6,490 research outputs found
Bounds on the Average Sensitivity of Nested Canalizing Functions
Nested canalizing Boolean (NCF) functions play an important role in
biological motivated regulative networks and in signal processing, in
particular describing stack filters. It has been conjectured that NCFs have a
stabilizing effect on the network dynamics. It is well known that the average
sensitivity plays a central role for the stability of (random) Boolean
networks. Here we provide a tight upper bound on the average sensitivity for
NCFs as a function of the number of relevant input variables. As conjectured in
literature this bound is smaller than 4/3 This shows that a large number of
functions appearing in biological networks belong to a class that has very low
average sensitivity, which is even close to a tight lower bound.Comment: revised submission to PLOS ON
Learning DNFs under product distributions via {\mu}-biased quantum Fourier sampling
We show that DNF formulae can be quantum PAC-learned in polynomial time under
product distributions using a quantum example oracle. The best classical
algorithm (without access to membership queries) runs in superpolynomial time.
Our result extends the work by Bshouty and Jackson (1998) that proved that DNF
formulae are efficiently learnable under the uniform distribution using a
quantum example oracle. Our proof is based on a new quantum algorithm that
efficiently samples the coefficients of a {\mu}-biased Fourier transform.Comment: 17 pages; v3 based on journal version; minor corrections and
clarification
Formulas vs. Circuits for Small Distance Connectivity
We give the first super-polynomial separation in the power of bounded-depth
boolean formulas vs. circuits. Specifically, we consider the problem Distance
Connectivity, which asks whether two specified nodes in a graph of size
are connected by a path of length at most . This problem is solvable
(by the recursive doubling technique) on {\bf circuits} of depth
and size . In contrast, we show that solving this problem on {\bf
formulas} of depth requires size for all . As corollaries:
(i) It follows that polynomial-size circuits for Distance Connectivity
require depth for all . This matches the
upper bound from recursive doubling and improves a previous lower bound of Beame, Pitassi and Impagliazzo [BIP98].
(ii) We get a tight lower bound of on the size required to
simulate size- depth- circuits by depth- formulas for all and . No lower bound better than
was previously known for any .
Our proof technique is centered on a new notion of pathset complexity, which
roughly speaking measures the minimum cost of constructing a set of (partial)
paths in a universe of size via the operations of union and relational
join, subject to certain density constraints. Half of our proof shows that
bounded-depth formulas solving Distance Connectivity imply upper bounds
on pathset complexity. The other half is a combinatorial lower bound on pathset
complexity
Approximate resilience, monotonicity, and the complexity of agnostic learning
A function is -resilient if all its Fourier coefficients of degree at
most are zero, i.e., is uncorrelated with all low-degree parities. We
study the notion of of Boolean
functions, where we say that is -approximately -resilient if
is -close to a -valued -resilient function in
distance. We show that approximate resilience essentially characterizes the
complexity of agnostic learning of a concept class over the uniform
distribution. Roughly speaking, if all functions in a class are far from
being -resilient then can be learned agnostically in time and
conversely, if contains a function close to being -resilient then
agnostic learning of in the statistical query (SQ) framework of Kearns has
complexity of at least . This characterization is based on the
duality between approximation by degree- polynomials and
approximate -resilience that we establish. In particular, it implies that
approximation by low-degree polynomials, known to be sufficient for
agnostic learning over product distributions, is in fact necessary.
Focusing on monotone Boolean functions, we exhibit the existence of
near-optimal -approximately
-resilient monotone functions for all
. Prior to our work, it was conceivable even that every monotone
function is -far from any -resilient function. Furthermore, we
construct simple, explicit monotone functions based on and that are close to highly resilient functions. Our constructions are
based on a fairly general resilience analysis and amplification. These
structural results, together with the characterization, imply nearly optimal
lower bounds for agnostic learning of monotone juntas
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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