American Institute of Mathematical Sciences

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March  2022, 21(3): 1093-1107. doi: 10.3934/cpaa.2022011

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 Published  March 2022 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 and Applied Analysis, 2022, 21 (3) : 1093-1107. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[9]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[26]

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

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[9]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[26]

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

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