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November  2021, 17(6): 3525-3550. doi: 10.3934/jimo.2020131

Continuity, differentiability and semismoothness of generalized tensor functions

Xia Li , Yong Wang , and  Zheng-Hai Huang

School of Mathematics, Tianjin University, Tianjin 300350, China

* Corresponding author: Yong Wang

Received  February 2020 Revised  May 2020 Published  November 2021 Early access  August 2020

A large number of real-world problems can be transformed into mathematical problems by means of third-order real tensors. Recently, as an extension of the generalized matrix function, the generalized tensor function over the third-order real tensor space was introduced with the aid of a scalar function based on the T-product for third-order tensors and the tensor singular value decomposition; and some useful algebraic properties of the function were investigated. In this paper, we show that the generalized tensor function can inherit a lot of good properties from the associated scalar function, including continuity, directional differentiability, Fréchet differentiability, Lipschitz continuity and semismoothness. These properties provide an important theoretical basis for the studies of various mathematical problems with generalized tensor functions, and particularly, for the studies of tensor optimization problems with generalized tensor functions.

Keywords: T-product, tensor singular value decomposition, generalized tensor function, semismoothness, differentiability.
Mathematics Subject Classification: Primary: 15A69, 49J50, 49J52.
Citation: Xia Li, Yong Wang, Zheng-Hai Huang. Continuity, differentiability and semismoothness of generalized tensor functions. Journal of Industrial and Management Optimization, 2021, 17 (6) : 3525-3550. doi: 10.3934/jimo.2020131
References:
[1]

B. P. W. Ames and H. S. Sendov, Derivatives of compound matrix valued functions, Journal of Mathematical Analysis and Applications, 433 (2016), 1459-1485.  doi: 10.1016/j.jmaa.2015.08.029.

[2]

F. Andersson, M. Carlsson and K. M. Perfekt, Operator-Lipschitz estimates for the singular value functional calculus, Proceedings of the American Mathematical Society, 144 (2016), 1867-1875.  doi: 10.1090/proc/12843.

[3]

F. Arrigo, M. Benzi and C. Fenu, Computation of generalized matrix functions, SIAM Journal on Matrix Analysis and Applications, 37 (2016), 836-860.  doi: 10.1137/15M1049634.

[4]

J. L. Aurentz, A. P. Austin, M. Benzi and V. Kalantzis, Stable computation of generalized matrix functions via polynomial interpolation, SIAM Journal on Matrix Analysis and Applications, 40 (2019), 210-234.  doi: 10.1137/18M1191786.

[5]

M. Benzi and R. Huang, Some matrix properties preserved by generalized matrix functions, Special Matrices, 7 (2019), 27-37.  doi: 10.1515/spma-2019-0003.

[6]

R. Bhatia, Matrix Analysis, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0653-8.

[7]

K. Braman, Third-order tensors as linear operators on a space of matrices, Linear Algebra and its Applications, 433 (2010), 1241-1253.  doi: 10.1016/j.laa.2010.05.025.

[8]

R. H. F. Chan and X. Q. Jin, An Introduction to Iterative Toeplitz Solvers, SIAM, Philadelphia, PA, 2007. doi: 10.1137/1.9780898718850.

[9]

X. Chen, H. Qi and P. Tseng, Analysis of nonsmooth symmetric-matrix-valued functions with applications to semidefinite complementarity problems, SIAM Journal on Optimization, 13 (2003), 960-985.  doi: 10.1137/S1052623400380584.

[10]

X. Chen and P. Tseng, Non-interior continuation methods for solving semidefinite complementarity problems, Mathematical Programming, 95 (2003), 431-474.  doi: 10.1007/s10107-002-0306-1.

[11]

F. H. Clarke, Optimization and Nonsmooth Analysis, Second edition, SIAM, Philadelphia, PA, 1990. doi: 10.1137/1.9781611971309.

