American Institue of Mathematical Sciences

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

* 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, C. Ciliberto, Weakly defective varieties, Transactions of the American Mathematical Society, 354 (2002), 151-178. doi: 10.1090/S0002-9947-01-02810-0. [2] L. Chiantini, 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. [3] L. Chiantini, G. Ottaviani, 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. [4] A. Cichocki, D. Mandic, L. De Lathauwer, G. Zhou, Q. Zhao, C. Caiafa, 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. [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. [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. [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. [8] L. DeLathauwer, 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. [9] I. Domanov, 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. [10] I. Domanov, 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. [11] J. Draisma, J. Kuttler, Bounded-rank tensors are defined in bounded degree, Duke Mathematical Journal, 163 (2014), 35-63. doi: 10.1215/00127094-2405170. [12] J. D. Hauenstein, A. J. Sommese, Witness sets of projections, Applied Mathematics and Computation, Elsevier, 217 (2010), 3349-3354. doi: 10.1016/j.amc.2010.08.067. [13] J. D. Hauenstein, 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. [14] R. Ke, W. Li, 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. [15] T. G. Kolda, B. W. Bader, Tensor decompositions and applications, SIAM Review, 51 (2009), 455-500. doi: 10.1137/07070111X. [16] J. M. Landsberg, Tensors: Geometry and Applications, AMS, Providence, Rhode Island, USA, 2012. [17] M. W. Mahoney, L.-H. Lim, 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. [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. [19] Y. Matsushima (E. Kobayashi, translator), Differentiable Manifolds, Marcel Dekker, Inc. , North-Holland Publishing Co. , North Miami Beach, FL, U. S. A. , 1972. [20] E. E. Papalexakis, C. Faloutsos, 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.

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References:
 [1] L. Chiantini, C. Ciliberto, Weakly defective varieties, Transactions of the American Mathematical Society, 354 (2002), 151-178. doi: 10.1090/S0002-9947-01-02810-0. [2] L. Chiantini, 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. [3] L. Chiantini, G. Ottaviani, 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. [4] A. Cichocki, D. Mandic, L. De Lathauwer, G. Zhou, Q. Zhao, C. Caiafa, 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. [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. [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. [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. [8] L. DeLathauwer, 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. [9] I. Domanov, 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. [10] I. Domanov, 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. [11] J. Draisma, J. Kuttler, Bounded-rank tensors are defined in bounded degree, Duke Mathematical Journal, 163 (2014), 35-63. doi: 10.1215/00127094-2405170. [12] J. D. Hauenstein, A. J. Sommese, Witness sets of projections, Applied Mathematics and Computation, Elsevier, 217 (2010), 3349-3354. doi: 10.1016/j.amc.2010.08.067. [13] J. D. Hauenstein, 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. [14] R. Ke, W. Li, 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. [15] T. G. Kolda, B. W. Bader, Tensor decompositions and applications, SIAM Review, 51 (2009), 455-500. doi: 10.1137/07070111X. [16] J. M. Landsberg, Tensors: Geometry and Applications, AMS, Providence, Rhode Island, USA, 2012. [17] M. W. Mahoney, L.-H. Lim, 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. [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. [19] Y. Matsushima (E. Kobayashi, translator), Differentiable Manifolds, Marcel Dekker, Inc. , North-Holland Publishing Co. , North Miami Beach, FL, U. S. A. , 1972. [20] E. E. Papalexakis, C. Faloutsos, 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.
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