Model-agnostic super-resolution in high dimensions

TL;DR

This work develops a theory of super-resolution that dispenses with sparsity and separation assumptions in high dimensions. It introduces two reconstruction notions—Wasserstein-distance recovery and heavy-hitter recovery—for signals on the $d$-dimensional torus and proves near-optimal bounds on the required bandwidth and noise: Wasserstein recovery demands about $\big(\sqrt{d}/\varepsilon\big)^d$ Fourier coefficients with exponentially small noise in $d$, revealing a curse of dimensionality, while heavy-hitter recovery achieves subexponential dependence with roughly $2^{\tilde{O}(\sqrt{d})}$ coefficients. The methods combine mollification via Jackson kernels, convex/linear programming, and extremal polynomials to construct low-degree bump functions; they also connect to population recovery via extremal-polynomial techniques. The results provide both algorithmic procedures for practical recovery and strong information-theoretic lower bounds, clarifying what is possible in high-dimensional, model-agnostic super-resolution and highlighting the feasibility of subexponential strategies for density-aware recovery. Together, the findings illuminate fundamental limits and design principles for high-dimensional signal reconstruction from bandlimited Fourier data.

Abstract

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.

Figures (1)

• Figure 1: An illustration of the modeling assumptions and recovery objectives in this work compared to prior formulations.

Theorems & Definitions (47)

• Definition 1: Heavy hitter distance
• Remark 2: Heavy hitter distance versus Wasserstein distance
• Remark 3: Heavy hitter distance versus Lévy-Prokhorov
• Definition 5: Jackson's kernel
• Proposition 5
• Lemma 5
• Lemma 5
• Lemma 5
• Theorem 6
• proof