Model-agnostic super-resolution in high dimensions

2025-11-11

AI Summary

This paper studies model-agnostic super-resolution on the high-dimensional torus $[0,1)^d$, abandoning conventional assumptions on point-source structure or spatial separation, and instead addressing arbitrary unknown signals—including nonnegative probability distributions. It formulates two reconstruction objectives: (1) global approximation in Wasserstein distance; and (2) “heavy hitter”-style local high-accuracy recovery for nonnegative signals—i.e., precise identification of regions where signal mass concentrates. Theoretically, it establishes tight sample complexity bounds: $exp(Theta(d))$ Fourier coefficients suffice for global approximation, while only $exp(Theta(sqrt{d}))$ are required for heavy hitter recovery—substantially alleviating the curse of dimensionality. The method integrates Fourier analysis, probabilistic measure approximation, and noise-robust estimation theory, with matching upper and lower bounds rigorously proved. This work provides the first characterization of the fundamental difficulty of high-dimensional super-resolution without any modeling assumptions.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Non-convex Optimization
• Computer Vision: Other Foundations of Computer Vision

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding

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. - {f Reconstruction in Wasserstein distance:} We give essentially matching upper and lower bounds on the cutoff frequency $T$ and the magnitude $kappa$ 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. - {f"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 $kappa$ 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.

Problem

• Generalizing super-resolution for arbitrary d-dimensional signals
• Establishing cutoff frequency bounds for Wasserstein distance reconstruction
• Developing heavy hitter reconstruction for nonnegative probability distributions

Innovation

• Model-agnostic super-resolution without spatial separation assumptions
• Wasserstein distance reconstruction requiring exp(d) Fourier coefficients
• Heavy hitter reconstruction using exp(sqrt(d)) Fourier coefficients