[12]

S. Gandy, B. Recht and I. Yamada, Tensor completion and low-n-rank tensor recovery via convex optimization, Inverse Problems, 27 (2011), 025010, 19 pp. doi: 10.1088/0266-5611/27/2/025010.

[13]

D. Goldfarb and Z. Qin, Robust low-rank tensor recovery: Models and algorithms, SIAM Journal on Matrix Analysis and Applications, 35 (2014), 225-253.  doi: 10.1137/130905010.

[14]

G. H. Golub and C. F. Van Loan, Matrix Computations, 4th edition, Johns Hopkins University Press, Baltimore, MD, 2013.

[15]

N. Hao, M. E. Kilmer, K. Braman and R. C. Hoover, Facial recognition using tensor-tensor decompositions, SIAM Journal on Imaging Sciences, 6 (2013), 437-463.  doi: 10.1137/110842570.

[16]

J. B. Hawkins and A. Ben-Israel, On generalized matrix functions, Linear and Multilinear Algebra, 1 (1973), 163-171.  doi: 10.1080/03081087308817015.

[17]

N. J. Higham, Functions of Matrices: Theory and Computation, SIAM, Philadelphia, PA, 2008. doi: 10.1137/1.9780898717778.

[18]

A. Hjorungnes and D. Gesbert, Complex-valued matrix differentiation: Techniques and key results, IEEE Transactions on Signal Processing, 55 (2007), 2740-2746.  doi: 10.1109/TSP.2007.893762.

[19]

Z. H. Huang and L. Qi, Formulating an n-person noncooperative game as a tensor complementarity problem, Computational Optimization and Applications, 66 (2017), 557-576.  doi: 10.1007/s10589-016-9872-7.

[20]

M. E. Kilmer, K. Braman, N. Hao and R. C. Hoover, Third-order tensors as operators on matrices: A theoretical and computational framework with applications in imaging, SIAM Journal on Matrix Analysis and Applications, 34 (2013), 148-172.  doi: 10.1137/110837711.

[21]

M. E. Kilmer and C. D. Martin, Factorization strategies for third-order tensors, Linear Algebra and its Applications, 435 (2011), 641-658.  doi: 10.1016/j.laa.2010.09.020.

[22]

J. Liu, P. Musialski, P. Wonka and J. Ye, Tensor completion for estimating missing values in visual data, IEEE Transactions on Pattern Analysis and Machine Intelligence, 35 (2013), 208-220.  doi: 10.1109/TPAMI.2012.39.

[23]

C. Lu, J. Feng, Y. Chen, W. Liu, Z. Lin and S. Yan, Tensor robust principal component analysis: Exact recovery of corrupted low-rank tensors via convex optimization, Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, (2016), 5249–5257. doi: 10.1109/CVPR.2016.567.

[24]

C. Lu, J. Feng, Y. Chen, W. Liu, Z. Lin and S. Yan, Tensor robust principal component analysis with a new tensor nuclear norm, IEEE Transactions on Pattern Analysis and Machine Intelligence, 42 (2020), 925-938. 

[25]

K. Lund, The tensor t-function: a definition for functions of third-order tensors, Numerical Linear Algebra with Applications, 27 (2020), e2288. doi: 10.1002/nla.2288.

[26]

C. D. Martin, R. Shafer and B. LaRue, An order-p tensor factorization with applications in imaging, SIAM Journal on Scientific Computing, 35 (2013), A474–A490. doi: 10.1137/110841229.

[27]

Y. Miao, L. Qi and Y. Wei, Generalized tensor function via the tensor singular value decomposition based on the T-product, Linear Algebra and its Applications, 590 (2020), 258-303.  doi: 10.1016/j.laa.2019.12.035.

[28]

Y. Miao, L. Qi and Y. Wei, T-Jordan canonical form and T-Drazin inverse based on the T-product, Communications on Applied Mathematics and Computation, (2020). doi: 10.1007/s42967-019-00055-4.

