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

Code

• github.com/mrfanuel/denoising-modulo-samples-k-NN-regression
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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 y_i = (f(x_i) + _i) 1; i=1,,n where _i denotes noise. Assuming _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(( 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(x_i) 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(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.

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