  Since its introduction by Valiant in 1984, PAC learning of DNF expressions
remains one of the central problems in learning theory. We consider this
problem in the setting where the underlying distribution is uniform, or more
generally, a product distribution. Kalai, Samorodnitsky and Teng (2009) showed
that in this setting a DNF expression can be efficiently approximated from its
"heavy" low-degree Fourier coefficients alone. This is in contrast to previous
approaches where boosting was used and thus Fourier coefficients of the target
function modified by various distributions were needed. This property is
crucial for learning of DNF expressions over smoothed product distributions, a
learning model introduced by Kalai et al. (2009) and inspired by the seminal
smoothed analysis model of Spielman and Teng (2001).
  We introduce a new approach to learning (or approximating) a polynomial
threshold functions which is based on creating a function with range [-1,1]
that approximately agrees with the unknown function on low-degree Fourier
coefficients. We then describe conditions under which this is sufficient for
learning polynomial threshold functions. Our approach yields a new, simple
algorithm for approximating any polynomial-size DNF expression from its "heavy"
low-degree Fourier coefficients alone. Our algorithm greatly simplifies the
proof of learnability of DNF expressions over smoothed product distributions.
We also describe an application of our algorithm to learning monotone DNF
expressions over product distributions. Building on the work of Servedio
(2001), we give an algorithm that runs in time $\poly((s \cdot
\log{(s/\eps)})^{\log{(s/\eps)}}, n)$, where $s$ is the size of the target DNF
expression and $\eps$ is the accuracy. This improves on $\poly((s \cdot
\log{(ns/\eps)})^{\log{(s/\eps)} \cdot \log{(1/\eps)}}, n)$ bound of Servedio
(2001).
