High Dimensional Expanders and Property Testing

We show that the high dimensional expansion property as defined by Gromov, Linial and Meshulam, for simplicial complexes is a form of testability. Namely, a simplicial complex is a high dimensional expander iff a suitable property is testable. Using this connection, we derive several testability results

Lower bounds for adaptive linearity tests

Linearity tests are randomized algorithms which have oracle access to the truth table of some function f, and are supposed to distinguish between linear functions and functions which are far from linear. Linearity tests were first introduced by (Blum, Luby and Rubenfeld, 1993), and were later used in the PCP theorem, among other applications. The quality of a linearity test is described by its correctness c - the probability it accepts linear functions, its soundness s - the probability it accepts functions far from linear, and its query complexity q - the number of queries it makes. Linearity tests were studied in order to decrease the soundness of linearity tests, while keeping the query complexity small (for one reason, to improve PCP constructions). Samorodnitsky and Trevisan (Samorodnitsky and Trevisan 2000) constructed the Complete Graph Test, and prove that no Hyper Graph Test can perform better than the Complete Graph Test. Later in (Samorodnitsky and Trevisan 2006) they prove, among other results, that no non-adaptive linearity test can perform better than the Complete Graph Test. Their proof uses the algebraic machinery of the Gowers Norm. A result by (Ben-Sasson, Harsha and Raskhodnikova 2005) allows to generalize this lower bound also to adaptive linearity tests. We also prove the same optimal lower bound for adaptive linearity test, but our proof technique is arguably simpler and more direct than the one used in (Samorodnitsky and Trevisan 2006). We also study, like (Samorodnitsky and Trevisan 2006), the behavior of linearity tests on quadratic functions. However, instead of analyzing the Gowers Norm of certain functions, we provide a more direct combinatorial proof, studying the behavior of linearity tests on random quadratic functions..

Testing List H-Homomorphisms

Let $H$ be an undirected graph. In the List $H$-Homomorphism Problem, given an undirected graph $G$ with a list constraint $L(v) \subseteq V(H)$ for each variable $v \in V(G)$, the objective is to find a list $H$-homomorphism $f:V(G) \to V(H)$, that is, $f(v) \in L(v)$ for every $v \in V(G)$ and $(f(u),f(v)) \in E(H)$ whenever $(u,v) \in E(G)$. We consider the following problem: given a map $f:V(G) \to V(H)$ as an oracle access, the objective is to decide with high probability whether $f$ is a list $H$-homomorphism or \textit{far} from any list $H$-homomorphisms. The efficiency of an algorithm is measured by the number of accesses to $f$. In this paper, we classify graphs $H$ with respect to the query complexity for testing list $H$-homomorphisms and show the following trichotomy holds: (i) List $H$-homomorphisms are testable with a constant number of queries if and only if $H$ is a reflexive complete graph or an irreflexive complete bipartite graph. (ii) List $H$-homomorphisms are testable with a sublinear number of queries if and only if $H$ is a bi-arc graph. (iii) Testing list $H$-homomorphisms requires a linear number of queries if $H$ is not a bi-arc graph

Locally Testable Codes and Cayley Graphs

We give two new characterizations of (\F_2-linear) locally testable error-correcting codes in terms of Cayley graphs over \F_2^h: \begin{enumerate} \item A locally testable code is equivalent to a Cayley graph over \F_2^h whose set of generators is significantly larger than $h$ and has no short linear dependencies, but yields a shortest-path metric that embeds into $\ell_1$ with constant distortion. This extends and gives a converse to a result of Khot and Naor (2006), which showed that codes with large dual distance imply Cayley graphs that have no low-distortion embeddings into $\ell_1$. \item A locally testable code is equivalent to a Cayley graph over \F_2^h that has significantly more than $h$ eigenvalues near 1, which have no short linear dependencies among them and which "explain" all of the large eigenvalues. This extends and gives a converse to a recent construction of Barak et al. (2012), which showed that locally testable codes imply Cayley graphs that are small-set expanders but have many large eigenvalues. \end{enumerate}Comment: 22 page

Property Testing via Set-Theoretic Operations

Given two testable properties $\mathcal{P}_{1}$ and $\mathcal{P}_{2}$, under what conditions are the union, intersection or set-difference of these two properties also testable? We initiate a systematic study of these basic set-theoretic operations in the context of property testing. As an application, we give a conceptually different proof that linearity is testable, albeit with much worse query complexity. Furthermore, for the problem of testing disjunction of linear functions, which was previously known to be one-sided testable with a super-polynomial query complexity, we give an improved analysis and show it has query complexity O(1/\eps^2), where \eps is the distance parameter.Comment: Appears in ICS 201

Some closure features of locally testable affine-invariant properties

Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2013.Cataloged from PDF version of thesis.Includes bibliographical references (p. 31-32).We prove that the class of locally testable affine-invariant properties is closed under sums, intersections and "lifts". The sum and intersection are two natural operations on linear spaces of functions, where the sum of two properties is simply their sum as a vector space. The "lift" is a less well-studied property, which creates some interesting affine-invariant properties over large domains, from properties over smaller domains. Previously such results were known for "single-orbit characterized" affine-invariant properties, which are known to be a subclass of locally testable ones, and are potentially a strict subclass. The fact that the intersection of locally-testable affine-invariant properties are locally testable could have been derived from previously known general results on closure of property testing under set-theoretic operations, but was not explicitly observed before. The closure under sum and lifts is implied by an affirmative answer to a central question attempting to characterize locally testable affine-invariant properties, but the status of that question remains wide open. Affine-invariant properties are clean abstractions of commonly studied, and extensively used, algebraic properties such linearity and low-degree. Thus far it is not known what makes affine-invariant properties locally testable - no characterizations are known, and till this work it was not clear if they satisfied any closure properties. This work shows that the class of locally testable affine-invariant properties are closed under some very natural operations. Our techniques use ones previously developed for the study of "single-orbit characterized" properties, but manage to apply them to the potentially more general class of all locally testable ones via a simple connection that may be of broad interest in the study of affine-invariant properties.by Alan Xinyu Guo.S.M