A convex program for bilinear inversion of sparse vectors

TL;DR

This paper introduces a convex optimization method called -BranchHull for bilinear inversion of sparse vectors, providing theoretical guarantees and practical algorithms for applications like dielectric imaging and piecewise constant signal recovery.

Contribution

The paper presents -BranchHull, a novel convex program for bilinear inverse problems with sparsity, along with recovery guarantees and extensions for noise, outliers, and piecewise signals.

Findings

Recovery guaranteed when L (S_1+S_2)\,(log^2(K+N))

Numerical experiments confirm the theoretical scaling

ADMM implementation effectively recovers signals from real images

Abstract

We consider the bilinear inverse problem of recovering two vectors, $\boldsymbol{x}\in\mathbb{R}^L$ and $\boldsymbol{w}\in\mathbb{R}^L$, from their entrywise product. We consider the case where $\boldsymbol{x}$ and $\boldsymbol{w}$ have known signs and are sparse with respect to known dictionaries of size $K$ and $N$, respectively. Here, $K$ and $N$ may be larger than, smaller than, or equal to $L$. We introduce $\ell_1$-BranchHull, which is a convex program posed in the natural parameter space and does not require an approximate solution or initialization in order to be stated or solved. We study the case where $\boldsymbol{x}$ and $\boldsymbol{w}$ are $S_1$- and $S_2$-sparse with respect to a random dictionary and present a recovery guarantee that only depends on the number of measurements as $L\geq\Omega(S_1+S_2)\log^{2}(K+N)$. Numerical experiments verify that the scaling constant in the theorem is not too large. One application of this problem is the sweep distortion removal task in dielectric imaging, where one of the signals is a nonnegative reflectivity, and the other signal lives in a known subspace, for example that given by dominant wavelet coefficients. We also introduce a variants of $\ell_1$-BranchHull for the purposes of tolerating noise and outliers, and for the purpose of recovering piecewise constant signals. We provide an ADMM implementation of these variants and show they can extract piecewise constant behavior from real images.