Fine-Grained Complexity of Earth Mover's Distance under Translation

2024-03-07

International Symposium on Computational Geometry

AI Summary

This work studies the translation-invariant Earth Mover’s Distance (EMDuT)—the minimum EMD between two point sets under optimal translation. For $L_1$ and $L_infty$ metrics in both one-dimensional and high-dimensional Euclidean spaces, we establish tight computational complexity bounds. In $mathbb{R}^1$, we present a near-optimal $ ilde{O}(n^2)$-time algorithm and prove its optimality via an Orthogonal Vectors Hypothesis (OVH) reduction. In $mathbb{R}^d$, we give the first fixed-dimensional polynomial-time exact algorithm running in $ ilde{O}(n^{2d+2})$ time and show, under the Exponential Time Hypothesis (ETH), that the exponential dependence on dimension is unavoidable. These results unify the theoretical understanding of EMDuT’s computational hardness and provide foundational algorithms and complexity characterizations for translation-robust similarity computation.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Combinatorial Optimization
• Machine Learning: Learning with Manifolds

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets
• Security and Privacy: Large-scale security measurements

Abstract

The Earth Mover's Distance is a popular similarity measure in several branches of computer science. It measures the minimum total edge length of a perfect matching between two point sets. The Earth Mover's Distance under Translation ($mathrm{EMDuT}$) is a translation-invariant version thereof. It minimizes the Earth Mover's Distance over all translations of one point set. For $mathrm{EMDuT}$ in $mathbb{R}^1$, we present an $widetilde{mathcal{O}}(n^2)$-time algorithm. We also show that this algorithm is nearly optimal by presenting a matching conditional lower bound based on the Orthogonal Vectors Hypothesis. For $mathrm{EMDuT}$ in $mathbb{R}^d$, we present an $widetilde{mathcal{O}}(n^{2d+2})$-time algorithm for the $L_1$ and $L_infty$ metric. We show that this dependence on $d$ is asymptotically tight, as an $n^{o(d)}$-time algorithm for $L_1$ or $L_infty$ would contradict the Exponential Time Hypothesis (ETH). Prior to our work, only approximation algorithms were known for these problems.

Problem

• Computing Earth Mover's Distance under translation for point sets
• Developing efficient exact algorithms for EMDuT in various dimensions
• Establishing tight complexity bounds for translation-invariant similarity measures

Innovation

• Developed near-optimal algorithm for 1D EMDuT
• Presented efficient algorithm for high-dimensional EMDuT
• Proved tight complexity bounds using ETH hypothesis