We study the problem of reconstructing an unknown probability distribution to high accuracy, given noisy information about its low-degree Fourier coefficients. Prior results for this basic problem have typically imposed strong modeling assumptions on the distribution, often requiring that it is a linear combination of spatially separated point sources. In this work, we analyze a very general version of the high-dimensional recovery problem, by considering completely general distributions over the $d$-dimensional torus $[0,1)^d$; we do not assume any spatial separation between point sources, or even that the distribution is a finite linear combination of point sources. The question naturally arises: what can be said about distribution recovery in such a general setting? - As a warm-up, we first give results for reconstructing distributions with respect to the Wasserstein distance. We establish essentially matching upper and lower bounds on the cutoff frequency $T$ and the magnitude $κ$ of the noise for which accurate reconstruction is possible. Roughly speaking, our results here show that for $d$-dimensional distributions, estimates of $\approx \exp(d)$ many Fourier coefficients are necessary and sufficient for accurate reconstruction under the Wasserstein distance. - As our main result, we define a new notion of "heavy hitter" reconstruction for distributions, which essentially amounts to achieving high-accuracy reconstruction of all "sufficiently dense" regions of the distribution. We give essentially matching upper and lower bounds on the cutoff frequency $T$ and the magnitude $κ$ of the noise for which accurate reconstruction is possible under this notion. Our results show that (in sharp contrast with Wasserstein reconstruction) accurate estimates of only $\approx \exp(\sqrt{d})$ many Fourier coefficients are necessary and sufficient for heavy hitter reconstruction.

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