Empirical Chaos Processes and Blind Deconvolution (1608.08370v2)

Abstract: 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.