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doi: 10.3934/jimo.2020131

Continuity, differentiability and semismoothness of generalized tensor functions

School of Mathematics, Tianjin University, Tianjin 300350, China

* Corresponding author: Yong Wang

Received  February 2020 Revised  May 2020 Published  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.

Citation: Xia Li, Yong Wang, Zheng-Hai Huang. Continuity, differentiability and semismoothness of generalized tensor functions. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2020131
References:
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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.  Google Scholar

[2]

F. AnderssonM. 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.  Google Scholar

[3]

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

[4]

J. L. AurentzA. P. AustinM. 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.  Google Scholar

[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.  Google Scholar

[6]

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

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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.  Google Scholar

[8]

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

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X. ChenH. 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.  Google Scholar

[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.  Google Scholar

[11]

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

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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.  Google Scholar

[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.  Google Scholar

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N. HaoM. E. KilmerK. Braman and R. C. Hoover, Facial recognition using tensor-tensor decompositions, SIAM Journal on Imaging Sciences, 6 (2013), 437-463.  doi: 10.1137/110842570.  Google Scholar

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

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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.  Google Scholar

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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.  Google Scholar

[20]

M. E. KilmerK. BramanN. 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.  Google Scholar

[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.  Google Scholar

[22]

J. LiuP. MusialskiP. 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.  Google Scholar

[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.  Google Scholar

[24]

C. LuJ. FengY. ChenW. LiuZ. 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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

Y. MiaoL. 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.  Google Scholar

[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.  Google Scholar

[29]

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

[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.  Google Scholar

[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.  Google Scholar

[32]

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

[33]

N. D. SidiropoulosL. De LathauwerX. FuK. HuangE. 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.  Google Scholar

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

[36]

Y. XieD. TaoW. ZhangY. LiuL. 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.  Google Scholar

[37]

Y. XuZ. WuJ. 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.  Google Scholar

[38]

Y. XuL. YuH. XuH. 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.  Google Scholar

[39]

L. YangZ. H. HuangS. 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.  Google Scholar

[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.  Google Scholar

[41]

L. YangZ. 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.  Google Scholar

[42]

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

[43]

M. YinJ. GaoS. 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.  Google Scholar

[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.  Google Scholar

[45]

M. ZhangL. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[48]

P. ZhouC. LuZ. 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.  Google Scholar

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.  Google Scholar

[2]

F. AnderssonM. 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.  Google Scholar

[3]

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

[4]

J. L. AurentzA. P. AustinM. 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.  Google Scholar

[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.  Google Scholar

[6]

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

[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.  Google Scholar

[8]

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

[9]

X. ChenH. 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.  Google Scholar

[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.  Google Scholar

[11]

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

[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.  Google Scholar

[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.  Google Scholar

[14]

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

[15]

N. HaoM. E. KilmerK. Braman and R. C. Hoover, Facial recognition using tensor-tensor decompositions, SIAM Journal on Imaging Sciences, 6 (2013), 437-463.  doi: 10.1137/110842570.  Google Scholar

[16]

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

[17]

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

[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.  Google Scholar

[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.  Google Scholar

[20]

M. E. KilmerK. BramanN. 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.  Google Scholar

[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.  Google Scholar

[22]

J. LiuP. MusialskiP. 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.  Google Scholar

[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.  Google Scholar

[24]

C. LuJ. FengY. ChenW. LiuZ. 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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

Y. MiaoL. 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.  Google Scholar

[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.  Google Scholar

[29]

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

[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.  Google Scholar

[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.  Google Scholar

[32]

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

[33]

N. D. SidiropoulosL. De LathauwerX. FuK. HuangE. 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.  Google Scholar

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

[36]

Y. XieD. TaoW. ZhangY. LiuL. 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.  Google Scholar

[37]

Y. XuZ. WuJ. 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.  Google Scholar

[38]

Y. XuL. YuH. XuH. 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.  Google Scholar

[39]

L. YangZ. H. HuangS. 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.  Google Scholar

[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.  Google Scholar

[41]

L. YangZ. 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.  Google Scholar

[42]

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

[43]

M. YinJ. GaoS. 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.  Google Scholar

[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.  Google Scholar

[45]

M. ZhangL. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[48]

P. ZhouC. LuZ. 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.  Google Scholar

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