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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 thevectors have known signs and belong to known subspaces, we introduce the convexprogram BranchHull, which is posed in the natural parameter space that does notrequire an approximate solution or initialization in order to be stated orsolved. Under the structural assumptions that $x$ and $w$ are members of known$K$ and $N$ dimensional random subspaces, we present a recovery guarantee forthe noiseless case and a noisy case. In the noiseless case, we prove that theBranchHull recovers $x$ and $w$ up to the inherent scaling ambiguity with highprobability when $L\ \gg\ 2(K+N)$. The analysis provides a precise upper boundon the coefficient for the sample complexity. In a noisy case, we show thatwith high probability the BranchHull is robust to small dense noise when $L =\Omega(K+N)$. BranchHull is motivated by the sweep distortion removal task indielectric imaging, where one of the signals is a nonnegative reflectivity, andthe other signal lives in a known wavelet subspace. Additional potentialapplications are blind deconvolution and self-calibration.

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