  Consider an unknown smooth function $f: [0,1]^d \rightarrow \mathbb{R}$, and
say 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. Given
the samples $(x_i,y_i)_{i=1}^{n}$, our goal is to recover smooth, robust
estimates of the clean samples $f(x_i) \bmod 1$. We formulate a natural
approach for solving this problem, which works with angular embeddings of the
noisy mod 1 samples over the unit circle, inspired by the angular
synchronization framework. This amounts to solving a smoothness regularized
least-squares problem -- a quadratically constrained quadratic program (QCQP)
-- where the variables are constrained to lie on the unit circle. Our approach
is based on solving its relaxation, which is a trust-region sub-problem and
hence solvable efficiently. We provide theoretical guarantees demonstrating its
robustness to noise for adversarial, and random Gaussian and Bernoulli noise
models. To the best of our knowledge, these are the first such theoretical
results for this problem. We demonstrate the robustness and efficiency of our
approach via extensive numerical simulations on synthetic data, along with a
simple least-squares solution for the unwrapping stage, that recovers the
original samples of $f$ (up to a global shift). It is shown to perform well at
high levels of noise, when taking as input the denoised modulo $1$ samples.
  Finally, we also consider two other approaches for denoising the modulo 1
samples that leverage tools from Riemannian optimization on manifolds,
including a Burer-Monteiro approach for a semidefinite programming relaxation
of our formulation. For the two-dimensional version of the problem, which has
applications in radar interferometry, we are able to solve instances of
real-world data with a million sample points in under 10 seconds, on a personal
laptop.
