October  2016, 1(4): 391-401. doi: 10.3934/bdia.2016017

On identifiability of 3-tensors of multilinear rank $(1,\ L_{r},\ L_{r})$

1. 

Department of Computer Science, Southern Illinois University-Carbondale, Carbondale, IL 62901, USA

2. 

Department of Mathematics, Lamar University, Beaumont, TX 77710, USA

3. 

Department of Mathematics, Southern Illinois University-Carbondale, Carbondale, IL 62901, USA

* Corresponding author: Qiang Cheng, Mingqing Xiao. This research is supported in part by NSF-DMS 1419028 and NSF-IIS 1218712 of United States

Revised  May 2017 Published  May 2017

In this paper, we study a specific big data model via multilinear rank tensor decompositions. The model approximates to a given tensor by the sum of multilinear rank $(1, \ L_{r}, \ L_{r})$ terms. And we characterize the identifiability property of this model from a geometric point of view. Our main results consists of exact identifiability and generic identifiability. The arguments of generic identifiability relies on the exact identifiability, which is in particular closely related to the well-known "trisecant lemma" in the context of algebraic geometry (see Proposition 2.6 in [1]). This connection discussed in this paper demonstrates a clear geometric picture of this model.

Citation: Ming Yang, Dunren Che, Wen Liu, Zhao Kang, Chong Peng, Mingqing Xiao, Qiang Cheng. On identifiability of 3-tensors of multilinear rank $(1,\ L_{r},\ L_{r})$. Big Data & Information Analytics, 2016, 1 (4) : 391-401. doi: 10.3934/bdia.2016017
References:
[1]

L. Chiantini and C. Ciliberto, Weakly defective varieties, Transactions of the American Mathematical Society, 354 (2002), 151-178.  doi: 10.1090/S0002-9947-01-02810-0.  Google Scholar

[2]

L. Chiantini and G. Ottaviani, On generic identifiability of 3-tensors of small rank, SIAM Journal on Matrix Analysis and Applications, 33 (2012), 1018-1037.  doi: 10.1137/110829180.  Google Scholar

[3]

L. ChiantiniG. Ottaviani and N. Vannieuwenhoven, An algorithm for generic and low-rank specific identifiability of complex tensors, SIAM Journal on Matrix Analysis and Applications, 35 (2014), 1265-1287.  doi: 10.1137/140961389.  Google Scholar

[4]

A. CichockiD. MandicL. De LathauwerG. ZhouQ. ZhaoC. Caiafa and H. Phan, Tensor decompositions for signal processing applications: From two-way to multiway component analysis, Signal Processing Magazine, IEEE, 32 (2015), 145-163.  doi: 10.1109/MSP.2013.2297439.  Google Scholar

[5]

P. Comon, Tensor decompositions, State of the art and applications, J. G. McWhirter and I. K. Proudler. Mathematics in Signal Processing V, Clarendon Press, Oxford, 71 (2002), 1-24.   Google Scholar

[6]

L. DeLathauwer, Decompositions of a higher-order tensor in block terms-part Ⅰ: Lemmas for partitioned matrices, SIAM Journal on Matrix Analysis and Applications, 30 (2008), 1022-1032.  doi: 10.1137/060661685.  Google Scholar

[7]

L. DeLathauwer, Decompositions of a higher-order tensor in block terms-part Ⅱ: Definitions and uniqueness, SIAM Journal on Matrix Analysis and Applications, 30 (2008), 1033-1066.  doi: 10.1137/070690729.  Google Scholar

[8]

L. DeLathauwer and D. Nion, Decompositions of a higher-order tensor in block terms-part Ⅲ: Alternating least squares algorithms, SIAM Journal on Matrix Analysis and Applications, 30 (2008), 1067-1083.  doi: 10.1137/070690730.  Google Scholar

[9]

I. Domanov and L. DeLathauwer, Generic uniqueness of a structured matrix factorization and applications in blind source separation, IEEE Journal of Selected Topics in Signal Processing, 10 (2016), 701-711.  doi: 10.1109/JSTSP.2016.2526971.  Google Scholar

