使用深度网络学习不变黎曼几何表示
Learning Invariant Riemannian Geometric Representations Using Deep Nets
摘要：
非欧几里德约束是计算机视觉和机器学习中许多类型数据所固有的，通常是由于在高级推理期间需要遵守的特定不变性要求。通常，这些几何约束可以用黎曼几何的语言来表达，其中传统的向量空间机器学习不直接应用。本文讨论的中心问题是：如何训练最终输出是黎曼流形上元素的深度神经网络？为了回答这个问题，我们提出了一个用于深度神经网络的流形感知训练的通用框架-我们利用切空间和指数映射将所提出的问题转换为一种形式，使我们能够将当前深度学习的进展应用于这个问题。我们描述了两个具体的应用程序来证明这种方法：预测的概率分布多类图像分类，并预测照明不变的子空间从一个单一的人脸图像通过回归的格拉斯曼。这些应用程序显示了所提出的框架的一般性，并导致在基线，忽略了输出空间的几何形状，在改进的性能。除了解决这个特定的问题，我们相信本文开辟了新的调查路线集中在深层架构的黎曼几何的影响。
英文摘要：
Non-Euclidean constraints are inherent in many kinds of data in computer vision and machine learning, typically as a result of specific invariance requirements that need to be respected during high-level inference. Often, these geometric constraints can be expressed in the language of Riemannian geometry, where conventional vector space machine learning does not apply directly. The central question this paper deals with is: How does one train deep neural nets whose final outputs are elements on a Riemannian manifold? To answer this, we propose a general framework for manifold-aware training of deep neural networks -- we utilize tangent spaces and exponential maps in order to convert the proposed problem into a form that allows us to bring current advances in deep learning to bear upon this problem. We describe two specific applications to demonstrate this approach: prediction of probability distributions for multi-class image classification, and prediction of illumination-invariant subspaces from a single face-image via regression on the Grassmannian. These applications show the generality of the proposed framework, and result in improved performance over baselines that ignore the geometry of the output space. In addition to solving this specific problem, we believe this paper opens new lines of enquiry centered on the implications of Riemannian geometry on deep architectures.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
