Denoising modulo samples: k-NN regression and tightness of SDP relaxation

Abstract: Many modern applications involve the acquisition of noisy modulo samples of a function $f$, with the goal being to recover estimates of the original samples of $f$. For a Lipschitz function $f:[0,1]^d \to \mathbb{R}$, suppose we are given the samples $y_i = (f(x_i) + \eta_i)\bmod 1; \quad i=1,\dots,n$ where $\eta_i$ denotes noise. Assuming $\eta_i$ are zero-mean i.i.d Gaussian's, and $x_i$'s form a uniform grid, we derive a two-stage algorithm that recovers estimates of the samples $f(x_i)$ with a uniform error rate $O((\frac{\log n}{n})^{\frac{1}{d+2}})$ holding with high probability. The first stage involves embedding the points on the unit complex circle, and obtaining denoised estimates of $f(x_i)\bmod 1$ via a $k$NN (nearest neighbor) estimator. The second stage involves a sequential unwrapping procedure which unwraps the denoised mod $1$ estimates from the first stage. The estimates of the samples $f(x_i)$ can be subsequently utilized to construct an estimate of the function $f$, with the aforementioned uniform error rate. Recently, Cucuringu and Tyagi proposed an alternative way of denoising modulo $1$ data which works with their representation on the unit complex circle. They formulated a smoothness regularized least squares problem on the product manifold of unit circles, where the smoothness is measured with respect to the Laplacian of a proximity graph $G$ involving the $x_i$'s. This is a nonconvex quadratically constrained quadratic program (QCQP) hence they proposed solving its semidefinite program (SDP) based relaxation. We derive sufficient conditions under which the SDP is a tight relaxation of the QCQP. Hence under these conditions, the global solution of QCQP can be obtained in polynomial time.