  In many applications, we are given access to noisy modulo samples of a smooth
function with the goal being to robustly unwrap the samples, i.e., to estimate
the original samples of the function. In a recent work, Cucuringu and Tyagi
proposed denoising the modulo samples by first representing them on the unit
complex circle and then solving a smoothness regularized least squares problem
-- the smoothness measured w.r.t the Laplacian of a suitable proximity graph
$G$ -- on the product manifold of unit circles. This problem is a quadratically
constrained quadratic program (QCQP) which is nonconvex, hence they proposed
solving its sphere-relaxation leading to a trust region subproblem (TRS). In
terms of theoretical guarantees, $\ell_2$ error bounds were derived for (TRS).
These bounds are however weak in general and do not really demonstrate the
denoising performed by (TRS).
  In this work, we analyse the (TRS) as well as an unconstrained relaxation of
(QCQP). For both these estimators we provide a refined analysis in the setting
of Gaussian noise and derive noise regimes where they provably denoise the
modulo observations w.r.t the $\ell_2$ norm. The analysis is performed in a
general setting where $G$ is any connected graph.
