[Paper Review] Submodular hamming metrics

Jennifer Gillenwater, Rishabh Iyer+3 more|arXiv (Cornell University)|Dec 7, 2015

Complexity and Algorithms in GraphsComputer Science28 references4 citations

TL;DR

This paper introduces submodular Hamming metrics—functions derived from positive polymatroids—that define tractable discrete metrics over binary vectors. By leveraging submodularity, the authors establish hardness results, approximation algorithms, and empirically validate their effectiveness in clustering and diverse k-best list generation.

ABSTRACT

We show that there is a largely unexplored class of functions (positive polymatroids) that can define proper discrete metrics over pairs of binary vectors and that are fairly tractable to optimize over. By exploiting submodularity, we are able to give hardness results and approximation algorithms for optimizing over such metrics. Additionally, we demonstrate empirically the effectiveness of these metrics and associated algorithms on both a metric minimization task (a form of clustering) and also a metric maximization task (generating diverse k-best lists).

Motivation & Objective

• To explore the potential of positive polymatroids in defining proper discrete metrics over binary vectors.
• To develop tractable optimization methods for these metrics using submodularity.
• To demonstrate empirical effectiveness in metric minimization (clustering) and metric maximization (diverse k-best lists).

Proposed method

• The paper defines a class of discrete metrics using positive polymatroids, which are submodular functions.
• It exploits submodularity to derive approximation algorithms for optimization over these metrics.
• The approach formulates metric minimization and maximization tasks as submodular optimization problems.
• The authors design algorithms that efficiently handle both clustering and diverse list generation using the submodular Hamming framework.
• Theoretical analysis includes hardness results and approximation guarantees based on submodular function properties.
• Empirical evaluation is conducted on real-world clustering and diverse list generation tasks.

Experimental results

Research questions

• RQ1 Can positive polymatroids define proper discrete metrics over binary vectors?
• RQ2 How tractable is optimization over such submodular Hamming metrics?
• RQ3 What approximation guarantees can be derived for optimization over these metrics?
• RQ4 How effective are these metrics in practical tasks like clustering and diverse list generation?
• RQ5 What are the theoretical hardness bounds for optimizing over submodular Hamming metrics?

Key findings

• The paper identifies a new class of discrete metrics based on positive polymatroids that are both proper and amenable to optimization.
• Submodularity enables the development of approximation algorithms with theoretical guarantees for metric optimization.
• Empirical results show strong performance in clustering tasks using metric minimization.
• The framework effectively generates diverse k-best lists in metric maximization tasks.
• Hardness results demonstrate inherent computational limits for certain optimization variants.
• The approach balances theoretical tractability with practical effectiveness across multiple learning tasks.