Living Reference Work Entry

Encyclopedia of Algorithms

pp 1-6

Date: Latest Version

Testing Juntas and Related Properties of Boolean Functions

Keywords

Juntas Property testing Sublinear-time algorithms Dimension reduction

Years and Authors of Summarized Original Work

2004;

Fischer, Kindler, Ron, Safra, Samorodnitsky

2009;

Blais

Problem Definition

Fix positive integers n and k with n ≥ k. The function f : { 0, 1}n → { 0, 1} is a k-junta if it depends on at most k of the input coordinates. Formally, f is a k-junta if there exists a set \(J \subseteq \{ 1,2,\ldots ,n\}\) of size | J | ≤ k such that for all inputs x, y ∈ { 0, 1}n that satisfy xi = yi for each i ∈ J, we have f(x) = f(y). Juntas play an important role in different areas of computer science. In machine learning, juntas provide an elegant framework for studying the problem of learning with datasets that contain many irrelevant attributes [9, 10]. In the analysis of Boolean functions, they essentially capture the set of functions of low complexity under natural measures such as total influence [19] and noise sensitivity [12].

How efficiently can we distinguish k-juntas from functions that are far from being k-juntas? We can formalize this q ...

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