Compressive sampling of non-negative signals

Traditional Nyquist-Shannon sampling dictates that a continuous time signal be sampled at twice its bandwidth to achieve perfect recovery. However, It has been recently demonstrated that by exploiting the structure of the signal, it is possible to sample a signal below the Nyquist rate and achieve perfect reconstruction using a random projection, sparse representation and an lscr₁-norm minimisation. These methods constitute a new and emerging theory known as Compressive Sampling (or Compressed sensing). Here, we apply Compressive Sampling to non-negative signals, and propose an algorithm-non-negative under-determined iteratively reweighted least squares (NUIRLS)-for signal recovery. NUIRLS is derived within the framework of Non-negative Matrix Factorisation (NMF) and utilises Iteratively Reweighted Least Squares as its objective, recovering non-negative minimum lscr_p-norm solutions, 0 les p les 1. We demonstrate that-for sufficiently sparse non-negative signals-the signals recovered by NUIRLS and NMF are essentially the same, which suggests that a non-negativity constraint is enough to recover sufficiently sparse signals.