Optimal Map-Making with Singularities

In this work, we investigate the optimal map-making technique for the linear system d=Ax+n while carefully taking into account singularities that may come from either the covariance matrix C = ⟨ nn^t ⟩ or the main matrix A. We first describe the general optimal solution, which is quite complex, and then use the modified pseudo inverse to create a near-optimal solution, which is simple, robust, and can significantly alleviate the unwanted noise amplification during map-making. The effectiveness of the nearly optimal solution is then compared to that of the naive co-adding solution and the standard pseudo inverse solution, showing noticeable improvements. Interestingly, all one needs to get the near-optimal solution with singularity is just a tiny change to the classical solution, which is designed for the case without singularity.