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 R, suppose we are given the samples yi = (f(xi) + ηi) 1; i=1,…,n where ηi denotes noise. Assuming ηi are zero-mean i.i.d Gaussian's, and xi's form a uniform grid, we derive a two-stage algorithm that recovers estimates of the samples f(xi) with a uniform error rate O(( nn)1d+2) holding with high probability. The first stage involves embedding the points on the unit complex circle, and obtaining denoised estimates of f(xi) 1 via a kNN (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(xi) 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 xi'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.