A Global Geometric Analysis of Maximal Coding Rate Reduction

2024-06-04

International Conference on Machine Learning

AI Summary

This work addresses the lack of theoretical foundations for the Maximum Coding Rate Reduction (MCR²) objective. We present the first complete global landscape analysis of its loss function. By integrating tools from differential geometry, information geometry, and nonconvex optimization theory, we rigorously prove that all critical points of MCR² are either local maxima or strict saddle points—non-degenerate local minima do not exist. This benign landscape property ensures that first-order optimization algorithms converge efficiently to high-quality representations that simultaneously exhibit discriminability, diversity, and low dimensionality. Our theoretical analysis further reveals that MCR² is inherently compatible with gradient-based methods. Extensive experiments on both synthetic and real-world datasets validate the effectiveness and robustness of MCR² in learning compact, informative representations. Collectively, this work establishes a rigorous geometric and optimization-theoretic foundation for representation learning grounded in the MCR² principle.

Technology Category

• Machine Learning: Representation Learning
• Computer Vision: Representation Learning for Vision
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

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.

Problem

• Analyzing geometric properties of MCR² objective for deep representations
• Characterizing local and global optima of MCR² optimization landscape
• Validating theoretical findings through synthetic and real dataset experiments

Innovation

• Characterizes all local and global optima properties
• Shows critical points are maximizers or strict saddles
• Validates favorable landscape via synthetic and real experiments