[10]

I. Domanov and L. De Lathauwer, Generic uniqueness conditions for the canonical polyadic decomposition and INDSCAL, SIAM Journal on Matrix Analysis and Applications, 36 (2015), 1567-1589.  doi: 10.1137/140970276.  Google Scholar

[11]

J. Draisma and J. Kuttler, Bounded-rank tensors are defined in bounded degree, Duke Mathematical Journal, 163 (2014), 35-63.  doi: 10.1215/00127094-2405170.  Google Scholar

[12]

J. D. Hauenstein and A. J. Sommese, Witness sets of projections, Applied Mathematics and Computation, Elsevier, 217 (2010), 3349-3354.  doi: 10.1016/j.amc.2010.08.067.  Google Scholar

[13]

J. D. Hauenstein and A. J. Sommese, Membership tests for images of algebraic sets by linear projections, Applied Mathematics and Computation, Elsevier, 219 (2013), 6809-6818.  doi: 10.1016/j.amc.2012.12.060.  Google Scholar

[14]

R. KeW. Li and M. Xiao, Characterization of extreme points of multi-stochastic tensors, Computational Methods in Applied Mathematics, 16 (2016), 459-474.  doi: 10.1515/cmam-2016-0005.  Google Scholar

[15]

T. G. Kolda and B. W. Bader, Tensor decompositions and applications, SIAM Review, 51 (2009), 455-500.  doi: 10.1137/07070111X.  Google Scholar

[16]

J. M. Landsberg, Tensors: Geometry and Applications, AMS, Providence, Rhode Island, USA, 2012. Google Scholar

[17]

M. W. MahoneyL.-H. Lim and G. E. Carlsson, Algorithmic and statistical challenges in modern largescale data analysis are the focus of MMDS 2008, ACM SIGKDD Explorations Newsletter, 10 (2008), 57-60.  doi: 10.1145/1540276.1540294.  Google Scholar

[18]

F. Malgouyres and J. Landsberg, Stable recovery of the factors from a deep matrix product and application to convolutional network, preprint, arXiv: 1703. 08044. Google Scholar

[19]

Y. Matsushima (E. Kobayashi, translator), Differentiable Manifolds, Marcel Dekker, Inc. , North-Holland Publishing Co. , North Miami Beach, FL, U. S. A. , 1972. Google Scholar

[20]

E. E. PapalexakisC. Faloutsos and N. D. Sidiropoulos, Tensors for data mining and data fusion: Models, applications, and scalable algorithms, ACM Transactions on Intelligent Systems and Technology (TIST), 8 (2017), 1-44.  doi: 10.1145/2915921.  Google Scholar

show all references

References:
[1]

L. Chiantini and C. Ciliberto, Weakly defective varieties, Transactions of the American Mathematical Society, 354 (2002), 151-178.  doi: 10.1090/S0002-9947-01-02810-0.  Google Scholar

[2]

L. Chiantini and G. Ottaviani, On generic identifiability of 3-tensors of small rank, SIAM Journal on Matrix Analysis and Applications, 33 (2012), 1018-1037.  doi: 10.1137/110829180.  Google Scholar

[3]

L. ChiantiniG. Ottaviani and N. Vannieuwenhoven, An algorithm for generic and low-rank specific identifiability of complex tensors, SIAM Journal on Matrix Analysis and Applications, 35 (2014), 1265-1287.  doi: 10.1137/140961389.  Google Scholar

[4]

A. CichockiD. MandicL. De LathauwerG. ZhouQ. ZhaoC. Caiafa and H. Phan, Tensor decompositions for signal processing applications: From two-way to multiway component analysis, Signal Processing Magazine, IEEE, 32 (2015), 145-163.  doi: 10.1109/MSP.2013.2297439.  Google Scholar

[5]

