178 research outputs found
Improved Lower Bounds for Testing Triangle-freeness in Boolean Functions via Fast Matrix Multiplication
Understanding the query complexity for testing linear-invariant properties
has been a central open problem in the study of algebraic property testing.
Triangle-freeness in Boolean functions is a simple property whose testing
complexity is unknown. Three Boolean functions , and are said to be triangle free if there is no such that . This property
is known to be strongly testable (Green 2005), but the number of queries needed
is upper-bounded only by a tower of twos whose height is polynomial in 1 /
\epsislon, where \epsislon is the distance between the tested function
triple and triangle-freeness, i.e., the minimum fraction of function values
that need to be modified to make the triple triangle free. A lower bound of for any one-sided tester was given by Bhattacharyya and
Xie (2010). In this work we improve this bound to .
Interestingly, we prove this by way of a combinatorial construction called
\emph{uniquely solvable puzzles} that was at the heart of Coppersmith and
Winograd's renowned matrix multiplication algorithm
Nondeterministic graph property testing
A property of finite graphs is called nondeterministically testable if it has
a "certificate" such that once the certificate is specified, its correctness
can be verified by random local testing. In this paper we study certificates
that consist of one or more unary and/or binary relations on the nodes, in the
case of dense graphs. Using the theory of graph limits, we prove that
nondeterministically testable properties are also deterministically testable.Comment: Version 2: 11 pages; we allow orientation in the certificate,
describe new application
Random local algorithms
Consider the problem when we want to construct some structure on a bounded
degree graph, e.g. an almost maximum matching, and we want to decide about each
edge depending only on its constant radius neighbourhood. We show that the
information about the local statistics of the graph does not help here. Namely,
if there exists a random local algorithm which can use any local statistics
about the graph, and produces an almost optimal structure, then the same can be
achieved by a random local algorithm using no statistics.Comment: 9 page
Testing Booleanity and the Uncertainty Principle
Let f:{-1,1}^n -> R be a real function on the hypercube, given by its
discrete Fourier expansion, or, equivalently, represented as a multilinear
polynomial. We say that it is Boolean if its image is in {-1,1}.
We show that every function on the hypercube with a sparse Fourier expansion
must either be Boolean or far from Boolean. In particular, we show that a
multilinear polynomial with at most k terms must either be Boolean, or output
values different than -1 or 1 for a fraction of at least 2/(k+2)^2 of its
domain.
It follows that given oracle access to f, together with the guarantee that
its representation as a multilinear polynomial has at most k terms, one can
test Booleanity using O(k^2) queries. We show an \Omega(k) queries lower bound
for this problem.
Our proof crucially uses Hirschman's entropic version of Heisenberg's
uncertainty principle.Comment: 15 page
Hypotheses testing on infinite random graphs
Drawing on some recent results that provide the formalism necessary to
definite stationarity for infinite random graphs, this paper initiates the
study of statistical and learning questions pertaining to these objects.
Specifically, a criterion for the existence of a consistent test for complex
hypotheses is presented, generalizing the corresponding results on time series.
As an application, it is shown how one can test that a tree has the Markov
property, or, more generally, to estimate its memory
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