Following up on this morning's entry, the phase transition with some noise seems to show that a non least squares solver does very well in recovering positive sparse signals with no L1 regularization. Let us note that there is something peculiar about being positive see Dustin's blog entry on A variant on the compressed sensing of Yves Meyer.
As a side note, I wonder if some of the work by Phil Schniter et al's investigation or Justin et al results in the group sparse setting should (or not) be included in the references of this preprint. Let us also note that using a 0/1 ensemble also impart a non negativity constraint of the measurement ensemble that may help the recovery.
Robust Nonnegative Sparse Recovery and the Nullspace Property of 0/1 Measurements by Richard Kueng, Peter Jung
We investigate recovery of nonnegative vectors from non--adaptive compressive measurements in the presence of noise of unknown power. It is known in literature that under additional assumptions on the measurement design recovery is possible in the noiseless setting with nonnegative least squares without any regularization. We show that such known uniquenes results carry over to the noisy setting. We present guarantees which hold instantaneously by establishing the relation to the robust nullspace property. As an important example, we establish that an m x n random iid. 0/1-valued Bernoulli matrix has with overwhelming probability the robust nullspace property for m=O(s log(n)) and is applicable in the nonnegative case. Our analysis is motivated by applications in wireless network activity detection.
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