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A new initialization method based on normed statistical spaces in deep networks
Fast non-convex low-rank matrix decomposition for separation of potential field data using minimal memory
1. | Institute of Geophysics & Geomatics, China University of Geosciences (Wuhan), Wuhan, MO 430074, China |
2. | School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287-1804, USA |
3. | School of Mathematics & Physics, China University of Geosciences (Wuhan), Wuhan, MO 430074, China |
A fast non-convex low-rank matrix decomposition method for potential field data separation is presented. The singular value decomposition of the large size trajectory matrix, which is also a block Hankel matrix, is obtained using a fast randomized singular value decomposition algorithm in which fast block Hankel matrix-vector multiplications are implemented with minimal memory storage. This fast block Hankel matrix randomized singular value decomposition algorithm is integrated into the $\text{Altproj}$ algorithm, which is a standard non-convex method for solving the robust principal component analysis optimization problem. The integration of this improved estimation for the partial singular value decomposition avoids the construction of the trajectory matrix in the robust principal component analysis optimization problem. Hence, gravity and magnetic data matrices of large size can be computed and potential field data separation is achieved with better computational efficiency. The presented algorithm is also robust and, hence, algorithm-dependent parameters are easily determined. The performance of the algorithm, with and without the efficient estimation of the low rank matrix, is contrasted for the separation of synthetic gravity and magnetic data matrices of different sizes. These results demonstrate that the presented algorithm is not only computationally more efficient but it is also more accurate. Moreover, it is possible to solve far larger problems. As an example, for the adopted computational environment, matrices of sizes larger than $ 205 \times 205 $ generate "out of memory" exceptions without the improvement, whereas a matrix of size $ 2001\times 2001 $ can now be calculated in $ 1062.29 $s. Finally, the presented algorithm is applied to separate real gravity and magnetic data in the Tongling area, Anhui province, China. Areas which may exhibit mineralizations are inferred based on the separated anomalies.
References:
[1] |
W. B. Agocs,
Least squares residual anomaly determination, Geophysics, 16 (1951), 686-696.
doi: 10.1190/1.1437720. |
[2] |
A. Beck and M. Teboulle,
A fast iterative shrinkage-thresholding algorithm for linear inverse problems, SIAM Journal on Imaging Sciences, 2 (2009), 183-202.
doi: 10.1137/080716542. |
[3] |
D. S. Broomhead and G. P. King,
Extracting qualitative dynamics from experimental data, Physica D: Nonlinear Phenomena, 20 (1986), 217-236.
doi: 10.1016/0167-2789(86)90031-X. |
[4] |
H.Q. Cai, J.-F. Cai, T. Wang, and G. Yin, Accelerated Structured Alternating Projections for Robust Spectrally Sparse Signal Recovery, preprint, http://arXiv.org/abs/1910.05859, (2020). |
[5] |
E. J. Candès, X. D. Li, Y. Ma and J. Wright, Robust principal component analysis?, Journal of the ACM (JACM), 58 (2011), Art. 11, 37 pp.
doi: 10.1145/1970392.1970395. |
[6] |
K. C. Clarke,
Optimum second-derivative and downward-continuation filters, Geophysics, 34 (1969), 424-437.
|
[7] |
M. Fedi and T. Quarta,
Wavelet analysis for the regional-residual and local separation of potential field anomalies, Geophysical Prospecting, 46 (1998), 507-525.
doi: 10.1046/j.1365-2478.1998.00105.x. |
[8] |
G. H. Golub and C. F. Van Loan, Matrix Computations, Johns Hopkins Press, Baltimore, 1996.
![]() ![]() |
[9] |
N. Golyandina, I. Florinsky and K. Usevich,
Filtering of digital terrain models by 2D singular spectrum analysis, International Journal of Ecology & Development, 8 (2007), 81-94.
|
[10] |
Z. Z. Hou and W. C. Yang,
Wavelet transform and multi-scale analysis on gravity anomalies of China, Chinese Journal of Geophysics, 40 (1997), 85-95.
|
[11] |
N. Halko, P. Martinsson and J. A. Tropp,
Finding structure with randomness: Probabilistic algorithms for constructing approximate matrix decompositions, SIAM Review, 53 (2011), 217-288.
doi: 10.1137/090771806. |
[12] |
E. Liberty, F. Woolfe, P. Martinsson, V. Rokhlin and M. Tygert,
Randomized algorithms for the low-rank approximation of matrices, Proceedings of the National Academy of Sciences, 104 (2007), 20167-20172.
doi: 10.1073/pnas.0709640104. |
[13] |
Z. C. Lin and H. Y. Zhang, Low-rank Models in Visual Analysis, , Elsevier Science Publishing Co Inc, New York, 2017. |
[14] |
Z. C. Lin, M. M. Chen, and Y. Ma, The augmented Lagrange multiplier method for exact recovery of corrupted low-rank matrices, preprint, arXiv: 1009.5055, (2013). |
[15] |
L. Lu, W. Xu and S. Z. Qiao,
A fast SVD for multilevel block Hankel matrices with minimal memory storage, Numerical Algorithms, 69 (2015), 875-891.
