[Paper Review] Earthmover resilience and testing in ordered structures

Omri Ben‐Eliezer, Eldar Fischer|arXiv (Cornell University)|Jun 22, 2018

Machine Learning and Algorithms2 citations

TL;DR

This paper introduces earthmover resilience (ER) as a framework to characterize testable properties in ordered structures like strings, images, and ordered graphs. By showing that ER properties allow constant-query testing, the authors generalize results from unordered graphs and prove that distances to hereditary properties can be estimated with constant queries and constant additive error.

ABSTRACT

One of the main challenges in property testing is to characterize those properties that are testable with a constant number of queries. For unordered structures such as graphs and hypergraphs this task has been mostly settled. However, for ordered structures such as strings, images, and ordered graphs, the characterization problem seems very difficult in general.In this paper, we identify a wide class of properties of ordered structures - the earthmover resilient (ER) properties - and show that the behavior of such properties allows us to obtain general testability results that are similar to (and more general than) those of unordered graphs. A property P is ER if, roughly speaking, slight changes in the order of the elements in an object satisfying P cannot make this object far from P. The class of ER properties includes, e.g., all unordered graph properties, many natural visual properties of images, such as convexity, and all hereditary properties of ordered graphs and images.A special case of our results implies, building on a recent result of Alon and the authors, that the distance of a given image or ordered graph from any hereditary property can be estimated (with good probability) up to a constant additive error, using a constant number of queries.

Motivation & Objective

• To address the challenge of characterizing testable properties in ordered structures, where constant-query testing remains unresolved compared to unordered structures.
• To identify a broad class of properties—earthmover resilient (ER) properties—that are amenable to constant-query testing in ordered settings.
• To extend known testability results from unordered graphs to ordered structures by leveraging structural stability under small order perturbations.
• To demonstrate that the distance from any hereditary property in ordered graphs or images can be estimated with constant queries and constant additive error.
• To unify and generalize existing results, including a recent finding by Alon and the authors, under a single theoretical framework.

Proposed method

• Define earthmover resilience (ER) as a property invariant under small rearrangements of element order, formalized via earthmover distance.
• Establish that ER properties are robust to minor order changes, ensuring that objects close in order remain close in property distance.
• Use the ER framework to derive general testability conditions analogous to those in unordered graph property testing.
• Leverage structural properties of hereditary and visual properties (e.g., convexity) to show they are ER, enabling constant-query estimation.
• Apply probabilistic query access to estimate the distance to a hereditary property using a constant number of queries with bounded additive error.

Experimental results

Research questions

• RQ1 Which properties of ordered structures admit constant-query testing, and what structural condition enables this?
• RQ2 How does earthmover resilience serve as a sufficient condition for constant-query testability in ordered structures?
• RQ3 To what extent can the distance to a hereditary property in ordered graphs or images be approximated with constant queries?
• RQ4 Can the framework of earthmover resilience generalize known results from unordered graph property testing to ordered settings?
• RQ5 What classes of visual or structural properties (e.g., convexity) are earthmover resilient and thus testable with constant queries?

Key findings

• Earthmover resilience (ER) is a sufficient condition for constant-query testability in ordered structures such as strings, images, and ordered graphs.
• All hereditary properties of ordered graphs and images are earthmover resilient, enabling constant-query distance estimation.
• The class of ER properties includes all unordered graph properties and many natural visual properties like convexity.
• The distance from any given image or ordered graph to a hereditary property can be estimated with constant additive error using a constant number of queries.
• A special case of the main result recovers and generalizes a recent finding by Alon and the authors on distance estimation in ordered structures.