  We study identifiability for bilinear inverse problems under sparsity and
subspace constraints. We show that, up to a global scaling ambiguity, almost
all such maps are injective on the set of pairs of sparse vectors if the number
of measurements $m$ exceeds $2(s_1+s_2)-2$, where $s_1$ and $s_2$ denote the
sparsity of the two input vectors, and injective on the set of pairs of vectors
lying in known subspaces of dimensions $n_1$ and $n_2$ if $m\geq 2(n_1+n_2)-4$.
We also prove that both these bounds are tight in the sense that one cannot
have injectivity for a smaller number of measurements. Our proof technique
draws from algebraic geometry. As an application we derive optimal
identifiability conditions for the deconvolution problem, thus improving on
recent work of Li et al. [1].
