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doi: 10.3934/cpaa.2022011
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Uniform approximation property of frames with applications to erasure recovery

Ting Chen , Fusheng Lv and  Wenchang Sun ,

School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, China

* Corresponding author

Received  April 2021 Revised  November 2021 Early access December 2021

Fund Project: This work was partially supported by the National Natural Science Foundation of China (11801282, 12101329 and 12171250) and the Fundamental Research Funds for the Central Universities

In this paper, we introduce frames with the uniform approximation property (UAP). We show that with a UAP frame, it is efficient to recover a signal from its frame coefficients with one erasure while the recovery error is smaller than that with the ordinary recovery algorithm. In fact, our approach gives a balance between the recovery accuracy and the computational complexity.

Keywords: Uniform approximation property, UAP frames, frames, erasures.
Mathematics Subject Classification: Primary: 42C15, 42C40.
Citation: Ting Chen, Fusheng Lv, Wenchang Sun. Uniform approximation property of frames with applications to erasure recovery. Communications on Pure & Applied Analysis, doi: 10.3934/cpaa.2022011
References:
[1]

A. S. Bandeira, M. Fickus, D. G. Mixon and P. Wong, The road to deterministic matrices with the restricted isometry property, J. Fourier Anal. Appl., 19 (2013), 1123-1149.  doi: 10.1007/s00041-013-9293-2.  Google Scholar

[2]

B. G. Bodmann, P. G. Casazza, D. Edidin and R. Balan, Frames for linear reconstruction without phase, in 2008 42nd Annual Conference on Information Sciences and Systems, 2008,721–726. Google Scholar

[3]

B. G. Bodmann, Optimal linear transmission by loss-insensitive packet encoding, Appl. Comput. Harmon. Anal., 22 (2007), 274-285.  doi: 10.1016/j.acha.2006.07.003.  Google Scholar

[4]

B. G. Bodmann and N. Hammen, Stable phase retrieval with low-redundancy frames, Adv. Comput. Math., 41 (2015), 317-331.  doi: 10.1007/s10444-014-9359-y.  Google Scholar

[5]

B. G. Bodmann and V. I. Paulsen, Frames, graphs and erasures, Linear Algebra Appl., 404 (2005), 118-146.  doi: 10.1016/j.laa.2005.02.016.  Google Scholar

[6]

B. G. Bodmann, V. I. Paulsen and M. Tomforde, Equiangular tight frames from complex Seidel matrices containing cube roots of unity, Linear Algebra Appl., 430 (2009), 396-417.  doi: 10.1016/j.laa.2008.08.002.  Google Scholar

[7]

P. G. Casazza and J. Kovačević, Equal-norm tight frames with erasures, Adv. Comput. Math., 18 (2003), 387-430.  doi: 10.1023/A:1021349819855.  Google Scholar

[8]

Q. Cheng, F. Lv and W. Sun, Frames of uniform subframe bounds with applications to erasures, Linear Algebra Appl., 555 (2018), 186-200.  doi: 10.1016/j.laa.2018.05.025.  Google Scholar

[9]

O. Christensen, An Introduction to Frames and Riesz Bases, Birkhäuser/Springer, Cham, 2016. doi: 10.1007/978-3-319-25613-9.  Google Scholar

[10]

I. Daubechies, Ten lectures on wavelets, vol. 61 of CBMS-NSF Regional Conference Series in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992. doi: 10.1137/1.9781611970104.  Google Scholar

[11]

M. Fickus, J. Jasper, E. J. King and D. G. Mixon, Equiangular tight frames that contain regular simplices, Linear Algebra Appl., 555 (2018), 98-138.  doi: 10.1016/j.laa.2018.06.004.  Google Scholar

[12]

M. Fickus, J. Jasper, D. G. Mixon, J. D. Peterson and C. E. Watson, Equiangular tight frames with centroidal symmetry, Appl. Comput. Harmon. Anal., 44 (2018), 476-496.  doi: 10.1016/j.acha.2016.06.004.  Google Scholar

[13]

B. Han and Z. Xu, Robustness properties of dimensionality reduction with Gaussian random matrices, Sci. China Math., 60 (2017), 1753-1778.  doi: 10.1007/s11425-016-9018-x.  Google Scholar

[14]

D. Han, D. Larson, S. Scholze and W. Sun, Erasure recovery matrices for encoder protection, Appl. Comput. Harmon. Anal., 48 (2020), 766-786.  doi: 10.1016/j.acha.2018.09.004.  Google Scholar

[15]

