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 [11] 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.

在许多应用中，我们可以访问平滑函数的噪声模样本，其目标是稳健地展开样本，即估计函数的原始样本。在最近的一项工作中，Cucuringu 和 Tyagi [11] 提出通过首先在单位复圆上表示模样本，然后解决平滑度正则化最小二乘问题——通过合适的邻近图G的拉普拉斯算子测量的平滑度——对模样本进行去噪。单位圆的积流形。这个问题是一个非凸的二次约束二次规划（QCQP），因此他们提出解决导致信任域子问题（TRS）的球面松弛问题。在理论保证方面，${\ell }_2$为 (TRS) 推导出错误界限。然而，这些界限通常很弱，并没有真正证明 (TRS) 执行的去噪。

在这项工作中，我们分析了 (TRS) 以及 (QCQP) 的无约束松弛。对于这两个估计器，我们在高斯噪声的设置中提供了精细的分析，并推导出噪声状态，在那里它们可证明对模观测值进行降噪${\ell }_2$规范。分析是在一般设置中执行的，其中G是任何连通图。