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Quantization for Spectral Super-Resolution

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Abstract

We show that the method of distributed noise-shaping beta-quantization offers superior performance for the problem of spectral super-resolution with quantization whenever there is redundancy in the number of measurements. More precisely, we define the over-sampling ratio \(\lambda \) as the largest integer such that \(\lfloor M/\lambda \rfloor - 1\ge 4/\Delta \), where M denotes the number of Fourier measurements and \(\Delta \) is the minimum separation distance associated with the atomic measure to be resolved. We prove that for any number \(K\ge 2\) of quantization levels available for the real and imaginary parts of the measurements, our quantization method combined with either TV-min/BLASSO or ESPRIT guarantees reconstruction accuracy of order \(O(M^{1/4}\lambda ^{5/4} K^{- \lambda /2})\) and \(O(M^{3/2} \lambda ^{1/2} K^{- \lambda })\), respectively, where the implicit constants are independent of M, K and \(\lambda \). In contrast, naive rounding or memoryless scalar quantization for the same alphabet offers a guarantee of order \(O(M^{-1}K^{-1})\) only, regardless of the reconstruction algorithm.

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Notes

  1. Some super-resolution papers assume that the Fourier samples are indexed by \(\{-N,\dots ,N\}\) for some integer N, whereas we assume the frequencies lie in \(\{0,\dots ,M-1\}\). Both settings are equivalent, but we need to be careful with the transcription process: Our number of measurements is M, which corresponds to \(2N+1\) in some other papers.

  2. The numerical constant 0.3298 that appears in (3.1) is double the one found in [19], again due to our convention that the Fourier samples are from \(\{0,1,\dots ,M-1\}\).

  3. When the noise energy is sufficiently small, there is a unique optimal permutation. In the subsequent results, our assumptions guarantee that this is the case.

References

  1. Aubel, C., Bölcskei, H.: Deterministic performance analysis of subspace methods for cisoid parameter estimation. In: 2016 IEEE International Symposium on Information Theory (ISIT), pp. 1551–1555. IEEE (2016)

  2. Azaïs, J.-M., De Castro, Y., Gamboa, F.: Spike detection from inaccurate samplings. Appl. Comput. Harmon. Anal. 38(2), 177–195 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  3. Benedetto, J.J., Li, W.: Super-resolution by means of beurling minimal extrapolation. Appl. Comput. Harmon. Anal. 48(1), 218–241 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  4. Benedetto, J.J., Powell, A.M., Yılmaz, Ö.: Second-order sigma-delta (\(\Sigma \Delta \)) quantization of finite frame expansions. Appl. Comput. Harmon. Anal. 20(1), 126–148 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  5. Benedetto, J.J., Powell, A.M., Yilmaz, O.: Sigma-delta (\(\Sigma \Delta \)) quantization and finite frames. IEEE Trans. Inf. Theory 52(5), 1990–2005 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  6. Blum, J., Lammers, M., Powell, A.M., Yılmaz, Ö.: Sobolev duals in frame theory and sigma-delta quantization. J. Fourier Anal. Appl. 16(3), 365–381 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  7. Candès, E.J., Fernandez-Granda, C.: Super-resolution from noisy data. J. Fourier Anal. Appl. 19(6), 1229–1254 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  8. Candès, E.J., Fernandez-Granda, C.: Towards a mathematical theory of super-resolution. Commun. Pure Appl. Math. 67(6), 906–956 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  9. Chou, E.: Beta-duals of frames and applications to problems in quantization. PhD thesis, New York University (2013)

  10. Chou, E., Güntürk, C.S.: Distributed noise-shaping quantization: I. Beta duals of finite frames and near-optimal quantization of random measurements. Constr. Approx. 44(1), 1–22 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  11. Chou, E., Güntürk, C.S.: Distributed noise-shaping quantization: II. Classical frames. In: Excursions in Harmonic Analysis, vol. 5, pp. 179–198. Springer (2017)

  12. Chou, E., Güntürk, C.S., Krahmer, F., Saab, R., Yılmaz, Ö.: Noise-shaping quantization methods for frame-based and compressive sampling systems. In: Sampling Theory, a Renaissance, pp. 157–184. Springer (2015)

