  This paper investigates conditions under which certain kinds of systems of
bilinear equations have a unique structured solution. In particular, we look at
when we can recover vectors $\boldsymbol{w},\boldsymbol{q}$ from observations
of the form \[ y_{\ell} =
<\boldsymbol{w},\boldsymbol{b}_{\ell}><\boldsymbol{c}_{\ell},\boldsymbol{q}>,
\quad \ell = 1,\ldots,L, \] where $\boldsymbol{b}_\ell,\boldsymbol{c}_\ell$ are
known. We show that if $\boldsymbol{w}\in\mathbb{C}^{M_1}$ and
$\boldsymbol{q}\in\mathbb{C}^{M_2}$ are sparse, with no more than $K$ and $N$
nonzero entries, respectively, and the
$\boldsymbol{b}_\ell,\boldsymbol{c}_\ell$ are generic, selected as independent
Gaussian random vectors, then $\boldsymbol{w},\boldsymbol{q}$ are uniquely
determined from \[
  L \geq \mathrm{Const}\cdot (K+N)\log^5(M_1M_2) \] such equations with high
probability.
  The key ingredient in our analysis is a uniform probabilistic bound on how
far a random process of the form \[Z(\boldsymbol{X}) =
\sum_{\ell=1}^L|\boldsymbol{b}_\ell^*\boldsymbol{X}\boldsymbol{c}_\ell|^2 \]
deviates from its mean over a set of structured matrices
$\boldsymbol{X}\in\mathcal{X}$. As both $\boldsymbol{b}_\ell$ and
$\boldsymbol{c}_\ell$ are random, this is a specialized type of $4$th order
chaos; we refer to $Z(\boldsymbol{X})$ as an {\em empirical chaos process}.
Bounding this process yields a set of general conditions for when the map
$\boldsymbol{X}\rightarrow
\{\boldsymbol{b}_\ell^*\boldsymbol{X}\boldsymbol{c}_\ell\}_{\ell=1}^L$ is a
restricted isometry over the set of matrices $\mathcal{X}$. The conditions are
stated in terms of general geometric properties of the set $\mathcal{X}$, and
are explicitly computed for the case where $\mathcal{X}$ is the set of matrices
that are simultaneously sparse and low rank.
