Sigma 流用于图像与数据标注及结构化预测学习
Sigma Flows for Image and Data Labeling and Learning Structured Prediction
AI总结：
本文提出 sigma 流模型，结合 Laplace-Beltrami 与分配流方法，通过黎曼梯度流和几何 PDE 实现结构化预测，并指出其与 transformer 的相似性，实验验证了其表现力。
摘要：
本文介绍了 sigma 流模型，用于预测在黎曼流形上观测到的数据的结构化标注，其中欧几里得图像域作为特例。该方法结合了约 25 年前由 Sochen、Kimmel 和 Malladi 提出的用于图像去噪和增强的 Laplace-Beltrami 框架，以及由作者引入和研究的分配流方法。sigma 流作为广义调和能量的黎曼梯度流出现，因此受一个非线性几何偏微分方程支配，该方程确定了一个从闭黎曼域流形到统计流形的调和映射，其中统计流形配备了来自信息几何的 Fisher-Rao 度量。sigma 流的一个特定成分是域流形的黎曼度量与演化状态之间的相互依赖。这使得该方法以特定方式适用于机器学习，通过一个具有紧凑时变参数化的映射来实现这种依赖，该映射可以从数据中学习。概念验证实验展示了 sigma 流模型的表现力和预测性能。文中指出了 sigma 流与 transformer 网络架构以及通过 sigma 流的几何积分生成的网络之间的结构相似性，这突出了与深度学习的联系，并且反过来可能促进在其他科学机器学习领域中使用几何设计原则进行结构化预测。
英文摘要：
This paper introduces the sigma flow model for the prediction of structured labelings of data observed on Riemannian manifolds, including Euclidean image domains as special case. The approach combines the Laplace-Beltrami framework for image denoising and enhancement, introduced by Sochen, Kimmel and Malladi about 25 years ago, and the assignment flow approach introduced and studied by the authors. The sigma flow arises as Riemannian gradient flow of generalized harmonic energies and thus is governed by a nonlinear geometric PDE which determines a harmonic map from a closed Riemannian domain manifold to a statistical manifold, equipped with the Fisher-Rao metric from information geometry. A specific ingredient of the sigma flow is the mutual dependency of the Riemannian metric of the domain manifold on the evolving state. This makes the approach amenable to machine learning in a specific way, by realizing this dependency through a mapping with compact time-variant parametrization that can be learned from data. Proof of concept experiments demonstrate the expressivity of the sigma flow model and prediction performance. Structural similarities to transformer network architectures and networks generated by the geometric integration of sigma flows are pointed out, which highlights the connection to deep learning and, conversely, may stimulate the use of geometric design principles for structured prediction in other areas of scientific machine learning.
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