The problem of \emph{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 signals over the $d$-dimensional torus $[0,1)^d$; we do not assume any spatial separation between point sources, or even that the signal is a finite linear combination of point sources. We obtain two sets of results, corresponding to two natural notions of reconstruction. - {\bf Reconstruction in Wasserstein distance:} We give essentially matching upper and lower bounds on the cutoff frequency $T$ and the magnitude $κ$ of the noise for which accurate reconstruction in Wasserstein distance is possible. Roughly speaking, our results here show that for $d$-dimensional signals, estimates of $\approx \exp(d)$ many Fourier coefficients are necessary and sufficient for accurate reconstruction under the Wasserstein distance. - {\bf "Heavy hitter" reconstruction:} For nonnegative signals (equivalently, probability distributions), we introduce a new notion of "heavy hitter" reconstruction that essentially amounts to achieving high-accuracy reconstruction of all "sufficiently dense" regions of the distribution. Here too 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. 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.

翻译：超分辨率问题，粗略而言，是在给定（可能含噪声的）低阶傅里叶系数信息的情况下，以高精度重构未知信号。先前关于超分辨率的研究对信号施加了强烈的建模假设，通常要求信号是空间分离点源的线性组合。在这项工作中，我们分析了一个非常广义的超分辨率问题版本，考虑定义在 $d$ 维环面 $[0,1)^d$ 上的完全一般信号；我们既不假设点源之间存在空间分离，甚至不要求信号是点源的有限线性组合。我们得到了两组结果，分别对应两种自然的重构概念。\n- **Wasserstein 距离下的重构：** 我们给出了关于截止频率 $T$ 和噪声幅度 $κ$ 的本质上匹配的上界和下界，以确定在 Wasserstein 距离下实现精确重构的条件。粗略地说，我们的结果表明，对于 $d$ 维信号，需要约 $\exp(d)$ 个傅里叶系数的估计是实现 Wasserstein 距离下精确重构的必要且充分条件。\n- **“重击者”重构：** 对于非负信号（等价于概率分布），我们引入了一种新的“重击者”重构概念，其本质在于实现对分布所有“足够稠密”区域的高精度重构。在此，我们也给出了关于截止频率 $T$ 和噪声幅度 $κ$ 的本质上匹配的上界和下界，以确定实现精确重构的条件。我们的结果表明——与 Wasserstein 重构形成鲜明对比——仅需约 $\exp(\sqrt{d})$ 个傅里叶系数的精确估计，即是实现重击者重构的必要且充分条件。