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.

翻译：在许多应用中,我们有机会获得一个光滑功能的噪音模版样本,目的是严格地解包样品,即估计该功能的原始样品。在最近的一项工作中,Cucuringu和Tyagi提议在单位复杂圆圈上首先代表模版样本,然后解决一个平滑的固定最小方块问题 -- -- 在单位圆的多个产品中测量出一个适当的近距离图($G$)的光滑度,目标是严格地解包样品,即估计该功能的原始样品。Cucuringu和Tyagi提议通过首先在单位复杂圆圈中代表它们,然后解决模版样品样品的解密性,然后解决平滑性,然后解决平滑性最小方块问题 -- -- 在单位圆的多个产品中测量到的光滑度(TRS) $G$($G$ $) 。在这项工作中,我们分析(TRS) 以及不受限制的松动的四面调(QQP),因此,他们建议解决其球体松动,导致一个信任区域的子区域分质区域(TRS) 分析。在高压分析中,我们进行了高压的平面的平调。