2017/02/14 by Alireza Aghasi, Aghasi, Alireza, Ali Ahmed +5
Engineering · #FOS: Computer and information sciences #FOS: Mathematics #Information Theory (cs.IT) #Microwave Imaging and Scattering Analysis #Optimization and Control (math.OC) #Sparse and Compressive Sensing Techniques #Ultrasonics and Acoustic Wave Propagation
paper · pdf · doi:10.48550/arxiv.1702.04342
openalex publication_date 2017/02/14 · openalex created_date 2022/10/06 · openalex updated_date 2026/07/28
We consider the bilinear inverse problem of recovering two vectors, x and\nw, in \ℝL from their entrywise product. For the case where the\nvectors have known signs and belong to known subspaces, we introduce the convex\nprogram BranchHull, which is posed in the natural parameter space that does not\nrequire an approximate solution or initialization in order to be stated or\nsolved. Under the structural assumptions that x and w are members of known\nK and N dimensional random subspaces, we present a recovery guarantee for\nthe noiseless case and a noisy case. In the noiseless case, we prove that the\nBranchHull recovers x and w up to the inherent scaling ambiguity with high\nprobability when L \≫ 2(K+N). The analysis provides a precise upper bound\non the coefficient for the sample complexity. In a noisy case, we show that\nwith high probability the BranchHull is robust to small dense noise when L =\n\Ω(K+N). BranchHull is motivated by the sweep distortion removal task in\ndielectric imaging, where one of the signals is a nonnegative reflectivity, and\nthe other signal lives in a known wavelet subspace. Additional potential\napplications are blind deconvolution and self-calibration.\n