# Estimating the Distance from Testable Affine-Invariant Properties

@article{Hatami2013EstimatingTD, title={Estimating the Distance from Testable Affine-Invariant Properties}, author={Hamed Hatami and Shachar Lovett}, journal={2013 IEEE 54th Annual Symposium on Foundations of Computer Science}, year={2013}, pages={237-242} }

Let P be an affine invariant property of multivariate functions over a constant size finite field. We show that if P is locally testable with a constant number of queries, then one can estimate the distance of a function f from P with a constant number of queries. This was previously unknown even for simple properties such as cubic polynomials over the binary field. Our test is simple: take a restriction of f to a constant dimensional affine subspace, and measure its distance from P. We show…

## 14 Citations

### A characterization of locally testable affine-invariant properties via decomposition theorems

- Mathematics, Computer ScienceSTOC
- 2014

A characterization of affine-invariant properties that are (two-sided error) testable with a constant number of queries and an algorithm that tests whether the structured part of the input function has a specific form is given.

### On testing affine-invariant properties over finite fields ∗

- Mathematics
- 2013

An affine-invariant property over a finite field is a property of functions over Fp that is closed under all affine transformations of the domain. This class of properties includes such well-known…

### Guest column: on testing affine-invariant properties over finite fields

- MathematicsSIGA
- 2013

The last few years has seen rapid progress in characterizing the affine-invariant properties which are testable with a constant number of queries, and the current state of this project is surveyed.

### Testing properties of functions on finite groups

- MathematicsRandom Struct. Algorithms
- 2016

It is shown that conjugate invariance, homomorphism, and the property of being proportional to an irreducible character is testable with a constant number of queries to f, where a character is a crucial notion in representation theory.

### On Higher-Order Fourier Analysis over Non-Prime Fields

- Mathematics, Computer ScienceAPPROX-RANDOM
- 2016

The tools of higher-order Fourier analysis are extended to analyze functions over general finite fields and it is proved that all locally characterized affine-invariant properties of functions f: K^n -> K are testable with one-sided error.

### Gowers Norm, Function Limits, and Parameter Estimation

- MathematicsSODA
- 2016

A metric over limits of function sequences is introduced, and it is shown that the property of being a function of a constant number of low-degree polynomials and a constants number of factored polynOMials is constant-query testable if it is closed under blowing-up.

### C C ] 1 2 A ug 2 01 3 Correlation Testing for Affine Invariant Properties on F n p in the High Error Regime ∗

- Mathematics, Computer Science
- 2021

It is shown that any general property which is affine invariant and which is correlation testable using a constant number of queries can in fact be tested by Gowers uniformity tests, and hence having correlation with the property is equivalent toHaving correlation with degree d polynomials for some fixed d.

### Approximating Sumset Size

- Computer Science, MathematicsSODA
- 2022

An algorithm whose query complexity depends only on ε and is completely independent of the ambient dimension n is given, which is a sublinear-time algorithm for the problem of sumset size estimation.

### D S ] 2 6 Ju l 2 02 1 Approximating Sumset Size

- Computer Science, Mathematics
- 2021

An algorithm whose query complexity depends only on ε and is completely independent of the ambient dimension n is given, which is a sublinear-time algorithm for the problem of sumset size estimation.

### Using higher-order Fourier analysis over general fields

- Mathematics, Computer ScienceElectron. Colloquium Comput. Complex.
- 2015

The tools of higher-order Fourier analysis are extended to analyze functions over general fields to prove that all locally characterized affine-invariant properties of functions are testable with one-sided error.

## 22 References

### Every locally characterized affine-invariant property is testable

- MathematicsSTOC '13
- 2013

It is shown that all affine-invariant properties having local characterizations are testable, and it is proved that any property that can be described as the property of decomposing into a known structure of low-degree polynomials is locally characterized and is, hence, testable.

### Testing Low Complexity Affine-Invariant Properties

- MathematicsSODA
- 2013

The main result is that for any fixed prime p ≥ 2 and fixed integer R ≥ 2, any affine-invariant property P of functions f is any affinities p, i.e., having low degree is preserved by composition with affine maps.

### Correlation testing for affine invariant properties on Fpn in the high error regime

- Mathematics, Computer ScienceSTOC '11
- 2011

It is shown that any such property can in fact be tested by the Gowers uniformity test, and hence having correlation with the property is equivalent toHaving correlation with degree d polynomials for some fixed d, and the result holds also for non-linear properties which are affine invariant.

### A Unified Framework for Testing Linear-Invariant Properties

- Mathematics2010 IEEE 51st Annual Symposium on Foundations of Computer Science
- 2010

This work shows that every linear-invariant property that can be tested with one-sided error can be characterized by forbidding induced solutions to a (possibly infinite) set of linear equations, and conjecture that this result can be extended to cover systems oflinear equations.

### Testing Fourier Dimensionality and Sparsity

- Computer ScienceSIAM J. Comput.
- 2011

The results show that the property of a Boolean function having a concise Fourier representation is locally testable and lower bounds are proved showing that any testing algorithm must have query complexity within a polynomial factor of one of the algorithms, which are nonadaptive.

### Proximity Oblivious Testing and the Role of Invariances

- Mathematics, Computer ScienceStudies in Complexity and Cryptography
- 2010

It is shown that easy testability is not guaranteed even when the property is characterized by local conditions that are invariant under a 1-transitive group of permutations.

### Worst Case to Average Case Reductions for Polynomials

- Mathematics, Computer Science2008 49th Annual IEEE Symposium on Foundations of Computer Science
- 2008

It is shown that a polynomial that can be approximated by a few polynomials of bounded degree (i.e. a poynomial with non negligible correlation with a function of few bounded degree polynmials), can be computed by a many polynoms of boundeddegree.

### Algebraic property testing: the role of invariance

- MathematicsElectron. Colloquium Comput. Complex.
- 2007

This work considers (F-)linear properties that are invariant under linear transformations of the domain and proves that an O(1)-local "characterization" is a necessary and sufficient condition for O( 1)-local testability, and shows that local formal characterizations essentially imply local testability.

### Green's conjecture and testing linear-invariant properties

- MathematicsSTOC '09
- 2009

This paper confirms the conjecture that for any system of homogenous linear equations Mx=0 and for any ε >0 there is a constant time algorithm that can distinguish with high probability between sets of integers that are (M,0)-free from sets that are ε-far from being ( M,0-free), by showing that such a testing algorithm exists even for non-homogenouslinear equations.

### Testing versus estimation of graph properties

- Computer Science, MathematicsSTOC '05
- 2005

It is shown here that in the setting of the dense graph model, all testable properties are not only tolerantly testable, but also admit a constant query size algorithm that estimates the distance from the property up to any fixed additive constant.