doi: 10.1007/s11075-014-9930-0. |
[16] |
A. Mandal and and S. Niyogi,
Filter assisted bi-dimensional empirical mode decomposition: a hybrid approach for regional-residual separation of gravity anomaly, Journal of Applied Geophysics, 159 (2018), 218-227.
|
[17] |
K. L. Mickus, C. L. V. Aiken and W. D. Kennedy,
Regional-residual gravity anomaly separation using the minimum-curvature technique, Geophysics, 56 (1991), 279-283.
doi: 10.1190/1.1443041. |
[18] |
P. Netrapalli, U. N. Niranjan, S. Sanghavi, A. Anandkumar, and P. Jain, Non-convex robust PCA, Advances in Neural Information Processing Systems, (2014), 1107–1115. |
[19] |
R. S. Pawlowski,
Preferential continuation for potential-field anomaly enhancement, Geophysics, 60 (1995), 390-398.
doi: 10.1190/1.1443775. |
[20] |
R. S. Pawlowski and R. O. Hansen,
Gravity anomaly separation by Wiener filtering, Geophysics, 55 (1990), 539-548.
doi: 10.1190/1.1442865. |
[21] |
A. Spector and F. S. Grant,
Statistical models for interpreting aeromagnetic data, Geophysics, 35 (1970), 293-302.
doi: 10.1190/1.1440092. |
[22] |
F. Takens, Detecting strange attractors in turbulence, Dynamical Systems and Turbulence, Warwick 1980, (1981), 366–381. |
[23] |
W. M. Telford, L. P. Geldart and R. E. Sheriff, Applied Geophysics,, Cambridge University Press, Cambridge, 2003.
doi: 10.1017/CBO9781139167932.![]() ![]() |
[24] |
A. A. Tsonis and J. B. Elsner,
Mapping the channels of communication between the tropics and higher latitudes in the atmosphere, Physica D: Nonlinear Phenomena, 92 (1996), 237-244.
doi: 10.1016/0167-2789(95)00265-0. |
[25] |
S. Vatankhah, R. A. Renaut and V. E. Ardestani, A fast algorithm for regularized focused 3D inversion of gravity data using randomized singular-value decomposition, Geophysics, 83 (2018), G25–G34.
doi: 10.1190/geo2017-0386.1. |
[26] |
S. Vatankhah, S. Liu, R.A. Renaut, X. Hu and J. Baniamerian, Improving the use of the randomized singular value decomposition for the inversion of gravity and magnetic data, Geophysics, 85 (2020), G93–G107.
doi: 10.1190/geo2019-0603.1. |
[27] |
C. R. Vogel, Computational Methods for Inverse Problems, Society for Industrial and Applied Mathematics, Philadelphia, 2002.
doi: 10.1137/1.9780898717570. |
[28] |
J. Wright, Y. Ma, J. Mairal, G. Sapiro, T. Huang and S. C. Yan,
Sparse representation for computer vision and pattern recognition, Proceedings of the IEEE, 98 (2010), 1031-1044.
doi: 10.21236/ADA513248. |
[29] |
W. C. Yang, Z. Q. Shi, Z. Z. Hou and Z. Y. Cheng,
Discrete wavelet transform for multiple decomposition of gravity anomalies, Chinese Journal of Geophysics, 44 (2001), 529-537.
doi: 10.1002/cjg2.171. |
[30] |
L. L. Zhang, T. Y. Hao and W. W. Jiang,
Separation of potential field data using 3-D principal component analysis and textural analysis, Geophysical Journal International, 179 (2009), 1397-1413.
doi: 10.1111/j.1365-246X.2009.04357.x. |
[31] |
S. Zhang and M. Wang, Correction of simultaneous bad measurements by exploiting the low-rank Hankel structure, 2018 IEEE International Symposium on Information Theory (ISIT), (2018), 646–650.
doi: 10.1109/ISIT.2018.8437340. |
[32] |
D. Zhu, H. W. Li, T. Y. Liu, L. H. Fu and S. H. Zhang, Low-rank matrix decomposition method for potential field data separation, Geophysics, 85 (2020), G1–G16.
doi: 10.1190/geo2019-0016.1. |
show all references
References:
[1] |
W. B. Agocs,
Least squares residual anomaly determination, Geophysics, 16 (1951), 686-696.
doi: 10.1190/1.1437720. |
[2] |
A. Beck and M. Teboulle,
A fast iterative shrinkage-thresholding algorithm for linear inverse problems, SIAM Journal on Imaging Sciences, 2 (2009), 183-202.