[29]

E. Newman, L. Horesh, H. Avron and M. E. Kilmer, Stable tensor neural networks for rapid deep learning, preprint, arXiv: 1811.06569

[30]

V. Noferini, A formula for the Fréchet derivative of a generalized matrix function, SIAM Journal on Matrix Analysis and Applications, 38 (2017), 434-457.  doi: 10.1137/16M1072851.

[31]

R. F. Rinehart, The equivalence of definitions of a matric function, American Mathematical Monthly, 62 (1955), 395-414.  doi: 10.1080/00029890.1955.11988651.

[32]

R. T. Rockafellar and R. J. B. Wets, Variational Analysis, Springer, Heidelberg, 2009. doi: 10.1007/978-3-642-02431-3.

[33]

N. D. Sidiropoulos, L. De Lathauwer, X. Fu, K. Huang, E. E. Papalexakis and C. Faloutsos, Tensor decomposition for signal processing and machine learning, IEEE Transactions on Signal Processing, 65 (2017), 3551-3582.  doi: 10.1109/TSP.2017.2690524.

[34] G. W. Stewart and J. Sun, Matrix Perturbation Theory, Academic Press, Boston, MA, 1990. 
[35]

D. Sun and J. Sun, Semismooth matrix-valued functions, Mathematics of Operations Research, 27 (2002), 150-169.  doi: 10.1287/moor.27.1.150.342.

[36]

Y. Xie, D. Tao, W. Zhang, Y. Liu, L. Zhang and Y. Qu, On unifying multi-view self-representations for clustering by tensor multi-rank minimization, International Journal of Computer Vision, 126 (2018), 1157-1179.  doi: 10.1007/s11263-018-1086-2.

[37]

Y. Xu, Z. Wu, J. Chanussot and Z. Wei, Joint reconstruction and anomaly detection from compressive hyperspectral images using Mahalanobis distance-regularized tensor RPCA, IEEE Transactions on Geoscience and Remote Sensing, 56 (2018), 2919-2930.  doi: 10.1109/TGRS.2017.2786718.

[38]

Y. Xu, L. Yu, H. Xu, H. Zhang and T. Nguyen, Vector sparse representation of color image using quaternion matrix analysis, IEEE Transactions on Image Processing, 24 (2015), 1315-1329.  doi: 10.1109/TIP.2015.2397314.

[39]

L. Yang, Z. H. Huang, S. Hu and J. Han, An iterative algorithm for third-order tensor multi-rank minimization, Computational Optimization and Applications, 63 (2016), 169-202.  doi: 10.1007/s10589-015-9769-x.

[40]

L. Yang, Z. H. Huang and Y. F. Li, A splitting augmented Lagrangian method for low multilinear-rank tensor recovery, Asia-Pacific Journal of Operational Research, 32 (2015), 1540008. doi: 10.1142/S0217595915400084.

[41]

L. Yang, Z. H. Huang and X. Shi, A fixed point iterative method for low n-rank tensor pursuit, IEEE Transactions on Signal Processing, 61 (2013), 2952-2962.  doi: 10.1109/TSP.2013.2254477.

[42]

Z. Yang, A Study on Nonsymmetric Matrix-valued Functions, Master's thesis, Department of Mathematics, National University of Singapore, 2009.

[43]

M. Yin, J. Gao, S. Xie and Y. Guo, Multiview subspace clustering via tensorial t-product representation, IEEE Transactions on Neural Networks and Learning Systems, 30 (2019), 851-864.  doi: 10.1109/TNNLS.2018.2851444.

[44]

M. Yuan and C. H. Zhang, On tensor completion via nuclear norm minimization, Foundations of Computational Mathematics, 16 (2016), 1031-1068.  doi: 10.1007/s10208-015-9269-5.

[45]

M. Zhang, L. Yang and Z. H. Huang, Minimum n-rank approximation via iterative hard thresholding, Applied Mathematics and Computation, 256 (2015), 860-875.  doi: 10.1016/j.amc.2015.01.099.

