Perfect Recovery Conditions For Non-Negative Sparse Modeling

Sparse modeling has been widely and successfully used in many applications such as computer vision, machine learning, and pattern recognition and, accompanied with those applications, significant research has studied the theoretical limits and algorithm design for convex relaxations in sparse modeling. However, only little has been done for theoretical limits of non-negative versions of sparse modeling. The behavior is expected to be similar as the general sparse modeling, but a precise analysis has not been explored. This paper studies the performance of non-negative sparse modeling, especially for non-negativity constrained and $\ell_1$-penalized least squares, and gives an exact bound for which this problem can recover the correct signal elements. We pose two conditions to guarantee the correct signal recovery: minimum coefficient condition (MCC) and non-linearity vs. subset coherence condition (NSCC). The former defines the minimum weight for each of the correct atoms present in the signal and the latter defines the tolerable deviation from the linear model relative to the positive subset coherence (PSC), a novel type of "coherence" metric. We provide rigorous performance guarantees based on these conditions and experimentally verify their precise predictive power in a hyperspectral data unmixing application.