D. Han, F. Lv and W. Sun, Recovery of signals from unordered partial frame coefficients, Appl. Comput. Harmon. Anal., 44 (2018), 38-58.  doi: 10.1016/j.acha.2016.04.002.  Google Scholar

[16]

D. Han and W. Sun, Reconstruction of signals from frame coefficients with erasures at unknown locations, IEEE Trans. Inform. Theory, 60 (2014), 4013-4025.  doi: 10.1109/TIT.2014.2320937.  Google Scholar

[17]

T. Hoffman and J. P. Solazzo, Complex equiangular tight frames and erasures, Linear Algebra Appl., 437 (2012), 549-558.  doi: 10.1016/j.laa.2012.01.024.  Google Scholar

[18]

R. B. Holmes and V. I. Paulsen, Optimal frames for erasures, Linear Algebra Appl., 377 (2004), 31-51.  doi: 10.1016/j.laa.2003.07.012.  Google Scholar

[19]

D. Larson and S. Scholze, Signal reconstruction from frame and sampling erasures, J. Fourier Anal. Appl., 21 (2015), 1146-1167.  doi: 10.1007/s00041-015-9404-3.  Google Scholar

[20]

J. Leng and D. Han, Optimal dual frames for erasures II, Linear Algebra Appl., 435 (2011), 1464-1472.  doi: 10.1016/j.laa.2011.03.043.  Google Scholar

[21]

J. Leng, D. Han and T. Huang, Optimal dual frames for communication coding with probabilistic erasures, IEEE Trans. Signal Process., 59 (2011), 5380-5389.  doi: 10.1109/TSP.2011.2162955.  Google Scholar

[22]

J. Lopez and D. Han, Optimal dual frames for erasures, Linear Algebra Appl., 432 (2010), 471-482.  doi: 10.1016/j.laa.2009.08.031.  Google Scholar

[23]

P. G. Massey, M. A. Ruiz and D. Stojanoff, Optimal dual frames and frame completions for majorization, Appl. Comput. Harmon. Anal., 34 (2013), 201-223.  doi: 10.1016/j.acha.2012.03.011.  Google Scholar

[24]

P. M. Morillas, Optimal dual fusion frames for probabilistic erasures, Electron. J. Linear Algebra, 32 (2017), 191-203.  doi: 10.13001/1081-3810.3267.  Google Scholar

[25]

S. Pehlivan, D. Han and R. Mohapatra, Linearly connected sequences and spectrally optimal dual frames for erasures, J. Funct. Anal., 265 (2013), 2855-2876.  doi: 10.1016/j.jfa.2013.08.012.  Google Scholar

[26]

Y. Wang, Random matrices and erasure robust frames, J. Fourier Anal. Appl., 24 (2018), 1-16.  doi: 10.1007/s00041-016-9486-6.  Google Scholar

show all references

References:
[1]

A. S. Bandeira, M. Fickus, D. G. Mixon and P. Wong, The road to deterministic matrices with the restricted isometry property, J. Fourier Anal. Appl., 19 (2013), 1123-1149.  doi: 10.1007/s00041-013-9293-2.  Google Scholar

[2]

B. G. Bodmann, P. G. Casazza, D. Edidin and R. Balan, Frames for linear reconstruction without phase, in 2008 42nd Annual Conference on Information Sciences and Systems, 2008,721–726. Google Scholar

[3]

B. G. Bodmann, Optimal linear transmission by loss-insensitive packet encoding, Appl. Comput. Harmon. Anal., 22 (2007), 274-285.  doi: 10.1016/j.acha.2006.07.003.  Google Scholar

[4]

B. G. Bodmann and N. Hammen, Stable phase retrieval with low-redundancy frames, Adv. Comput. Math., 41 (2015), 317-331.  doi: 10.1007/s10444-014-9359-y.  Google Scholar

[5]

B. G. Bodmann and V. I. Paulsen, Frames, graphs and erasures, Linear Algebra Appl., 404 (2005), 118-146.  doi: 10.1016/j.laa.2005.02.016.  Google Scholar

[6]

B. G. Bodmann, V. I. Paulsen and M. Tomforde, Equiangular tight frames from complex Seidel matrices containing cube roots of unity, Linear Algebra Appl., 430 (2009), 396-417.  doi: 10.1016/j.laa.2008.08.002.  Google Scholar

[7]

P. G. Casazza and J. Kovačević, Equal-norm tight frames with erasures, Adv. Comput. Math., 18 (2003), 387-430.  doi: 10.1023/A:1021349819855.  Google Scholar

