Optimal Injectivity Conditions for Bilinear Inverse Problems with Applications to Identifiability of Deconvolution Problems

TL;DR

This paper establishes the minimal measurement requirements for injectivity in bilinear inverse problems with sparsity and subspace constraints, using algebraic geometry, and applies these results to deconvolution identifiability.

Contribution

It provides tight bounds on the number of measurements needed for injectivity in bilinear inverse problems under sparsity and subspace constraints, improving prior results.

Findings

Injectivity holds if measurements exceed 2(s1+s2)-2 for sparsity constraints.

Injectivity holds if measurements exceed 2(n1+n2)-4 for subspace constraints.

The bounds on measurements are proven to be tight and optimal.

Abstract

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].