2,591 research outputs found
The intersection of two halfspaces has high threshold degree
The threshold degree of a Boolean function f:{0,1}^n->{-1,+1} is the least
degree of a real polynomial p such that f(x)=sgn p(x). We construct two
halfspaces on {0,1}^n whose intersection has threshold degree Theta(sqrt n), an
exponential improvement on previous lower bounds. This solves an open problem
due to Klivans (2002) and rules out the use of perceptron-based techniques for
PAC learning the intersection of two halfspaces, a central unresolved challenge
in computational learning. We also prove that the intersection of two majority
functions has threshold degree Omega(log n), which is tight and settles a
conjecture of O'Donnell and Servedio (2003).
Our proof consists of two parts. First, we show that for any nonconstant
Boolean functions f and g, the intersection f(x)^g(y) has threshold degree O(d)
if and only if ||f-F||_infty + ||g-G||_infty < 1 for some rational functions F,
G of degree O(d). Second, we settle the least degree required for approximating
a halfspace and a majority function to any given accuracy by rational
functions.
Our technique further allows us to make progress on Aaronson's challenge
(2008) and contribute strong direct product theorems for polynomial
representations of composed Boolean functions of the form F(f_1,...,f_n). In
particular, we give an improved lower bound on the approximate degree of the
AND-OR tree.Comment: Full version of the FOCS'09 pape
The Dual Polynomial of Bipartite Perfect Matching
We obtain a description of the Boolean dual function of the Bipartite Perfect
Matching decision problem, as a multilinear polynomial over the Reals. We show
that in this polynomial, both the number of monomials and the magnitude of
their coefficients are at most exponential in . As an
application, we obtain a new upper bound of on the approximate degree of the bipartite perfect matching function,
improving the previous best known bound of . We deduce
that, beyond a factor, the polynomial method
cannot be used to improve the lower bound on the bounded-error quantum query
complexity of bipartite perfect matching
An Optimal Lower Bound on the Communication Complexity of Gap-Hamming-Distance
We prove an optimal lower bound on the randomized communication
complexity of the much-studied Gap-Hamming-Distance problem. As a consequence,
we obtain essentially optimal multi-pass space lower bounds in the data stream
model for a number of fundamental problems, including the estimation of
frequency moments.
The Gap-Hamming-Distance problem is a communication problem, wherein Alice
and Bob receive -bit strings and , respectively. They are promised
that the Hamming distance between and is either at least
or at most , and their goal is to decide which of these is the
case. Since the formal presentation of the problem by Indyk and Woodruff (FOCS,
2003), it had been conjectured that the naive protocol, which uses bits of
communication, is asymptotically optimal. The conjecture was shown to be true
in several special cases, e.g., when the communication is deterministic, or
when the number of rounds of communication is limited.
The proof of our aforementioned result, which settles this conjecture fully,
is based on a new geometric statement regarding correlations in Gaussian space,
related to a result of C. Borell (1985). To prove this geometric statement, we
show that random projections of not-too-small sets in Gaussian space are close
to a mixture of translated normal variables
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