## Consistency of l1 Regularized Maximum Likelihood for the Poisson Distributed Data

Consider a set of Poisson distributed random variables (RV). Assume a sparse vector ** w **which is shared among the RVs, such that the rate of the

*i*th random variable is

*. The goal is to estimate*

**w**^{T}**a**_{i}*using this set of RVs. This problem is closely to related to the Compressive Sensing problem. In fact*

**w****A**= [

**a**_{1}

**a**_{2}…

**a**_{n}]

^{T}can be considered as the sensing matrix. We will show that under the condition that

**A**satisfies Restricted Eigenvalue condition,

*l*

^{1}regularized maximum likelihood estimation of

*will converge to the ground truth.*

**w**- D. Motamedvaziri, V. Saligrama, M. H. Rohban, “Sparse Signal Recovery under Poisson Statistics for Online Marketing Applications,” ICASSP 2014, (accepted).
- D. Motamedvaziri, M. H. Rohban, V. Saligrama, “
*Sparse Signal Recovery under Poisson Statistics*,” submitted to 51st Allerton Conference on Communication, Control, and Computing, 2013 arXiv:1307.4666 [math.ST].