doi: 10.1137/080716542. |
[3] |
D. S. Broomhead and G. P. King,
Extracting qualitative dynamics from experimental data, Physica D: Nonlinear Phenomena, 20 (1986), 217-236.
doi: 10.1016/0167-2789(86)90031-X. |
[4] |
H.Q. Cai, J.-F. Cai, T. Wang, and G. Yin, Accelerated Structured Alternating Projections for Robust Spectrally Sparse Signal Recovery, preprint, http://arXiv.org/abs/1910.05859, (2020). |
[5] |
E. J. Candès, X. D. Li, Y. Ma and J. Wright, Robust principal component analysis?, Journal of the ACM (JACM), 58 (2011), Art. 11, 37 pp.
doi: 10.1145/1970392.1970395. |
[6] |
K. C. Clarke,
Optimum second-derivative and downward-continuation filters, Geophysics, 34 (1969), 424-437.
|
[7] |
M. Fedi and T. Quarta,
Wavelet analysis for the regional-residual and local separation of potential field anomalies, Geophysical Prospecting, 46 (1998), 507-525.
doi: 10.1046/j.1365-2478.1998.00105.x. |
[8] |
G. H. Golub and C. F. Van Loan, Matrix Computations, Johns Hopkins Press, Baltimore, 1996.
![]() ![]() |
[9] |
N. Golyandina, I. Florinsky and K. Usevich,
Filtering of digital terrain models by 2D singular spectrum analysis, International Journal of Ecology & Development, 8 (2007), 81-94.
|
[10] |
Z. Z. Hou and W. C. Yang,
Wavelet transform and multi-scale analysis on gravity anomalies of China, Chinese Journal of Geophysics, 40 (1997), 85-95.
|
[11] |
N. Halko, P. Martinsson and J. A. Tropp,
Finding structure with randomness: Probabilistic algorithms for constructing approximate matrix decompositions, SIAM Review, 53 (2011), 217-288.
doi: 10.1137/090771806. |
[12] |
E. Liberty, F. Woolfe, P. Martinsson, V. Rokhlin and M. Tygert,
Randomized algorithms for the low-rank approximation of matrices, Proceedings of the National Academy of Sciences, 104 (2007), 20167-20172.
doi: 10.1073/pnas.0709640104. |
[13] |
Z. C. Lin and H. Y. Zhang, Low-rank Models in Visual Analysis, , Elsevier Science Publishing Co Inc, New York, 2017. |
[14] |
Z. C. Lin, M. M. Chen, and Y. Ma, The augmented Lagrange multiplier method for exact recovery of corrupted low-rank matrices, preprint, arXiv: 1009.5055, (2013). |
[15] |
L. Lu, W. Xu and S. Z. Qiao,
A fast SVD for multilevel block Hankel matrices with minimal memory storage, Numerical Algorithms, 69 (2015), 875-891.
doi: 10.1007/s11075-014-9930-0. |
[16] |
A. Mandal and and S. Niyogi,
Filter assisted bi-dimensional empirical mode decomposition: a hybrid approach for regional-residual separation of gravity anomaly, Journal of Applied Geophysics, 159 (2018), 218-227.
|
[17] |
K. L. Mickus, C. L. V. Aiken and W. D. Kennedy,
Regional-residual gravity anomaly separation using the minimum-curvature technique, Geophysics, 56 (1991), 279-283.
doi: 10.1190/1.1443041. |
[18] |
P. Netrapalli, U. N. Niranjan, S. Sanghavi, A. Anandkumar, and P. Jain, Non-convex robust PCA, Advances in Neural Information Processing Systems, (2014), 1107–1115. |
[19] |
R. S. Pawlowski,
Preferential continuation for potential-field anomaly enhancement, Geophysics, 60 (1995), 390-398.
doi: 10.1190/1.1443775. |
[20] |
R. S. Pawlowski and R. O. Hansen,
Gravity anomaly separation by Wiener filtering, Geophysics, 55 (1990), 539-548.
doi: 10.1190/1.1442865. |
[21] |
A. Spector and F. S. Grant,
Statistical models for interpreting aeromagnetic data, Geophysics, 35 (1970), 293-302.
doi: 10.1190/1.1440092. |
[22] |
F. Takens, Detecting strange attractors in turbulence, Dynamical Systems and Turbulence, Warwick 1980, (1981), 366–381. |
[23] |
W. M. Telford, L. P. Geldart and R. E. Sheriff, Applied Geophysics,, Cambridge University Press, Cambridge, 2003.
doi: 10.1017/CBO9781139167932.![]() ![]() |
[24] |
A. A. Tsonis and J. B. Elsner,
Mapping the channels of communication between the tropics and higher latitudes in the atmosphere, Physica D: Nonlinear Phenomena, 92 (1996), 237-244.
