  Let $X$ be a closed $6-$dimensional manifold with a half-closed
$SU(3)-$structure. On the product manifold $X\times S^{1}$, with respect to the
product $G_{2}-$structure and on a pullback vector bundle from $X$, we show
that any $G_{2}-$instanton is equivalent to a Hermitian Yang-Mills connection
on $X$ via a "broken gauge". This result reveals the topological type of the
moduli of $G_{2}-$instantons on $X\times S^{1}$. In dimension $8$, similar
result holds for moduli of $Spin(7)-$instantons. A generalization and an
example are given.
