Location-Invariant Properties of Functions and Properties of Distributions: Aligned in Testing but Diverge in Verification

·2026.07.16 09:00

Key point

Testing location-invariant properties of functions and testing properties of distributions are similar, but the relationship between the two concepts does not hold in the verification stage.

Details

A location-invariant property of a function refers to a characteristic in which, regardless of where each value appears, the property can be characterized solely by the frequency with which values appear.

According to prior research, the Query Complexity of testing the location-invariant property of a function is known to be closely related to the Sample Complexity of testing the corresponding property of a distribution.

However, this study reveals that this close relationship does not hold in the Verification stage. In other words, the correlation observed in the testing stage does not carry over to the verification stage.

complexity-theory distribution-properties location-invariant-properties function-properties testing-vs-verification

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