Model-agnostic super-resolution in high dimensions
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arXiv
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| Main Authors: | , , , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866911699119898624 |
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| author | Chen, Xi De, Anindya Huang, Yizhi Nadimpalli, Shivam Servedio, Rocco A. Yang, Tianqi |
| author_facet | Chen, Xi De, Anindya Huang, Yizhi Nadimpalli, Shivam Servedio, Rocco A. Yang, Tianqi |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_07846 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Model-agnostic super-resolution in high dimensions Chen, Xi De, Anindya Huang, Yizhi Nadimpalli, Shivam Servedio, Rocco A. Yang, Tianqi Data Structures and Algorithms Computational Complexity Statistics Theory 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. |
| title | Model-agnostic super-resolution in high dimensions |
| topic | Data Structures and Algorithms Computational Complexity Statistics Theory |
| url | https://arxiv.org/abs/2511.07846 |