Sparse Power Factorization With Refined Peakiness Conditions

Many important signal processing tasks, like blind deconvolution and self-calibration, can be modeled as a bilinear inverse problem, meaning that the observation $y$ depends Iinearly on two unknown vectors $u$ and $v$. In many of these problems, at least one of the input vectors can be assumed to be sparse, i.e., to have only few non-zero entries. Sparse Power Factorization (SPF), proposed by Lee, Wu, and Bresler, aims to tackle this problem. Under the assumption that the measurements are random, they established recovery guarantees for signals with a significant portion of the mass concentrated in a single entry at a sampling rate, which scales with the intrinsic dimension of the signals. In this note we extend these recovery guarantees to a broader and more realistic class of signals, at the cost of a slightly increased number of measurements. Namely, we require that a significant portion of the mass is concentrated in a small set of entries (rather than just one entry).