Title: Bayesian Lasso Posterior Sampling via Parallelized Measure Transport,
Abstract: It is well known that the Lasso can be interpreted as a Bayesian posterior
mode estimate with a Laplacian prior. Obtaining samples from the full posterior
distribution, the Bayesian Lasso, confers major advantages in performance as
compared to having only the Lasso point estimate. Traditionally, the Bayesian
Lasso is implemented via Gibbs sampling methods which suffer from lack of
scalability, unknown convergence rates, and generation of samples that are
necessarily correlated. We provide a measure transport approach to generate
i.i.d samples from the posterior by constructing a transport map that
transforms a sample from the Laplacian prior into a sample from the posterior.
We show how the construction of this transport map can be parallelized into
modules that iteratively solve Lasso problems and perform closed-form linear
algebra updates. With this posterior sampling method, we perform maximum
likelihood estimation of the Lasso regularization parameter via the EM
algorithm. We provide comparisons to traditional Gibbs samplers using the
diabetes dataset of Efron et al. Lastly, we give an example implementation on a
computing system that leverages parallelization, a graphics processing unit,
whose execution time has much less dependence on dimension as compared to a
standard implementation.