病端到端分类验证了该方法，测试准确率达86.5%；在P3流形上的合成数据上，训练准确率达92.5%。

ABSTRACT

Convolutional neural networks are ubiquitous in Machine Learning applications for solving a variety of problems. They however can not be used in their native form when the domain of the data is 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 domains such as the 2-sphere has been reported by some research groups, which is referred to as the spherical CNNs (SCNNs). The key property of SCNNs distinct from CNNs is that they exhibit the rotational equivariance property that allows for sharing learned weights within a layer. In this paper, we theoretically generalize the CNNs to Riemannian homogeneous manifolds, that include but are not limited to the aforementioned example manifolds. Our key contributions in this work are: (i) A theorem stating that linear group equivariance systems are fully characterized by correlation of functions on the domain manifold and vice-versa. This is fundamental to the characterization of all linear group equivariant systems and parallels the widely used result in linear system theory for vector spaces. (ii) As a corrolary, we prove the equivariance of the correlation operation to group actions admitted by the input domains which are Riemannian homogeneous manifolds. (iii) We present the first end-to-end deep network architecture for classification of diffusion magnetic resonance image (dMRI) scans acquired from a cohort of 44 Parkinson Disease patients and 50 control/normal subjects. (iv) A proof of concept experiment involving synthetic data generated on the manifold of symmetric positive definite matrices is presented to demonstrate the applicability of our network to other types of domains.

研究动机与目标

• 将CNN在欧几里得空间中的成功推广至球面、格拉斯曼流形和对称正定矩阵等非欧几里得数据域。
• 为黎曼齐性流形上的群等变深度学习建立理论基础。
• 设计一种用于分类帕金森病患者与对照组的dMRI扫描的端到端深度网络架构。
• 在对称正定矩阵流形（P3）的合成数据上验证该框架。

提出的方法

• 通过相关性操作，对黎曼齐性流形上的线性群等变系统进行理论表征。
• 证明相关性在流形上群作用下保持等变性，推广了欧几里得空间中卷积的等价性。
• 设计基于群不变相关层、ReLU激活函数和全连接层的HCNN架构。
• 将HCNN应用于dMRI数据，分别使用原始扩散MR信号和基于球面反卷积的EAP表示。
• 在P3上使用高斯分布生成合成数据，以测试分类性能。
• 采用置换检验评估dMRI分类中组间与组内体素特征的统计显著性。

实验结果

研究问题

• RQ1 群等变系统与相关性的理论等价性，能否从向量空间推广至黎曼齐性流形？
• RQ2 在黎曼齐性流形上，相关性是否在群作用下保持等变性？
• RQ3 HCNN能否在帕金森病患者与对照组的dMRI扫描中实现高分类准确率？
• RQ4 组间与组内体素特征在dMRI数据中区分疾病状态时的贡献如何？
• RQ5 HCNN能否推广至非紧致黎曼齐性流形（如P3）？

主要发现

• HCNN在44名帕金森病患者与50名对照组的dMRI扫描中达到86.5%的测试准确率，证明其在神经影像分类中的可行性。
• 置换检验显示，组间特征具有统计显著性（p < 0.01），而仅使用组内特征时准确率仅为50%。
• 在原始dMR信号上使用3D空间卷积将分类准确率提升至66.67%，证实组内与组间特征学习的重要性。
• 基于EAP的表示仅在左侧腹侧被盖区域具有统计显著性（p = 0.01），表明其相比原始dMR信号具有更高表达能力。
• 在P3上的合成数据中，HCNN达到92.5%的训练准确率与86.5%的测试准确率，验证了该框架在非紧致齐性空间上的有效性。
• 相关性在群作用下保持等变性的理论证明，相较于先前工作更为简洁且具有独创性，且该性质在单个层内成立，而非整个网络层面。