[46]

Z. Zhang and S. Aeron, Exact tensor completion using t-SVD, IEEE Transactions on Signal Processing, 65 (2017), 1511-1526.  doi: 10.1109/TSP.2016.2639466.

[47]

Z. Zhang, G. Ely, S. Aeron, N. Hao and M. E. Kilmer, Novel methods for multilinear data completion and de-noising based on tensor-SVD, Preceedings of the IEEE Conference on Computer Vision and Pattern Recognition, (2014), 3842–3849. doi: 10.1109/CVPR.2014.485.

[48]

P. Zhou, C. Lu, Z. Lin and C. Zhang, Tensor factorization for low-rank tensor completion, IEEE Transactions on Image Processing, 27 (2018), 1152-1163.  doi: 10.1109/TIP.2017.2762595.

show all references

References:
[1]

B. P. W. Ames and H. S. Sendov, Derivatives of compound matrix valued functions, Journal of Mathematical Analysis and Applications, 433 (2016), 1459-1485.  doi: 10.1016/j.jmaa.2015.08.029.

[2]

F. Andersson, M. Carlsson and K. M. Perfekt, Operator-Lipschitz estimates for the singular value functional calculus, Proceedings of the American Mathematical Society, 144 (2016), 1867-1875.  doi: 10.1090/proc/12843.

[3]

F. Arrigo, M. Benzi and C. Fenu, Computation of generalized matrix functions, SIAM Journal on Matrix Analysis and Applications, 37 (2016), 836-860.  doi: 10.1137/15M1049634.

[4]

J. L. Aurentz, A. P. Austin, M. Benzi and V. Kalantzis, Stable computation of generalized matrix functions via polynomial interpolation, SIAM Journal on Matrix Analysis and Applications, 40 (2019), 210-234.  doi: 10.1137/18M1191786.

[5]

M. Benzi and R. Huang, Some matrix properties preserved by generalized matrix functions, Special Matrices, 7 (2019), 27-37.  doi: 10.1515/spma-2019-0003.

[6]

R. Bhatia, Matrix Analysis, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0653-8.

[7]

K. Braman, Third-order tensors as linear operators on a space of matrices, Linear Algebra and its Applications, 433 (2010), 1241-1253.  doi: 10.1016/j.laa.2010.05.025.

[8]

R. H. F. Chan and X. Q. Jin, An Introduction to Iterative Toeplitz Solvers, SIAM, Philadelphia, PA, 2007. doi: 10.1137/1.9780898718850.

[9]

X. Chen, H. Qi and P. Tseng, Analysis of nonsmooth symmetric-matrix-valued functions with applications to semidefinite complementarity problems, SIAM Journal on Optimization, 13 (2003), 960-985.  doi: 10.1137/S1052623400380584.

[10]

X. Chen and P. Tseng, Non-interior continuation methods for solving semidefinite complementarity problems, Mathematical Programming, 95 (2003), 431-474.  doi: 10.1007/s10107-002-0306-1.

[11]

F. H. Clarke, Optimization and Nonsmooth Analysis, Second edition, SIAM, Philadelphia, PA, 1990. doi: 10.1137/1.9781611971309.

[12]

S. Gandy, B. Recht and I. Yamada, Tensor completion and low-n-rank tensor recovery via convex optimization, Inverse Problems, 27 (2011), 025010, 19 pp. doi: 10.1088/0266-5611/27/2/025010.

[13]

D. Goldfarb and Z. Qin, Robust low-rank tensor recovery: Models and algorithms, SIAM Journal on Matrix Analysis and Applications, 35 (2014), 225-253.  doi: 10.1137/130905010.

[14]

G. H. Golub and C. F. Van Loan, Matrix Computations, 4th edition, Johns Hopkins University Press, Baltimore, MD, 2013.

[15]

N. Hao, M. E. Kilmer, K. Braman and R. C. Hoover, Facial recognition using tensor-tensor decompositions, SIAM Journal on Imaging Sciences, 6 (2013), 437-463.  doi: 10.1137/110842570.

