The problem of super-resolution, roughly speaking, is to reconstruct an unknown signal to high accuracy, given (potentially noisy) information about its low-degree Fourier coefficients. Prior results on super-resolution have imposed strong modeling assumptions on the signal, typically requiring that it is a linear combination of spatially separated point sources. In this work we analyze a very general version of the super-resolution problem by considering completely general non-negative signals (equivalently, 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 super-resolution in such a general setting? - As a warm-up, we first give a set of results for reconstructing distributions under 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: we show that for $d$-dimensional distributions, estimates of $\approx \exp(d)$ many Fourier coefficients are both necessary and sufficient for accurate Wasserstein reconstruction. - 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 both necessary and sufficient for heavy hitter reconstruction.

翻译：超分辨率问题，粗略而言，是指根据信号的（可能含噪的）低阶傅里叶系数信息，高精度地重建未知信号。以往的超分辨率研究结果对信号施加了很强的建模假设，通常要求信号是空间分离的点源的线性组合。在本研究中，我们通过考虑 $d$ 维环面 $[0,1)^d$ 上完全一般的非负信号（等价于分布），来分析一个非常通用的超分辨率问题版本；我们既未假设点源之间存在任何空间分离，也未假设该分布是有限多个点源的线性组合。一个自然的问题随之而来：在这种一般性设定下，关于超分辨率我们能得出什么结论？——作为铺垫，我们首先给出了一组关于在Wasserstein距离下重建分布的结果。我们建立了关于截止频率 $T$ 和噪声幅度 $κ$ 的基本匹配的上界与下界，从而确定了精确重建的可能性条件：我们发现，对于 $d$ 维分布，约 $\exp(d)$ 个傅里叶系数的估计值对于精确的Wasserstein重建既是必要的也是充分的。——作为主要结果，我们定义了分布的一种新概念，即“重击者”重建，其本质是实现对分布中所有“足够稠密”区域的高精度重建。我们给出了关于截止频率 $T$ 和噪声幅度 $κ$ 的基本匹配的上界与下界，以确定在此概念下精确重建的可能性。我们的结果表明，（与Wasserstein重建形成鲜明对比）对于重击者重建，约 $\exp(\sqrt{d})$ 个傅里叶系数的精确估计值既是必要的也是充分的。