论文阅读 [TPAMI-2022] ManifoldNet: A Deep Neural Network for Manifold-Valued Data With Applications

论文搜索(studyai.com)

搜索论文: ManifoldNet: A Deep Neural Network for Manifold-Valued Data With Applications

搜索论文: http://www.studyai.com/search/whole-site/?q=ManifoldNet:+A+Deep+Neural+Network+for+Manifold-Valued+Data+With+Applications

关键字(Keywords)

Manifolds; Computer vision; Computer architecture; Biomedical imaging; Neural networks; Measurement; Standards; Weighted fréchet mean; equivariance; group action; riemannian manifolds

机器学习

流形学习; 网络量化压缩

摘要(Abstract)

Geometric deep learning is a relatively nascent field that has attracted significant attention in the past few years.

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广泛关注。.

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.

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。.

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.

例如，计算机视觉中常见的姿势数据位于李群中，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位于对称正定矩阵的流形上。.

Much of this data is naturally represented as a grid of manifold-valued data.

这些数据中的大部分自然地表示为多值数据的网格。.

In this paper we present a novel theoretical framework for developing deep neural networks to cope with these grids of manifold-valued data inputs.

在本文中，我们提出了一个新的理论框架，用于开发深度神经网络，以处理这些流形值数据输入的网格。.

We also present a novel architecture to realize this theory and call it the ManifoldNet.

我们还提出了一种新的体系结构来实现这一理论，并称之为ManifoldNet。.

Analogous to vector spaces where convolutions are equivalent to computing weighted sums, manifold-valued data ‘convolutions’ can be defined using the weighted Fréchet Mean (wFM{\sf wFM}wFMwFM).

与向量空间类似，在向量空间中，卷积相当于计算加权和，流形值数据“卷积”可以使用加权Fréchet均值（wFM{\sf wFM}wFMwFM）定义。.

(This requires endowing the manifold with a Riemannian structure if it did not already come with one.) The hidden layers of ManifoldNet compute wFM{\sf wFM}wFMwFMs of their inputs, where the weights are to be learnt.

（如果歧管还没有黎曼结构，则需要赋予其黎曼结构。）ManifoldNet的隐藏层计算其输入的wFM{\sf wFM}wFMwFMs，在那里学习权重。.

This means the data remain manifold-valued as they propagate through the hidden layers.

这意味着，当数据通过隐藏层传播时，它们仍然具有多重价值。.

To reduce computational complexity, we present a provably convergent recursive algorithm for computing the wFM{\sf wFM}wFMwFM.

为了降低计算复杂度，我们提出了一种可证明收敛的递归算法来计算wFM{\sf wFM}wFMwFM。.

Further, we prove that on non-constant sectional curvature manifolds, each wFM{\sf wFM}wFMwFM layer is a contraction mapping and provide constructive evidence for its non-collapsibility when stacked in layers.

进一步，我们证明了在非常数截面曲率流形上，每个wFM{\sf wFM}wFMwFM层都是一个收缩映射，并为其分层时的不可折叠性提供了建设性证据。.

This captures the two fundamental properties of deep network layers.

这抓住了深层网络层的两个基本属性。.

Analogous to the equivariance of convolution in euclidean space to translations, we prove that the wFM{\sf wFM}wFMwFM is equivariant to the action of the group of isometries admitted by the Riemannian manifold on which the data reside.

类似于欧几里德空间中卷积与平移的等价性，我们证明wFM{\sf wFM}wFMwFM与数据所在的黎曼流形所承认的等距群的作用是等价的。.

To showcase the performance of ManifoldNet, we present several experiments using both computer vision and medical imaging data sets…

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。。.

作者(Authors)

[‘Rudrasis Chakraborty’, ‘Jose Bouza’, ‘Jonathan H. Manton’, ‘Baba C. Vemuri’]