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Journal Articles Information and Inference Year : 2021

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

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Hemant Tyagi
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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) + η_i) mod 1; i = 1,. .. , n where η_i denotes noise. Assuming η_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((log n / n)^{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) mod 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 [7] 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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Dates and versions

hal-03101740 , version 1 (07-01-2021)
hal-03101740 , version 2 (10-01-2022)

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Michaël Fanuel, Hemant Tyagi. Denoising modulo samples: k-NN regression and tightness of SDP relaxation. Information and Inference, 2021, ⟨10.1093/imaiai/iaab022⟩. ⟨hal-03101740v2⟩
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