[8]

Q. Cheng, F. Lv and W. Sun, Frames of uniform subframe bounds with applications to erasures, Linear Algebra Appl., 555 (2018), 186-200.  doi: 10.1016/j.laa.2018.05.025.  Google Scholar

[9]

O. Christensen, An Introduction to Frames and Riesz Bases, Birkhäuser/Springer, Cham, 2016. doi: 10.1007/978-3-319-25613-9.  Google Scholar

[10]

I. Daubechies, Ten lectures on wavelets, vol. 61 of CBMS-NSF Regional Conference Series in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992. doi: 10.1137/1.9781611970104.  Google Scholar

[11]

M. Fickus, J. Jasper, E. J. King and D. G. Mixon, Equiangular tight frames that contain regular simplices, Linear Algebra Appl., 555 (2018), 98-138.  doi: 10.1016/j.laa.2018.06.004.  Google Scholar

[12]

M. Fickus, J. Jasper, D. G. Mixon, J. D. Peterson and C. E. Watson, Equiangular tight frames with centroidal symmetry, Appl. Comput. Harmon. Anal., 44 (2018), 476-496.  doi: 10.1016/j.acha.2016.06.004.  Google Scholar

[13]

B. Han and Z. Xu, Robustness properties of dimensionality reduction with Gaussian random matrices, Sci. China Math., 60 (2017), 1753-1778.  doi: 10.1007/s11425-016-9018-x.  Google Scholar

[14]

D. Han, D. Larson, S. Scholze and W. Sun, Erasure recovery matrices for encoder protection, Appl. Comput. Harmon. Anal., 48 (2020), 766-786.  doi: 10.1016/j.acha.2018.09.004.  Google Scholar

[15]

D. Han, F. Lv and W. Sun, Recovery of signals from unordered partial frame coefficients, Appl. Comput. Harmon. Anal., 44 (2018), 38-58.  doi: 10.1016/j.acha.2016.04.002.  Google Scholar

[16]

D. Han and W. Sun, Reconstruction of signals from frame coefficients with erasures at unknown locations, IEEE Trans. Inform. Theory, 60 (2014), 4013-4025.  doi: 10.1109/TIT.2014.2320937.  Google Scholar

[17]

T. Hoffman and J. P. Solazzo, Complex equiangular tight frames and erasures, Linear Algebra Appl., 437 (2012), 549-558.  doi: 10.1016/j.laa.2012.01.024.  Google Scholar

[18]

R. B. Holmes and V. I. Paulsen, Optimal frames for erasures, Linear Algebra Appl., 377 (2004), 31-51.  doi: 10.1016/j.laa.2003.07.012.  Google Scholar

[19]

D. Larson and S. Scholze, Signal reconstruction from frame and sampling erasures, J. Fourier Anal. Appl., 21 (2015), 1146-1167.  doi: 10.1007/s00041-015-9404-3.  Google Scholar

[20]

J. Leng and D. Han, Optimal dual frames for erasures II, Linear Algebra Appl., 435 (2011), 1464-1472.  doi: 10.1016/j.laa.2011.03.043.  Google Scholar

[21]

J. Leng, D. Han and T. Huang, Optimal dual frames for communication coding with probabilistic erasures, IEEE Trans. Signal Process., 59 (2011), 5380-5389.  doi: 10.1109/TSP.2011.2162955.  Google Scholar

[22]

J. Lopez and D. Han, Optimal dual frames for erasures, Linear Algebra Appl., 432 (2010), 471-482.  doi: 10.1016/j.laa.2009.08.031.  Google Scholar

[23]

P. G. Massey, M. A. Ruiz and D. Stojanoff, Optimal dual frames and frame completions for majorization, Appl. Comput. Harmon. Anal., 34 (2013), 201-223.  doi: 10.1016/j.acha.2012.03.011.  Google Scholar

[24]

P. M. Morillas, Optimal dual fusion frames for probabilistic erasures, Electron. J. Linear Algebra, 32 (2017), 191-203.  doi: 10.13001/1081-3810.3267.  Google Scholar

[25]

S. Pehlivan, D. Han and R. Mohapatra, Linearly connected sequences and spectrally optimal dual frames for erasures, J. Funct. Anal., 265 (2013), 2855-2876.  doi: 10.1016/j.jfa.2013.08.012.  Google Scholar

[26]

Y. Wang, Random matrices and erasure robust frames, J. Fourier Anal. Appl., 24 (2018), 1-16.  doi: 10.1007/s00041-016-9486-6.  Google Scholar

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