  Using a Bayesian approach, we consider the problem of recovering sparse
signals under additive sparse and dense noise. Typically, sparse noise models
outliers, impulse bursts or data loss. To handle sparse noise, existing methods
simultaneously estimate the sparse signal of interest and the sparse noise of
no interest. For estimating the sparse signal, without the need of estimating
the sparse noise, we construct a robust Relevance Vector Machine (RVM). In the
RVM, sparse noise and ever present dense noise are treated through a combined
noise model. The precision of combined noise is modeled by a diagonal matrix.
We show that the new RVM update equations correspond to a non-symmetric
sparsity inducing cost function. Further, the combined modeling is found to be
computationally more efficient. We also extend the method to block-sparse
signals and noise with known and unknown block structures. Through simulations,
we show the performance and computation efficiency of the new RVM in several
applications: recovery of sparse and block sparse signals, housing price
prediction and image denoising.
