数据分类的深度学习度量拓扑
A Metric Topology of Deep Learning for Data Classification
AI总结：
本文从度量拓扑视角研究数据分类的深度学习，提出一种基于概率的距离度量，使分类性能相近的网络在商集上构成紧致度量空间，为理解DL提供新数学基础。
摘要：
经验上，深度学习（DL）在实际应用中已展现出前所未有的成功。然而，DL在很大程度上仍然是一个神秘的“黑箱”，这促使近期的理论研究为其构建数学基础。在本文中，我们通过度量拓扑的视角研究用于数据分类的深度学习。考虑到网络参数空间上传统的欧几里得度量通常无法根据分类结果区分DL网络，我们从概率的角度提出了一种有意义的距离度量，使得产生相似分类性能的DL网络彼此接近。所提出的距离度量在网络参数向量之间定义了这样一种等价关系，即表现同样好的网络属于同一等价类。有趣的是，我们提出的距离度量可以被证明在模等价关系的商集上作为一个度量。然后，在相当温和的条件下，我们证明了除了一个可能预测非唯一标签的、可忽略的小网络子集外，我们提出的度量空间是紧致的，并且与众所周知的商拓扑空间一致。我们的研究有助于对DL的基本理解，并开辟了利用丰富的度量空间理论研究DL的新途径。
英文摘要：
Empirically, Deep Learning (DL) has demonstrated unprecedented success in practical applications. However, DL remains by and large a mysterious "black-box", spurring recent theoretical research to build its mathematical foundations. In this paper, we investigate DL for data classification through the prism of metric topology. Considering that conventional Euclidean metric over the network parameter space typically fails to discriminate DL networks according to their classification outcomes, we propose from a probabilistic point of view a meaningful distance measure, whereby DL networks yielding similar classification performances are close. The proposed distance measure defines such an equivalent relation among network parameter vectors that networks performing equally well belong to the same equivalent class. Interestingly, our proposed distance measure can provably serve as a metric on the quotient set modulo the equivalent relation. Then, under quite mild conditions it is shown that, apart from a vanishingly small subset of networks likely to predict non-unique labels, our proposed metric space is compact, and coincides with the well-known quotient topological space. Our study contributes to fundamental understanding of DL, and opens up new ways of studying DL using fruitful metric space theory.
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。
