最大编码率缩减（MCR²）的全局几何分析
A Global Geometric Analysis of Maximal Coding Rate Reduction
AI总结：
本文针对最大编码率缩减（MCR²）目标缺乏全局优化景观理论支撑的问题，完整刻画其各类临界点性质，证明其具备利于一阶优化的良好景观，并通过实验验证理论结论。
摘要：
用于学习结构化、紧凑深度表示的最大编码率缩减（maximal coding rate reduction，MCR²）目标正受到越来越多的关注，尤其是近期它被用于推导完全可解释且高效的深度网络架构之后。但该目标目前缺乏完整的理论论证：人们仅了解其全局最优解的性质，尚未对其全局优化景观开展研究。本工作完整刻画了MCR²所有局部最优解、全局最优解以及其他类型临界点的性质。具体而言，我们证明MCR²问题的每一个（局部或全局）极大值点都对应低维、有判别力且多样化的表示，此外该目标的每一个临界点要么是局部极大值点，要么是严格鞍点。这种良好的优化景观使得MCR²成为通过一阶优化方法学习多样化、有判别力表示的理想目标选择。为验证我们的理论发现，我们在合成数据集与真实数据集上开展了大量实验。
英文摘要：
The maximal coding rate reduction (MCR$^2$) objective for learning structured and compact deep representations is drawing increasing attention, especially after its recent usage in the derivation of fully explainable and highly effective deep network architectures. However, it lacks a complete theoretical justification: only the properties of its global optima are known, and its global landscape has not been studied. In this work, we give a complete characterization of the properties of all its local and global optima, as well as other types of critical points. Specifically, we show that each (local or global) maximizer of the MCR$^2$ problem corresponds to a low-dimensional, discriminative, and diverse representation, and furthermore, each critical point of the objective is either a local maximizer or a strict saddle point. Such a favorable landscape makes MCR$^2$ a natural choice of objective for learning diverse and discriminative representations via first-order optimization methods. To validate our theoretical findings, we conduct extensive experiments on both synthetic and real data sets.
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。