  13. Daubechies, I., DeVore, R.: Approximating a bandlimited function using very coarsely quantized data: a family of stable sigma-delta modulators of arbitrary order. Ann. Math. 158, 679–710 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  14. Deift, P., Krahmer, F., Güntürk, C.S.: An optimal family of exponentially accurate one-bit sigma-delta quantization schemes. Commun. Pure Appl. Math. 64(7), 883–919 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  15. Demanet, L., Nguyen, N.: The recoverability limit for superresolution via sparsity. arXiv preprint arXiv:1502.01385 (2015)

  16. Donoho, D.L.: Superresolution via sparsity constraints. SIAM J. Math. Anal. 23(5), 1309–1331 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  17. Duval, V., Peyré, G.: Exact support recovery for sparse spikes deconvolution. Found. Comput. Math. 15(5), 1315–1355 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  18. Fannjiang, A.C.: Compressive spectral estimation with single-snapshot ESPRIT: Stability and resolution. arXiv preprint arXiv:1607.01827 (2016)

  19. Fernandez-Granda, C.: Support detection in super-resolution. In: Proceedings of the 10th International Conference on Sampling Theory and Applications, pp. 145–148 (2013)

  20. Fernandez-Granda, C.: Super-resolution of point sources via convex programming. In: Information and Inference, pp. 251–303 (2016)

  21. Güntürk, C.S.: One-bit sigma-delta quantization with exponential accuracy. Commun. Pure Appl. Math. 56(11), 1608–1630 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  22. Güntürk, C.S.: Approximating a bandlimited function using very coarsely quantized data: improved error estimates in sigma-delta modulation. J. Am. Math. Soc. 17(1), 229–242 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  23. Güntürk, C.S., Lammers, M., Powell, A.M., Saab, R., Yılmaz, Ö.: Sobolev duals for random frames and \(\Sigma \Delta \) quantization of compressed sensing measurements. Found. Comput. Math. 13(1), 1–36 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  24. Güntürk, C.S., Li, W.: High-performance quantization for spectral super-resolution. In: Proceedings to Sampling Theory and Applications (2019)

  25. Huynh, T., Saab, R.: Fast binary embeddings and quantized compressed sensing with structured matrices. Commun. Pure Appl. Math. 73(1), 110–149 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  26. Li, W.: Elementary \({L}^\infty \) error estimates for super-resolution de-noising. arXiv preprint arXiv:1702.03021 (2017)

  27. Li, W., Liao, W.: Conditioning of restricted Fourier matrices and super-resolution of MUSIC. In: Proceedings of Sampling Theory and Applications (2019)

  28. Li, W., Liao, W.: Stable super-resolution limit and smallest singular value of restricted Fourier matrices. Appl. Comput. Harmon. Anal. 51, 118–156 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  29. Li, W., Liao, W., Fannjiang, A.: Super-resolution limit of the esprit algorithm. IEEE Trans. Inf. Theory 66(7), 4593–4608 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  30. Liao, W., Fannjiang, A.: MUSIC for single-snapshot spectral estimation: stability and super-resolution. Appl. Comput. Harmon. Anal. 40(1), 33–67 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  31. Moitra, A.: Super-resolution, extremal functions and the condition number of Vandermonde matrices. In: Proceedings of the Forty-Seventh Annual ACM Symposium on Theory of Computing (2015)

  32. Saab, R., Wang, R., Yılmaz, Ö.: Quantization of compressive samples with stable and robust recovery. Appl. Comput. Harmon. Anal. 44(1), 123–143 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  33. Wang, R.: Sigma delta quantization with harmonic frames and partial Fourier ensembles. J. Fourier Anal. Appl. 24(6), 1460–1490 (2018)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

Weilin Li gratefully acknowledges support from the AMS-Simons Travel Grant.

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Correspondence to Weilin Li.

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Communicated by Ronald DeVore.

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Güntürk, C.S., Li, W. Quantization for Spectral Super-Resolution. Constr Approx 56, 619–648 (2022). https://doi.org/10.1007/s00365-022-09574-5

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