Compressed Data Separation With Redundant Dictionaries

Most of the data scientists face today might be classified as multimodal data, i.e., being composed of distinct subcomponents. One common task is to separate such data into appropriate single components for further analysis. In this paper, we consider data separation from fewer, linear, nonadaptive, and noisy measurements. We show that the distinct subcomponents, which are (approximately) sparse in morphologically different (redundant) dictionaries, can be reconstructed by solving the split-analysis algorithm, provided that the dictionaries satisfy a mutual coherence (between the different dictionaries) condition and the measurement matrix satisfies a restricted isometry property adapted to a composed dictionary. These conditions impose no incoherence restriction on the dictionaries themselves, and our main result may be the first of this kind.