Reconstruction under outliers for Fourier-sparse functions

We consider the problem of learning an unknown $f$ with a sparse Fourier spectrum in the presence of outlier noise. In particular, the algorithm has access to a noisy oracle for (an unknown) $f$ such that (i) the Fourier spectrum of $f$ is $k$-sparse; (ii) at any query point $x$, the oracle returns $y$ such that with probability $1-\rho$, $|y-f(x)| \le \epsilon$. However, with probability $\rho$, the error $y-f(x)$ can be arbitrarily large. We study Fourier sparse functions over both the discrete cube $\{0,1\}^n$ and the torus $[0,1)$ and for both these domains, we design efficient algorithms which can tolerate any $\rho<1/2$ fraction of outliers. We note that the analogous problem for low-degree polynomials has recently been studied in several works~[AK03, GZ16, KKP17] and similar algorithmic guarantees are known in that setting. While our main results pertain to the case where the location of the outliers, i.e., $x$ such that $|y-f(x)|>\epsilon$ is randomly distributed, we also study the case where the outliers are adversarially located. In particular, we show that over the torus, assuming that the Fourier transform satisfies a certain \emph{granularity} condition, there is a sample efficient algorithm to tolerate $\rho =\Omega(1)$ fraction of outliers and further, that this is not possible without such a granularity condition. Finally, while not the principal thrust, our techniques also allow us non-trivially improve on learning low-degree functions $f$ on the hypercube in the presence of adversarial outlier noise. Our techniques combine a diverse array of tools from compressive sensing, sparse Fourier transform, chaining arguments and complex analysis.