doi: 10.1016/0167-2789(95)00265-0. |
[25] |
S. Vatankhah, R. A. Renaut and V. E. Ardestani, A fast algorithm for regularized focused 3D inversion of gravity data using randomized singular-value decomposition, Geophysics, 83 (2018), G25–G34.
doi: 10.1190/geo2017-0386.1. |
[26] |
S. Vatankhah, S. Liu, R.A. Renaut, X. Hu and J. Baniamerian, Improving the use of the randomized singular value decomposition for the inversion of gravity and magnetic data, Geophysics, 85 (2020), G93–G107.
doi: 10.1190/geo2019-0603.1. |
[27] |
C. R. Vogel, Computational Methods for Inverse Problems, Society for Industrial and Applied Mathematics, Philadelphia, 2002.
doi: 10.1137/1.9780898717570. |
[28] |
J. Wright, Y. Ma, J. Mairal, G. Sapiro, T. Huang and S. C. Yan,
Sparse representation for computer vision and pattern recognition, Proceedings of the IEEE, 98 (2010), 1031-1044.
doi: 10.21236/ADA513248. |
[29] |
W. C. Yang, Z. Q. Shi, Z. Z. Hou and Z. Y. Cheng,
Discrete wavelet transform for multiple decomposition of gravity anomalies, Chinese Journal of Geophysics, 44 (2001), 529-537.
doi: 10.1002/cjg2.171. |
[30] |
L. L. Zhang, T. Y. Hao and W. W. Jiang,
Separation of potential field data using 3-D principal component analysis and textural analysis, Geophysical Journal International, 179 (2009), 1397-1413.
doi: 10.1111/j.1365-246X.2009.04357.x. |
[31] |
S. Zhang and M. Wang, Correction of simultaneous bad measurements by exploiting the low-rank Hankel structure, 2018 IEEE International Symposium on Information Theory (ISIT), (2018), 646–650.
doi: 10.1109/ISIT.2018.8437340. |
[32] |
D. Zhu, H. W. Li, T. Y. Liu, L. H. Fu and S. H. Zhang, Low-rank matrix decomposition method for potential field data separation, Geophysics, 85 (2020), G1–G16.
doi: 10.1190/geo2019-0016.1. |










Step | Cost in flops | Cost in storage | Cost in flops | Cost in storage |
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Matrix size | Rank |
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Matrix sizes | Time (seconds) | ||||
Matrix sizes | Time (seconds) | ||||
Geologic model | Shape | Central position | Model parameters | Density | Magnetization |
(length, width, depth extent)/radius | (g/cm |
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Geologic model | Shape | Central position | Model parameters | Density | Magnetization |
(length, width, depth extent)/radius | (g/cm |
(A/m) | |||
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model- |
sphere | ||||
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Time (s) | Time (s) | |||||||
Acronym | Description |
|
fast block Hankel matrix randomized SVD algorithm |
fast block Hankel matrix-matrix multiplication algorithm | |
fast block Hankel matrix-vector multiplication Algorithm | |
fast non-convex low-rank matrix decomposition algorithm for potential field separation | |
exact augmented Lagrange multiplier method | |
inexact augmented Lagrange multiplier method | |
low-rank matrix decomposition for potential field separation | |
robust principal component analysis | |
randomized singular value decomposition | |
singular value decomposition | |
root mean square error | |
reduce to the pole |
Acronym | Description |
|
fast block Hankel matrix randomized SVD algorithm |
fast block Hankel matrix-matrix multiplication algorithm | |
fast block Hankel matrix-vector multiplication Algorithm | |
fast non-convex low-rank matrix decomposition algorithm for potential field separation | |
exact augmented Lagrange multiplier method | |
inexact augmented Lagrange multiplier method | |
low-rank matrix decomposition for potential field separation | |
robust principal component analysis | |
randomized singular value decomposition | |
singular value decomposition | |
root mean square error | |
reduce to the pole |
Notation | Description |
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exchange matrix |
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Hankel matrix constructed from the |
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trajectory matrix of |
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first to |
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rank- |
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data matrices of regional and residual anomalies, respectively | |
trajectory matrices of |
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approximations of |
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element at |
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desired rank parameter in |
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oversampling parameter in |
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power iteration parameter in |
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desired rank parameter in |
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thresholding parameter in |
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weighting parameter in |
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Notation | Description |
|
exchange matrix |
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|
Hankel matrix constructed from the |
|
trajectory matrix of |
|
first to |
|
|
|
rank- |
|
data matrices of regional and residual anomalies, respectively | |
trajectory matrices of |
|
approximations of |
|
|
|
|
|
element at |
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|
|
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desired rank parameter in |
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oversampling parameter in |
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power iteration parameter in |
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desired rank parameter in |
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thresholding parameter in |
|
weighting parameter in |
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