Location-Invariant Properties of Functions Versus Properties of Distributions: United in Testing but Separated in Verification

2026-07-16 · Source: Apple Machine Learning Research · Field: Technology & Digital — Artificial Intelligence & Machine Learning, Mathematics & Computational Sciences, Research Methodology & Innovation · Depth: Expert, quick

Summary

Oded Goldreich and Guy N. Rothblum's work reveals a fundamental divergence in the verification of location-invariant properties between functions and their corresponding distributions, despite a known close relationship in testing complexity. A property is location-invariant if characterized by value frequencies, irrespective of their positions. The authors demonstrate that this close relationship, observed in query complexity for testing, does not extend to verification using Interactive Proofs of Proximity (IPPs), including doubly-sublinear IPPs (ds-IPPs). Specifically, they present ds-IPPs for functions where each value occurs m/n times, achieving a verifier query complexity of O(n^{0.5-α}) and a prover query complexity of e^{O(n^{0.5+α}/ε^2)} for α ∈ (0, 0.5). They also provide ds-IPPs for functions where values occur m/k times or not at all, with a verifier complexity of poly(1/ε) ⋅ k^{(2/3)-2α} and prover complexity of poly(1/ε) ⋅ e^{O(k^{(2/3)+α})} for α ∈ (0, 1/3). This contrasts sharply with the corresponding distribution properties, which lack doubly-efficient IPPs.

Key takeaway

For Research Scientists working on property testing and verification, this research highlights a critical distinction: efficient testing does not guarantee efficient verification. You should re-evaluate assumptions about the transferability of complexity bounds between testing and verification for location-invariant properties. Consider exploring doubly-sublinear IPPs as a viable approach for verifying specific function properties, such as uniformity, even when corresponding distribution properties lack efficient verification mechanisms. This shift in understanding can guide future protocol design.

Key insights

The close relationship between testing function and distribution properties breaks down in verification.

Principles

• Location-invariant properties are defined by value frequencies.
• Verification complexity differs significantly from testing complexity.
• Doubly-sublinear IPPs can verify specific function properties efficiently.

Method

The work constructs doubly-sublinear Interactive Proofs of Proximity (ds-IPPs) for specific location-invariant function properties, detailing verifier and prover query complexities.

In practice

• Apply ds-IPPs for efficient verification of function uniformity.
• Consider ds-IPPs for k-uniform function property verification.

Topics

• Property Testing
• Interactive Proofs of Proximity
• Doubly-Sublinear IPPs
• Location-Invariant Properties
• Function Verification
• Distribution Properties

Best for: AI Scientist, Research Scientist

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Editorial summary, takeaway, and curation by AIssential. Original article published by Apple Machine Learning Research.