论文标题
错误分析图表上平滑模量信号的错误分析
Error analysis for denoising smooth modulo signals on a graph
论文作者
论文摘要
在许多应用程序中，我们可以访问光滑功能的嘈杂模型样本，其目标是稳健地解开样品，即估计函数的原始样本。在最近的一项工作中，Cucuringu和Tyagi提出了通过在单位复合圆圈上表示模型样品，然后解决平滑度正则最小二乘问题问题 - 平滑度测量的w.r.t w.r.t w.r.t是合适的接近$ G $ G $的laplacian-在单位圈子的产品中。这个问题是一个四限制的二次程序（QCQP），它是非convex，因此他们提出了解决其球体 - 删除性的，导致信任区域子问题（TRS）。就理论保证而言，$ \ ell_2 $错误范围是（TRS）。但是，这些界限通常很弱，并且并不能真正证明（TRS）执行的脱氧。 在这项工作中，我们分析了（TRS）以及（QCQP）的不受约束的放松。对于这两个估计量，我们在高斯噪声的设置中提供了精致的分析，并得出了噪声制度，它们可以证明Modulo观测值W.R.T $ \ ELL_2 $ NORM。该分析是在$ g $的一般环境中进行的。
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.