[16]

J. B. Hawkins and A. Ben-Israel, On generalized matrix functions, Linear and Multilinear Algebra, 1 (1973), 163-171.  doi: 10.1080/03081087308817015.

[17]

N. J. Higham, Functions of Matrices: Theory and Computation, SIAM, Philadelphia, PA, 2008. doi: 10.1137/1.9780898717778.

[18]

A. Hjorungnes and D. Gesbert, Complex-valued matrix differentiation: Techniques and key results, IEEE Transactions on Signal Processing, 55 (2007), 2740-2746.  doi: 10.1109/TSP.2007.893762.

[19]

Z. H. Huang and L. Qi, Formulating an n-person noncooperative game as a tensor complementarity problem, Computational Optimization and Applications, 66 (2017), 557-576.  doi: 10.1007/s10589-016-9872-7.

[20]

M. E. Kilmer, K. Braman, N. Hao and R. C. Hoover, Third-order tensors as operators on matrices: A theoretical and computational framework with applications in imaging, SIAM Journal on Matrix Analysis and Applications, 34 (2013), 148-172.  doi: 10.1137/110837711.

[21]

M. E. Kilmer and C. D. Martin, Factorization strategies for third-order tensors, Linear Algebra and its Applications, 435 (2011), 641-658.  doi: 10.1016/j.laa.2010.09.020.

[22]

J. Liu, P. Musialski, P. Wonka and J. Ye, Tensor completion for estimating missing values in visual data, IEEE Transactions on Pattern Analysis and Machine Intelligence, 35 (2013), 208-220.  doi: 10.1109/TPAMI.2012.39.

[23]

C. Lu, J. Feng, Y. Chen, W. Liu, Z. Lin and S. Yan, Tensor robust principal component analysis: Exact recovery of corrupted low-rank tensors via convex optimization, Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, (2016), 5249–5257. doi: 10.1109/CVPR.2016.567.

[24]

C. Lu, J. Feng, Y. Chen, W. Liu, Z. Lin and S. Yan, Tensor robust principal component analysis with a new tensor nuclear norm, IEEE Transactions on Pattern Analysis and Machine Intelligence, 42 (2020), 925-938. 

[25]

K. Lund, The tensor t-function: a definition for functions of third-order tensors, Numerical Linear Algebra with Applications, 27 (2020), e2288. doi: 10.1002/nla.2288.

[26]

C. D. Martin, R. Shafer and B. LaRue, An order-p tensor factorization with applications in imaging, SIAM Journal on Scientific Computing, 35 (2013), A474–A490. doi: 10.1137/110841229.

[27]

Y. Miao, L. Qi and Y. Wei, Generalized tensor function via the tensor singular value decomposition based on the T-product, Linear Algebra and its Applications, 590 (2020), 258-303.  doi: 10.1016/j.laa.2019.12.035.

[28]

Y. Miao, L. Qi and Y. Wei, T-Jordan canonical form and T-Drazin inverse based on the T-product, Communications on Applied Mathematics and Computation, (2020). doi: 10.1007/s42967-019-00055-4.

[29]

E. Newman, L. Horesh, H. Avron and M. E. Kilmer, Stable tensor neural networks for rapid deep learning, preprint, arXiv: 1811.06569

[30]

V. Noferini, A formula for the Fréchet derivative of a generalized matrix function, SIAM Journal on Matrix Analysis and Applications, 38 (2017), 434-457.  doi: 10.1137/16M1072851.

[31]

R. F. Rinehart, The equivalence of definitions of a matric function, American Mathematical Monthly, 62 (1955), 395-414.  doi: 10.1080/00029890.1955.11988651.

[32]

R. T. Rockafellar and R. J. B. Wets, Variational Analysis, Springer, Heidelberg, 2009. doi: 10.1007/978-3-642-02431-3.

[33]

N. D. Sidiropoulos, L. De Lathauwer, X. Fu, K. Huang, E. E. Papalexakis and C. Faloutsos, Tensor decomposition for signal processing and machine learning, IEEE Transactions on Signal Processing, 65 (2017), 3551-3582.  doi: 10.1109/TSP.2017.2690524.

