Proximity Oblivious Testing and the Role of Invariances

@article{Goldreich2010ProximityOT,
  title={Proximity Oblivious Testing and the Role of Invariances},
  author={Oded Goldreich and Tali Kaufman},
  journal={Electron. Colloquium Comput. Complex.},
  year={2010},
  volume={17},
  pages={58}
}
We present a general notion of properties that are characterized by local conditions that are invariant under a sufficiently rich class of symmetries. Our framework generalizes two popular models of testing graph properties as well as the algebraic invariances studied by Kaufman and Sudan (STOC'08). Our focus is on the case that the property is characterized by a constant number of local conditions and a rich set of invariances. We show that, in the aforementioned models of testing graph… 
Every locally characterized affine-invariant property is testable
TLDR
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.
Algebraic Property Testing
TLDR
It is shown that sparsity and invariance under the affine group of permutations are sufficient conditions for a notion of very structured testing, and a new characterization of the extensively studied BCH codes is revealed.
GSF-locality is not sufficient for proximity-oblivious testing
TLDR
A property does not admit a POT, despite being GSF-local, by exploiting a recent work of the authors which constructed a first-order property that is not testable, and a new connection between FO properties and G SF-local properties via neighbourhood profiles.
Invariance in Property Testing
  • M. Sudan
  • Mathematics
    Electron. Colloquium Comput. Complex.
  • 2010
TLDR
This work points out the invariance classes associated with some the basic classes of testable properties, and focuses on "algebraic properties" which seem to be characterized by the fact that the properties are themselves vector spaces, while their domains are also vector spaces and the Properties are invariant under affine transformations of the domain.
Estimating the Distance from Testable Affine-Invariant Properties
TLDR
The analysis combines the approach of Fischer and Newman, who established a similar result for graph properties, with recently developed tools in higher order Fourier analysis, in particular those developed in Bhattacharyya et al.
On testing affine-invariant properties over finite fields ∗
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
Testing Hereditary Properties of Ordered Graphs and Matrices
TLDR
The proof bridges the gap between techniques related to the regularity lemma, used in the long chain of papers investigating graph testing, and string testing techniques and develops a Ramsey-type lemma for multipartite graphs with undesirable edges.
Guest column: on testing affine-invariant properties over finite fields
TLDR
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.
Erasure-Resilient Property Testing
TLDR
This work begins a study of property testers that are resilient to the presence of adversarially erased function values and identifies an $\alpha$-erasure-resilient $\var...$ that is resistant to being erased by an adversary.
On Sample-Based Testers
TLDR
This work advances the study of sample-based property testers by providing several general positive results as well as by revealing relations between variants of this testing model, and shows that certain types of query-based testers yield sample- based testers of sublinear sample complexity.
...
1
2
3
...

References

SHOWING 1-10 OF 26 REFERENCES
Algebraic property testing: the role of invariance
TLDR
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.
A combinatorial characterization of the testable graph properties: it's all about regularity
TLDR
One of the main open problems in the area of property-testing, which was raised in the 1996 paper of Goldreich, Goldwasser and Ron, is resolved by a purely combinatorial characterization of the graph properties that are testable with a constant number of queries.
On Proximity-Oblivious Testing
TLDR
It is shown that constant-query proximity-oblivious testers do not exist for many easily testable properties, and that even when they exist, repeating them does not necessarily yield the best standard testers for the corresponding property.
2-Transitivity Is Insufficient for Local Testability
TLDR
This paper refutes the conjecture that the presence of a single low weight code in the dual, and "2-transitivity" of the code suffice to get local testability, by giving a family of error correcting codes where the coordinates of the codewords form a large field of characteristic two, and the code is invariant under affine transformations of the domain.
Invariance in Property Testing
  • M. Sudan
  • Mathematics
    Electron. Colloquium Comput. Complex.
  • 2010
TLDR
This work points out the invariance classes associated with some the basic classes of testable properties, and focuses on "algebraic properties" which seem to be characterized by the fact that the properties are themselves vector spaces, while their domains are also vector spaces and the Properties are invariant under affine transformations of the domain.
Property Testing in Bounded Degree Graphs
Algorithmic and Analysis Techniques in Property Testing
  • D. Ron
  • Computer Science, Mathematics
    Found. Trends Theor. Comput. Sci.
  • 2009
TLDR
This monograph surveys results in property testing, where the emphasis is on common analysis and algorithmic techniques.
Property Testing: A Learning Theory Perspective
  • D. Ron
  • Computer Science
    Found. Trends Mach. Learn.
  • 2008
TLDR
This survey takes the learning-theory point of view and focuses on results for testing properties of functions that are of interest to the learning theory community, and covers results forTesting algebraic properties of function such as linearity, testing properties defined by concise representations, such as having a small DNF representation, and more.
On the query complexity of testing orientations for being Eulerian
We consider testing directed graphs Eulerianity in the orientation model introduced in Halevy et al. [2005]. Despite the local nature of the Eulerian property, it turns out to be significantly harder
Three theorems regarding testing graph properties
TLDR
Three theorems regarding testing graph properties in the adjacency matrix representation are presented, relating to the project of characterizing graph properties according to the complexity of testing them (in the adjacent matrix representation), which refers to the framework of graph partition problems.
...
1
2
3
...