P. Comon, Tensor decompositions, State of the art and applications, J. G. McWhirter and I. K. Proudler. Mathematics in Signal Processing V, Clarendon Press, Oxford, 71 (2002), 1-24.   Google Scholar

[6]

L. DeLathauwer, Decompositions of a higher-order tensor in block terms-part Ⅰ: Lemmas for partitioned matrices, SIAM Journal on Matrix Analysis and Applications, 30 (2008), 1022-1032.  doi: 10.1137/060661685.  Google Scholar

[7]

L. DeLathauwer, Decompositions of a higher-order tensor in block terms-part Ⅱ: Definitions and uniqueness, SIAM Journal on Matrix Analysis and Applications, 30 (2008), 1033-1066.  doi: 10.1137/070690729.  Google Scholar

[8]

L. DeLathauwer and D. Nion, Decompositions of a higher-order tensor in block terms-part Ⅲ: Alternating least squares algorithms, SIAM Journal on Matrix Analysis and Applications, 30 (2008), 1067-1083.  doi: 10.1137/070690730.  Google Scholar

[9]

I. Domanov and L. DeLathauwer, Generic uniqueness of a structured matrix factorization and applications in blind source separation, IEEE Journal of Selected Topics in Signal Processing, 10 (2016), 701-711.  doi: 10.1109/JSTSP.2016.2526971.  Google Scholar

[10]

I. Domanov and L. De Lathauwer, Generic uniqueness conditions for the canonical polyadic decomposition and INDSCAL, SIAM Journal on Matrix Analysis and Applications, 36 (2015), 1567-1589.  doi: 10.1137/140970276.  Google Scholar

[11]

J. Draisma and J. Kuttler, Bounded-rank tensors are defined in bounded degree, Duke Mathematical Journal, 163 (2014), 35-63.  doi: 10.1215/00127094-2405170.  Google Scholar

[12]

J. D. Hauenstein and A. J. Sommese, Witness sets of projections, Applied Mathematics and Computation, Elsevier, 217 (2010), 3349-3354.  doi: 10.1016/j.amc.2010.08.067.  Google Scholar

[13]

J. D. Hauenstein and A. J. Sommese, Membership tests for images of algebraic sets by linear projections, Applied Mathematics and Computation, Elsevier, 219 (2013), 6809-6818.  doi: 10.1016/j.amc.2012.12.060.  Google Scholar

[14]

R. KeW. Li and M. Xiao, Characterization of extreme points of multi-stochastic tensors, Computational Methods in Applied Mathematics, 16 (2016), 459-474.  doi: 10.1515/cmam-2016-0005.  Google Scholar

[15]

T. G. Kolda and B. W. Bader, Tensor decompositions and applications, SIAM Review, 51 (2009), 455-500.  doi: 10.1137/07070111X.  Google Scholar

[16]

J. M. Landsberg, Tensors: Geometry and Applications, AMS, Providence, Rhode Island, USA, 2012. Google Scholar

[17]

M. W. MahoneyL.-H. Lim and G. E. Carlsson, Algorithmic and statistical challenges in modern largescale data analysis are the focus of MMDS 2008, ACM SIGKDD Explorations Newsletter, 10 (2008), 57-60.  doi: 10.1145/1540276.1540294.  Google Scholar

[18]

F. Malgouyres and J. Landsberg, Stable recovery of the factors from a deep matrix product and application to convolutional network, preprint, arXiv: 1703. 08044. Google Scholar

[19]

Y. Matsushima (E. Kobayashi, translator), Differentiable Manifolds, Marcel Dekker, Inc. , North-Holland Publishing Co. , North Miami Beach, FL, U. S. A. , 1972. Google Scholar

[20]

E. E. PapalexakisC. Faloutsos and N. D. Sidiropoulos, Tensors for data mining and data fusion: Models, applications, and scalable algorithms, ACM Transactions on Intelligent Systems and Technology (TIST), 8 (2017), 1-44.  doi: 10.1145/2915921.  Google Scholar

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