  Dantzig Selector (DS) is widely used in compressed sensing and sparse
learning for feature selection and sparse signal recovery. Since the DS
formulation is essentially a linear programming optimization, many existing
linear programming solvers can be simply applied for scaling up. The DS
formulation can be explained as a basis pursuit denoising problem, wherein the
data matrix (or measurement matrix) is employed as the denoising matrix to
eliminate the observation noise. However, we notice that the data matrix may
not be the optimal denoising matrix, as shown by a simple counter-example. This
motivates us to pursue a better denoising matrix for defining a general DS
formulation. We first define the optimal denoising matrix through a minimax
optimization, which turns out to be an NPhard problem. To make the problem
computationally tractable, we propose a novel algorithm, termed as Optimal
Denoising Dantzig Selector (ODDS), to approximately estimate the optimal
denoising matrix. Empirical experiments validate the proposed method. Finally,
a novel sparse reinforcement learning algorithm is formulated by extending the
proposed ODDS algorithm to temporal difference learning, and empirical
experimental results demonstrate to outperform the conventional vanilla DS-TD
algorithm.
