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We consider the problem of estimating self-exciting generalized linear modelsfrom limited binary observations, where the history of the process serves asthe covariate. We analyze the performance of two classes of estimators, namelythe $\ell_1$-regularized maximum likelihood and greedy estimators, for acanonical self-exciting process and characterize the sampling tradeoffsrequired for stable recovery in the non-asymptotic regime. Our results extendthose of compressed sensing for linear and generalized linear models withi.i.d. covariates to those with highly inter-dependent covariates. We furtherprovide simulation studies as well as application to real spiking data from themouse's lateral geniculate nucleus and the ferret's retinal ganglion cellswhich agree with our theoretical predictions.

Robust Estimation of Self-Exciting Generalized Linear Models with Application to Neuronal Modeling