81 research outputs found
Mixing under monotone censoring
We initiate the study of mixing times of Markov chain under monotone
censoring. Suppose we have some Markov Chain on a state space with
stationary distribution and a monotone set . We
consider the chain which is the same as the chain started at some except that moves of of the form where and are {\em censored} and replaced by the move . If is
ergodic and is connected, the new chain converges to conditional on
. In this paper we are interested in the mixing time of the chain in
terms of properties of and . Our results are based on new connections
with the field of property testing. A number of open problems are presented.Comment: 6 page
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
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
A Hypergraph Dictatorship Test with Perfect Completeness
A hypergraph dictatorship test is first introduced by Samorodnitsky and
Trevisan and serves as a key component in their unique games based \PCP
construction. Such a test has oracle access to a collection of functions and
determines whether all the functions are the same dictatorship, or all their
low degree influences are Their test makes queries and has
amortized query complexity but has an inherent loss of
perfect completeness. In this paper we give an adaptive hypergraph dictatorship
test that achieves both perfect completeness and amortized query complexity
.Comment: Some minor correction
Quantum Property Testing
A language L has a property tester if there exists a probabilistic algorithm
that given an input x only asks a small number of bits of x and distinguishes
the cases as to whether x is in L and x has large Hamming distance from all y
in L. We define a similar notion of quantum property testing and show that
there exist languages with quantum property testers but no good classical
testers. We also show there exist languages which require a large number of
queries even for quantumly testing
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