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Consider an unknown smooth function $f: [0,1]^d \rightarrow \mathbb{R}$, andsay we are given $n$ noisy mod 1 samples of $f$, i.e., $y_i = (f(x_i) +\eta_i)\mod 1$, for $x_i \in [0,1]^d$, where $\eta_i$ denotes the noise. Giventhe samples $(x_i,y_i)_{i=1}^{n}$, our goal is to recover smooth, robustestimates of the clean samples $f(x_i) \bmod 1$. We formulate a naturalapproach for solving this problem, which works with angular embeddings of thenoisy mod 1 samples over the unit circle, inspired by the angularsynchronization framework. This amounts to solving a smoothness regularizedleast-squares problem -- a quadratically constrained quadratic program (QCQP)-- where the variables are constrained to lie on the unit circle. Our approachis based on solving its relaxation, which is a trust-region sub-problem andhence solvable efficiently. We provide theoretical guarantees demonstrating itsrobustness to noise for adversarial, and random Gaussian and Bernoulli noisemodels. To the best of our knowledge, these are the first such theoreticalresults for this problem. We demonstrate the robustness and efficiency of ourapproach via extensive numerical simulations on synthetic data, along with asimple least-squares solution for the unwrapping stage, that recovers theoriginal samples of $f$ (up to a global shift). It is shown to perform well athigh levels of noise, when taking as input the denoised modulo $1$ samples.  Finally, we also consider two other approaches for denoising the modulo 1samples that leverage tools from Riemannian optimization on manifolds,including a Burer-Monteiro approach for a semidefinite programming relaxationof our formulation. For the two-dimensional version of the problem, which hasapplications in radar interferometry, we are able to solve instances ofreal-world data with a million sample points in under 10 seconds, on a personallaptop.

Provably robust estimation of modulo 1 samples of a smooth function with applications to phase unwrapping