BranchHull: Bilinear Compressed Sensing
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- December 27, 2021
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We consider the bilinear inverse problem of recovering two vectors, x and w, RL from their entrywise product. For the case where the vectors have known signs and belong to known subspaces. An immediate formulation of the inverse problem leads to a non-convex optimization program. We use a geometrical insight to formulate a convex relaxation 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. Motivating applications include, blind multiplicative interference removal in signal processing, image restoration. Figure (left) shows the sign information restricts the solution to one of the branches of the hyperbola. Geometrically, this happens as the solution lies at the intersection of the l1-ball and the hyperbolic curve (constraint) as shown in figure (right).
Applications
Blind multiplicative interference removal in signal processing and image restoration
Related Papers
[1] A. Aghasi, A. Ahmed, P. Hand, and B. Joshi Bilinear Compressed Sensing under known Signs via Convex Programming, arXiv preprint arXiv:1906.11636, 2018.