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Existence of solutions for a class of quasilinear Schrödinger equation with a Kirchhoff-type
Uniform Approximation Property of Frames with Applications to Erasure Recovery
School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, China |
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.
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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