Gowers Norm, Function Limits, and Parameter Estimation

doi:10.1137/1.9781611974331.CH96

Paper

Gowers Norm, Function Limits, and Parameter Estimation

Published Oct 19, 2014 · Yuichi Yoshida

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Abstract

Let $\{f_i:\mathbb{F}_p^i \to \{0,1\}\}$ be a sequence of functions, where $p$ is a fixed prime and $\mathbb{F}_p$ is the finite field of order $p$. The limit of the sequence can be syntactically defined using the notion of ultralimit. Inspired by the Gowers norm, we introduce a metric over limits of function sequences, and study properties of it. One application of this metric is that it provides a characterization of affine-invariant parameters of functions that are constant-query estimable. Using this characterization, we show that the property of being a function of a constant number of low-degree polynomials and a constant number of factored polynomials (of arbitrary degrees) is constant-query testable if it is closed under blowing-up. Examples of this property include the property of having a constant spectral norm and degree-structural properties with rank conditions.

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Key takeawayThe Gowers norm metric can characterize affine-invariant parameters of functions, making them constant-query estimable and testable for properties like constant spectral norm and degree-structural properties.

PopulationOlder adults (50-71 years)

Sample size24

MethodsObservational

OutcomesBody Mass Index projections

ResultsSocial networks mitigate obesity in older groups.

Gowers Norm, Function Limits, and Parameter Estimation

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