BranchHull: Convex bilinear inversion from the entrywise product of signals with known signs

TL;DR

BranchHull is a convex method for bilinear inverse problems that guarantees exact recovery of signals with known signs in subspaces, even in noisy conditions, using minimal samples.

Contribution

This paper introduces BranchHull, a convex program that solves bilinear inverse problems without initialization, providing theoretical guarantees for exact and robust recovery.

Findings

Recovers signals up to scaling with high probability when L >> 2(K+N).

Provides explicit sample complexity bounds for noiseless recovery.

Demonstrates robustness to small dense noise when L = Omega(K+N).

Abstract

We consider the bilinear inverse problem of recovering two vectors, $x$ and $w$, in $\mathbb{R}^L$ from their entrywise product. For the case where the vectors have known signs and belong to known subspaces, we introduce the convex program BranchHull, which is posed in the natural parameter space that does not require an approximate solution or initialization in order to be stated or solved. Under the structural assumptions that $x$ and $w$ are members of known $K$ and $N$ dimensional random subspaces, we present a recovery guarantee for the noiseless case and a noisy case. In the noiseless case, we prove that the BranchHull recovers $x$ and $w$ up to the inherent scaling ambiguity with high probability when $L\ \gg\ 2(K+N)$. The analysis provides a precise upper bound on the coefficient for the sample complexity. In a noisy case, we show that with high probability the BranchHull is robust to small dense noise when $L = \Omega(K+N)$. BranchHull is motivated by 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 wavelet subspace. Additional potential applications are blind deconvolution and self-calibration.