  The l1-regularization is very popular in high dimensional statistics -- it
changes a combinatorial problem of choosing which subset of the parameter are
zero, into a simple continuous optimization. Using a continuous prior
concentrated near zero, the Bayesian counterparts are successful in quantifying
the uncertainty in the variable selection problems; nevertheless, the lack of
exact zeros makes it difficult for broader problems such as the change-point
detection and rank selection. Inspired by the duality of the l1-regularization
as a constraint onto an l1-ball, we propose a new prior by projecting a
continuous distribution onto the l1-ball. This creates a positive probability
on the ball boundary, which contains both continuous elements and exact zeros.
Unlike the spike-and-slab prior, this l1-ball projection is continuous and
differentiable almost surely, making the posterior estimation amenable to the
Hamiltonian Monte Carlo algorithm. We examine the properties, such as the
volume change due to the projection, the connection to the combinatorial prior,
the minimax concentration rate in the linear problem. We demonstrate the
usefulness of exact zeros that simplify the combinatorial problems, such as the
change-point detection in time series, the dimension selection of mixture model
and the low-rank-plus-sparse change detection in the medical images.
