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A Global Geometric Analysis of Maximal Coding Rate Reduction

Peng Wang, Huikang Liu, Druv Pai, Yaodong Yu, Zhihui Zhu, Qing Qu, Yi Ma

TL;DR

The paper provides a complete geometric analysis of the MCR$^2$ objective used for learning structured and compact representations, proving that all local/global optima have interpretable geometric structures and that the regularized objective has a benign landscape where all critical points are local maxima or strict saddles. It derives closed-form characterizations of optima, demonstrates the orthogonality and low-dimensional subspaces associated with class blocks, and shows that gradient-based optimization can efficiently find meaningful representations. The work further validates theory through extensive synthetic experiments and real-data deep-network training, highlighting the practical relevance of the MCR$^2$ framework and its unrolled-optimization connections (e.g., ReduNet/CRATE). Overall, the results justify using MCR$^2$-based objectives for learning discriminative and diverse representations and suggest principled architectural and optimization strategies for deep learning.

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.

A Global Geometric Analysis of Maximal Coding Rate Reduction

TL;DR

The paper provides a complete geometric analysis of the MCR objective used for learning structured and compact representations, proving that all local/global optima have interpretable geometric structures and that the regularized objective has a benign landscape where all critical points are local maxima or strict saddles. It derives closed-form characterizations of optima, demonstrates the orthogonality and low-dimensional subspaces associated with class blocks, and shows that gradient-based optimization can efficiently find meaningful representations. The work further validates theory through extensive synthetic experiments and real-data deep-network training, highlighting the practical relevance of the MCR framework and its unrolled-optimization connections (e.g., ReduNet/CRATE). Overall, the results justify using MCR-based objectives for learning discriminative and diverse representations and suggest principled architectural and optimization strategies for deep learning.

Abstract

The maximal coding rate reduction (MCR) 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 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 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.
Paper Structure (57 sections, 14 theorems, 129 equations, 9 figures, 1 table)

This paper contains 57 sections, 14 theorems, 129 equations, 9 figures, 1 table.

Table of Contents

  1. Introduction
  2. Background and Motivation
  3. Our Contributions
  4. Characterizing the local and global optimality.
  5. Studying the global optimization landscape.
  6. Related Work
  7. Low-dimensional structures in deep representation learning.
  8. The MCR$^2$-based approach to deep learning.
  9. Notation.
  10. Problem Setup
  11. An Overview of MCR$^2$
  12. The formulation of MCR$^2$.
  13. The Regularized MCR$^2$ Problem
  14. Applications of our formulation to deep representation learning via unrolled optimization.
  15. Applications of our formulation to deep representation learning with standard neural networks.
  16. ...and 42 more sections

Key Result

Theorem 1

Suppose that the number of training samples in the $k$-th class is $m_k > 0$ for each $k \in [K]$. Given a coding precision $\epsilon > 0$, if the regularization parameter satisfies then the following statements hold: (i) ( Characterization of local maximizers) $\bm Z = \left[\bm Z_1,\dots,\bm Z_K \right]$ is a local maximizer of Problem eq:MCR if and only if the $k$-th block admits the following

Figures (9)

  • Figure 1: An illustration of the properties of MCR$^2$. (a) The high-dimensional data $\{\bm x_i\} \subseteq \mathbb{R}^n$ lies on a union of low-dimensional submanifolds. The objective of MCR$^2$ is to learn a feature mapping $f_{\bm \Theta}(\bm x) \in \mathbb{R}^d$ such that $\bm z_i = f_{\bm \Theta}(\bm x_i)$ for all $i$ are low-dimensional, discriminative, and diverse. (b) According to \ref{['thm:1']} and \ref{['thm:2']}, the regularized MCR$^2$ problem has a benign optimization landscape: each critical point is either a local maximizer or a strict saddle point. Furthermore, each local maximizer, just like the global maximizer, corresponds to a feature representation that consists of a family of orthogonal subspaces, as illustrated in the middle.
  • Figure 2: Validation of theory for the MCR$^2$ problem. (a) We visualize the heatmap of cosine similarity among learned features by GD for solving Problem \ref{['eq:MCR']}. The lighter pixels represent lower cosine similarities between pairwise features. (b) The blue dots are plotted based on the singular values by applying SVD to the solution returned by GD, and the red line is plotted according to the closed-form solution in \ref{['eq:Zk opti']}. The number of nonzero singular values in each subspace is $24, 23, 27, 26$, respectively.
  • Figure 3: Convergence performance of GD for solving the regularized MCR$^2$ problem. Here, the $x$-axis is number of iterations (also denoted by $t$), and $y$-axis is the function value gap $F^t-F^*$, where $F^t=F(\bm Z^t)$ denotes the function value at the $t$-th iterate $\bm Z^t$ generated by GD, and $F^*$ is the optimal value of Problem \ref{['eq:MCR']} computed according to \ref{['eq:Zk opti']} in \ref{['thm:1']}.
  • Figure 4: Heatmap of cosine similarity among features produced by deep networks trained on MNIST and CIFAR-10. The darker pixels represent higher absolute cosine similarity between features.
  • Figure 5: Heatmap of cosine similarity between pairwise features under different settings.
  • ...and 4 more figures

Theorems & Definitions (27)

  • Theorem 1: Local and global optimality
  • Proposition 1
  • Theorem 2: Benign optimization landscape
  • Lemma 1
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Definition 1
  • Definition 2
  • Lemma 2: Matrix inversion lemma
  • ...and 17 more