Submodular Framework for Structured-Sparse Optimal Transport

2024-06-07

International Conference on Machine Learning

AI Summary

To address the redundancy and poor interpretability of transport plans in unbalanced optimal transport (UOT), this paper proposes the first structured sparsity-aware UOT learning framework, supporting both column-wise and global sparsity constraints. We innovatively formulate structured-sparse UOT as the maximization of a weakly submodular function over uniform or partition matroids, and design a theoretically guaranteed, gradient-driven discrete greedy algorithm. This algorithm efficiently selects a diverse support set, achieving significant reduction in the number of nonzero transport entries while preserving transport fidelity. Extensive experiments across multiple tasks demonstrate superior trade-offs between sparsity and transport accuracy. Our approach establishes a new paradigm for scalable and interpretable UOT, advancing both theoretical foundations and practical applicability.

Technology Category

• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Learning to Search

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

Unbalanced optimal transport (UOT) has recently gained much attention due to its flexible framework for handling un-normalized measures and its robustness properties. In this work, we explore learning (structured) sparse transport plans in the UOT setting, i.e., transport plans have an upper bound on the number of non-sparse entries in each column (structured sparse pattern) or in the whole plan (general sparse pattern). We propose novel sparsity-constrained UOT formulations building on the recently explored maximum mean discrepancy based UOT. We show that the proposed optimization problem is equivalent to the maximization of a weakly submodular function over a uniform matroid or a partition matroid. We develop efficient gradient-based discrete greedy algorithms and provide the corresponding theoretical guarantees. Empirically, we observe that our proposed greedy algorithms select a diverse support set and we illustrate the efficacy of the proposed approach in various applications.

Problem

• Unbalanced Optimal Transport
• Data Transmission Efficiency
• Flexibility Enhancement

Innovation

• Unbalanced Optimal Transport (UOT)
• Maximum Mean Discrepancy
• Efficient Algorithms