Dictionary learning from quadratic measurements in block sparse models

This paper introduces the problem of dictionary learning from quadratic measurements of block sparse observations. While existing literature considers dictionary learning directly from the measurements, the proposed approach shows that learning the dictionary from certain quadratic products of these measurements can offer unique advantages, especially with respect to the size of identified sparse support. The proposed results are valid under some assumptions on the structure of the unknown sparse coefficient matrix, which hold true in problems such as independent component analysis. Given an M × L observation matrix, it is shown that these assumptions can lead to the recovery of supports of size O(M²), along with the unknown dictionary, whereas existing literature in dictionary learning can only guarantee recovering sparse supports of size O(M)¹.