Growth bound for signs of m-term linear combinations

For $f={\sum }_{j=1}^m{\alpha }_j{g}_j$ with $g_j$ drawn from a binary class $\mathcal{G}$, fixing $(g_1,\dots ,g_m)$ leaves an $m$-dimensional linear sign class. Consequentlywhen $\mathcal{G}$ is finite, and in general

A bias-free network with $p$ inputs, $m$ sign-activated hidden nodes, and a thresholded linear output satisfiesThe hidden units contribute $s(\mathcal{G},n{)}^m\le (n+1{)}^{pm}$ sample patterns, and the output weights contribute $(n+1{)}^m$.