95 research outputs found
Maximum-order Complexity and Correlation Measures
We estimate the maximum-order complexity of a binary sequence in terms of its
correlation measures. Roughly speaking, we show that any sequence with small
correlation measure up to a sufficiently large order cannot have very small
maximum-order complexity
Weighted Polynomial Approximations: Limits for Learning and Pseudorandomness
Polynomial approximations to boolean functions have led to many positive
results in computer science. In particular, polynomial approximations to the
sign function underly algorithms for agnostically learning halfspaces, as well
as pseudorandom generators for halfspaces. In this work, we investigate the
limits of these techniques by proving inapproximability results for the sign
function.
Firstly, the polynomial regression algorithm of Kalai et al. (SIAM J. Comput.
2008) shows that halfspaces can be learned with respect to log-concave
distributions on in the challenging agnostic learning model. The
power of this algorithm relies on the fact that under log-concave
distributions, halfspaces can be approximated arbitrarily well by low-degree
polynomials. We ask whether this technique can be extended beyond log-concave
distributions, and establish a negative result. We show that polynomials of any
degree cannot approximate the sign function to within arbitrarily low error for
a large class of non-log-concave distributions on the real line, including
those with densities proportional to .
Secondly, we investigate the derandomization of Chernoff-type concentration
inequalities. Chernoff-type tail bounds on sums of independent random variables
have pervasive applications in theoretical computer science. Schmidt et al.
(SIAM J. Discrete Math. 1995) showed that these inequalities can be established
for sums of random variables with only -wise independence,
for a tail probability of . We show that their results are tight up to
constant factors.
These results rely on techniques from weighted approximation theory, which
studies how well functions on the real line can be approximated by polynomials
under various distributions. We believe that these techniques will have further
applications in other areas of computer science.Comment: 22 page
Bounds of triple exponential sums with mixed exponential and linear terms
We establish bounds of triple exponential sums with mixed exponential and
linear function. The method we use is by Shparlinski together with a bound of
additive energy from Roche-Newton, Rudnev and Shkredov.Comment: Corrected a typ
Pseudorandomness and Dynamics of Fermat Quotients
We obtain some theoretic and experimental results concerning various
properties (the number of fixed points, image distribution, cycle lengths) of
the dynamical system naturally associated with Fermat quotients acting on the
set . We also consider pseudorandom properties of Fermat
quotients such as joint distribution and linear complexity
Comparing Computational Entropies Below Majority (Or: When Is the Dense Model Theorem False?)
Computational pseudorandomness studies the extent to which a random variable
looks like the uniform distribution according to a class of tests
. Computational entropy generalizes computational pseudorandomness by
studying the extent which a random variable looks like a \emph{high entropy}
distribution. There are different formal definitions of computational entropy
with different advantages for different applications. Because of this, it is of
interest to understand when these definitions are equivalent.
We consider three notions of computational entropy which are known to be
equivalent when the test class is closed under taking majorities.
This equivalence constitutes (essentially) the so-called \emph{dense model
theorem} of Green and Tao (and later made explicit by Tao-Zeigler, Reingold et
al., and Gowers). The dense model theorem plays a key role in Green and Tao's
proof that the primes contain arbitrarily long arithmetic progressions and has
since been connected to a surprisingly wide range of topics in mathematics and
computer science, including cryptography, computational complexity,
combinatorics and machine learning. We show that, in different situations where
is \emph{not} closed under majority, this equivalence fails. This in
turn provides examples where the dense model theorem is \emph{false}.Comment: 19 pages; to appear in ITCS 202
Approximate Degree, Secret Sharing, and Concentration Phenomena
The epsilon-approximate degree deg~_epsilon(f) of a Boolean function f is the least degree of a real-valued polynomial that approximates f pointwise to within epsilon. A sound and complete certificate for approximate degree being at least k is a pair of probability distributions, also known as a dual polynomial, that are perfectly k-wise indistinguishable, but are distinguishable by f with advantage 1 - epsilon. Our contributions are:
- We give a simple, explicit new construction of a dual polynomial for the AND function on n bits, certifying that its epsilon-approximate degree is Omega (sqrt{n log 1/epsilon}). This construction is the first to extend to the notion of weighted degree, and yields the first explicit certificate that the 1/3-approximate degree of any (possibly unbalanced) read-once DNF is Omega(sqrt{n}). It draws a novel connection between the approximate degree of AND and anti-concentration of the Binomial distribution.
- We show that any pair of symmetric distributions on n-bit strings that are perfectly k-wise indistinguishable are also statistically K-wise indistinguishable with at most K^{3/2} * exp (-Omega (k^2/K)) error for all k < K <= n/64. This bound is essentially tight, and implies that any symmetric function f is a reconstruction function with constant advantage for a ramp secret sharing scheme that is secure against size-K coalitions with statistical error K^{3/2} * exp (-Omega (deg~_{1/3}(f)^2/K)) for all values of K up to n/64 simultaneously. Previous secret sharing schemes required that K be determined in advance, and only worked for f=AND. Our analysis draws another new connection between approximate degree and concentration phenomena.
As a corollary of this result, we show that for any d deg~_{1/3}(f). These upper and lower bounds were also previously only known in the case f=AND
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