Optimal map-making with singularities

TL;DR

This paper develops a near-optimal, robust map-making method that effectively handles singularities in the system, improving noise mitigation compared to traditional solutions.

Contribution

It introduces a simple modification to the classical pseudo-inverse approach to achieve near-optimal map-making in the presence of singularities.

Findings

Near-optimal solution reduces noise amplification

Modified pseudo-inverse outperforms naive co-adding and standard pseudo-inverse

Small change to classical solution suffices for handling singularities

Abstract

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 $C = \langle nn^t \rangle$ 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.