[34] G. W. Stewart and J. Sun, Matrix Perturbation Theory, Academic Press, Boston, MA, 1990. 
[35]

D. Sun and J. Sun, Semismooth matrix-valued functions, Mathematics of Operations Research, 27 (2002), 150-169.  doi: 10.1287/moor.27.1.150.342.

[36]

Y. Xie, D. Tao, W. Zhang, Y. Liu, L. Zhang and Y. Qu, On unifying multi-view self-representations for clustering by tensor multi-rank minimization, International Journal of Computer Vision, 126 (2018), 1157-1179.  doi: 10.1007/s11263-018-1086-2.

[37]

Y. Xu, Z. Wu, J. Chanussot and Z. Wei, Joint reconstruction and anomaly detection from compressive hyperspectral images using Mahalanobis distance-regularized tensor RPCA, IEEE Transactions on Geoscience and Remote Sensing, 56 (2018), 2919-2930.  doi: 10.1109/TGRS.2017.2786718.

[38]

Y. Xu, L. Yu, H. Xu, H. Zhang and T. Nguyen, Vector sparse representation of color image using quaternion matrix analysis, IEEE Transactions on Image Processing, 24 (2015), 1315-1329.  doi: 10.1109/TIP.2015.2397314.

[39]

L. Yang, Z. H. Huang, S. Hu and J. Han, An iterative algorithm for third-order tensor multi-rank minimization, Computational Optimization and Applications, 63 (2016), 169-202.  doi: 10.1007/s10589-015-9769-x.

[40]

L. Yang, Z. H. Huang and Y. F. Li, A splitting augmented Lagrangian method for low multilinear-rank tensor recovery, Asia-Pacific Journal of Operational Research, 32 (2015), 1540008. doi: 10.1142/S0217595915400084.

[41]

L. Yang, Z. H. Huang and X. Shi, A fixed point iterative method for low n-rank tensor pursuit, IEEE Transactions on Signal Processing, 61 (2013), 2952-2962.  doi: 10.1109/TSP.2013.2254477.

[42]

Z. Yang, A Study on Nonsymmetric Matrix-valued Functions, Master's thesis, Department of Mathematics, National University of Singapore, 2009.

[43]

M. Yin, J. Gao, S. Xie and Y. Guo, Multiview subspace clustering via tensorial t-product representation, IEEE Transactions on Neural Networks and Learning Systems, 30 (2019), 851-864.  doi: 10.1109/TNNLS.2018.2851444.

[44]

M. Yuan and C. H. Zhang, On tensor completion via nuclear norm minimization, Foundations of Computational Mathematics, 16 (2016), 1031-1068.  doi: 10.1007/s10208-015-9269-5.

[45]

M. Zhang, L. Yang and Z. H. Huang, Minimum n-rank approximation via iterative hard thresholding, Applied Mathematics and Computation, 256 (2015), 860-875.  doi: 10.1016/j.amc.2015.01.099.

[46]

Z. Zhang and S. Aeron, Exact tensor completion using t-SVD, IEEE Transactions on Signal Processing, 65 (2017), 1511-1526.  doi: 10.1109/TSP.2016.2639466.

[47]

Z. Zhang, G. Ely, S. Aeron, N. Hao and M. E. Kilmer, Novel methods for multilinear data completion and de-noising based on tensor-SVD, Preceedings of the IEEE Conference on Computer Vision and Pattern Recognition, (2014), 3842–3849. doi: 10.1109/CVPR.2014.485.

[48]

P. Zhou, C. Lu, Z. Lin and C. Zhang, Tensor factorization for low-rank tensor completion, IEEE Transactions on Image Processing, 27 (2018), 1152-1163.  doi: 10.1109/TIP.2017.2762595.

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