  The concept class of low-degree polynomial threshold functions (PTFs) plays a
fundamental role in machine learning. In this paper, we study PAC learning of
$K$-sparse degree-$d$ PTFs on $\mathbb{R}^n$, where any such concept depends
only on $K$ out of $n$ attributes of the input. Our main contribution is a new
algorithm that runs in time $({nd}/{\epsilon})^{O(d)}$ and under the Gaussian
marginal distribution, PAC learns the class up to error rate $\epsilon$ with
$O(\frac{K^{4d}}{\epsilon^{2d}} \cdot \log^{5d} n)$ samples even when an $\eta
\leq O(\epsilon^d)$ fraction of them are corrupted by the nasty noise of
Bshouty et al. (2002), possibly the strongest corruption model. Prior to this
work, attribute-efficient robust algorithms are established only for the
special case of sparse homogeneous halfspaces. Our key ingredients are: 1) a
structural result that translates the attribute sparsity to a sparsity pattern
of the Chow vector under the basis of Hermite polynomials, and 2) a novel
attribute-efficient robust Chow vector estimation algorithm which uses
exclusively a restricted Frobenius norm to either certify a good approximation
or to validate a sparsity-induced degree-$2d$ polynomial as a filter to detect
corrupted samples.
