On denoising modulo 1 samples of a function

Consider an unknown smooth function $f: [0,1] \rightarrow \mathbb{R}$, and say we are given $n$ noisy mod 1 samples of $f$, i.e., $y_i = (f(x_i) + \eta_i)\mod 1$ for $x_i \in [0,1]$, where $\eta_i$ denotes 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 complex circle, inspired by the angular synchronization framework. Our approach amounts to solving a quadratically constrained quadratic program (QCQP) which is NP-hard in its basic form, and therefore we consider its relaxation which is a trust region sub-problem and hence solvable efficiently. We demonstrate its robustness to noise via extensive numerical simulations on several synthetic examples, along with a detailed theoretical analysis. To the best of our knowledge, we provide the first algorithm for denoising mod 1 samples of a smooth function, which comes with robustness guarantees.