ManifoldNet: A Deep Neural Network for Manifold-Valued Data With Applications
ManifoldNet: A Deep Neural Network for Manifold-Valued Data With Applications
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DOI：
10.1109/tpami.2020.3003846
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发表时间：
2022-02-01
影响因子：
23.6
通讯作者：
Vemuri, Baba C.
中科院分区：
文献类型：
--
作者：
Chakraborty, Rudrasis;Bouza, Jose;Vemuri, Baba C.
几何深度学习是一个相对新兴的领域，在过去几年中引起了人们的极大关注。这部分是由于从非欧几里德域获取的数据或从驻留在光滑流形上的欧几里德空间数据中提取的特征的可用性。例如，在计算机视觉中通常遇到的姿态数据驻留在李群中，而在许多领域中普遍存在的协方差矩阵和在医学成像领域中遇到的扩散张量驻留在对称正定矩阵的流形上。这些数据中的大部分自然表示为流形值数据的网格。在本文中，我们提出了一种新的理论框架，用于开发深度神经网络来科普这些流形值数据输入网格。我们还提出了一种新的架构来实现这一理论，并称之为流形网。类似于卷积等价于计算加权和的向量空间，流形值数据的“卷积”可以使用加权弗雷歇均值（wFM）来定义。(This需要赋予流形一个黎曼结构，如果它还没有的话。）ManifoldNet的隐藏层计算其输入的wFM，其中要学习权重。这意味着数据在通过隐藏层传播时保持流形值。为了降低计算复杂度，我们提出了一个可证明收敛的递归算法计算wFM。此外，我们证明了在非常数截面曲率流形上，每一个wFM层是一个压缩映射，并提供了建设性的证据，其不可压缩时，堆叠层。这抓住了深层网络层的两个基本属性。类似的卷积在欧氏空间的平移的等变，我们证明了wFM是等变的行动组的等距承认的黎曼流形上的数据驻留。为了展示ManifoldNet的性能，我们使用计算机视觉和医学成像数据集进行了几个实验。
Geometric deep learning is a relatively nascent field that has attracted significant attention in the past few years. This is partly due to the availability of data acquired from non-euclidean domains or features extracted from euclidean-space data that reside on smooth manifolds. For instance, pose data commonly encountered in computer vision reside in Lie groups, while covariance matrices that are ubiquitous in many fields and diffusion tensors encountered in medical imaging domain reside on the manifold of symmetric positive definite matrices. Much of this data is naturally represented as a grid of manifold-valued data. In this paper we present a novel theoretical framework for developing deep neural networks to cope with these grids of manifold-valued data inputs. We also present a novel architecture to realize this theory and call it the ManifoldNet. Analogous to vector spaces where convolutions are equivalent to computing weighted sums, manifold-valued data 'convolutions' can be defined using the weighted Frechet Mean (wFM). (This requires endowing the manifold with a Riemannian structure if it did not already come with one.) The hidden layers of ManifoldNet compute wFMs of their inputs, where the weights are to be learnt. This means the data remain manifold-valued as they propagate through the hidden layers. To reduce computational complexity, we present a provably convergent recursive algorithm for computing the wFM. Further, we prove that on non-constant sectional curvature manifolds, each wFM layer is a contraction mapping and provide constructive evidence for its non-collapsibility when stacked in layers. This captures the two fundamental properties of deep network layers. Analogous to the equivariance of convolution in euclidean space to translations, we prove that the wFM is equivariant to the action of the group of isometries admitted by the Riemannian manifold on which the data reside. To showcase the performance of ManifoldNet, we present several experiments using both computer vision and medical imaging data sets.