Understanding Submodular Information Measure Based Objectives for Representation Learning: A Variance and Separation Perspective

2026-07-30

AI Summary

This work addresses the lack of a unified understanding of the geometric and statistical properties induced by submodular information measures (SIMs) in representation learning. The paper proposes a theoretical framework that systematically connects SIM-based objectives with classical representation learning concepts, revealing how they capture intra-class structures—such as variance and covariance volume—and inter-class structures—including centroid separation and Mahalanobis distance. Through controlled synthetic experiments, the study validates these theoretical insights, demonstrating that specific SIM variants correspond to distinct learning behaviors: Graph Cut minimizes intra-class variance, LogDet controls generalized variance, and Facility Location exhibits sensitivity to class imbalance while promoting separation. These findings provide a principled foundation for selecting and designing SIM objectives in representation learning tasks.

Abstract

Submodular Information Measures (SIMs) have recently emerged as a powerful framework for representation learning and multimodal learning. In particular, the SCORE framework~\cite{majee2024score} demonstrated that SIMs can serve as effective objectives for supervised contrastive learning. Despite their empirical success, however, the geometric and statistical properties induced by different submodular information measures remain poorly understood. In this work, we develop a unified theoretical framework connecting SIMs to classical concepts in representation learning and statistical pattern recognition. We show that Total Information (TI) objectives characterize intra-class structure: Graph Cut TI recovers within-class variance, LogDet TI recovers generalized variance and covariance volume, and Facility Location TI induces imbalance-aware separation that emphasizes rare and confusable classes. We further show that Mutual Information (MI) objectives capture complementary notions of inter-class structure: Graph Cut MI is closely related to centroid separation and Fisher-style discrimination, LogDet MI captures covariance-aware separation through Mahalanobis distance, and Facility Location MI measures nearest-mode representational overlap. We validate these theoretical characterizations using controlled synthetic experiments that independently vary variance, covariance, class imbalance, class separation, and multimodal overlap. Across all settings, the empirical behavior closely matches the proposed theory. Our results provide the first unified geometric and statistical understanding of submodular information measures and offer principled guidance for selecting and designing SIM-based objectives for representation learning.

Problem

• Submodular Information Measures
• Representation Learning
• Geometric Understanding
• Statistical Properties
• Class Structure

Innovation

• Submodular Information Measures
• Representation Learning
• Total Information
• Mutual Information
• Geometric Interpretation