A Dual Approach for Hierarchical Information-Theoretic Tree Abstractions

2025-12-01

AI Summary

This paper investigates the formal relationship between two tree abstraction problems—hard-constrained and soft-constrained—within the information bottleneck (IB) framework. Method: Leveraging Lagrangian duality theory, we derive the necessary and sufficient conditions for their equivalence and uncover an intrinsic connection between the Q-function and the dual function of the hard constraint in tree structures. Integrating insights from tree phase transitions and total unimodularity of constraint matrices, we propose an efficient variable selection method. Through Lagrangian relaxation and strong duality analysis of integer/linear programming formulations, we establish equivalence criteria and characterize the underlying phase-transition mechanism. Results: Empirical validation confirms the effectiveness of dual-variable optimization. Our work provides both theoretical foundations and algorithmic tools for hierarchical information compression and interpretable tree-structured representation learning.

Technology Category

• Machine Learning: Information Theory
• Constraint Satisfaction and Optimization: Constraint Learning and Acquisition
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

In this paper, we consider establishing a formal connection between two distinct tree-abstraction problems inspired by the information-bottleneck (IB) method. Specifically, we consider the hard- and soft-constrained formulations that have recently appeared in the literature to determine the conditions for which the two approaches are equivalent. Our analysis leverages concepts from Lagrangian relaxation and duality theory to relate the dual function of the hard-constrained problem to the Q-function employed in Q-tree search and shows the connection between tree phase transitions and solutions to the dual problem obtained by exploiting the problem structure. An algorithm is proposed that employs knowledge of the tree phase transitions to find a setting of the dual variable that solves the dual problem. Furthermore, we present an alternative approach to select the dual variable that leverages the integer programming formulation of the hard-constrained problem and the strong duality of linear programming. To obtain a linear program, we establish that a relaxation of the integer programming formulation of the hard-constrained tree-search problem has the integrality property by showing that the program constraint matrix is totally unimodular. Empirical results that corroborate the theoretical developments are presented and discussed throughout.

Problem

• Connects hard- and soft-constrained tree abstraction problems via information bottleneck
• Relates dual functions to Q-tree search and tree phase transitions
• Proposes algorithms using duality theory and integer programming relaxations

Innovation

• Leveraging Lagrangian relaxation and duality theory
• Proposing algorithm using tree phase transitions
• Employing integer programming with strong duality