Search Advanced

Big Data and Information Analytics

2016, Volume 1, Issue 4: 391-401. doi: 10.3934/bdia.2016017
Previous Article Next Article

On identifiability of 3-tensors of multilinear rank (1; Lr; Lr)

  • 1.
    Department of Computer Science Southern Illinois University-Carbondale Carbondale, IL 62901, USA;
  • 2.
    Department of Computer Science Southern Illinois University-Carbondale Carbondale, IL 62901, USA;
  • 3.
    Department of Mathematics Lamar University Beaumont, TX 77710, USA;
  • 4.
    Department of Mathematics Southern Illinois University-Carbondale Carbondale, IL 62901, USA
  • Published: 01 October 2016

  • 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; Lr; Lr) 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; Lr; Lr)[J]. Big Data and Information Analytics, 2016, 1(4): 391-401. doi: 10.3934/bdia.2016017

    Related Papers:

  • 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; Lr; Lr) 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.


    加载中
    [1] [ L. Chiantini and C. Ciliberto, Weakly defective varieties, Transactions of the American Mathematical Society, 354(2002), 151-178.
    [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.
    [3] [ L. Chiantini, G. 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.
    [4] [ A. Cichocki, D. Mandic, L. De Lathauwer, G. Zhou, Q. Zhao, C. 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.
    [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.
    [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.
    [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.
    [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.
    [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.
    [11] [ J. Draisma and J. Kuttler, Bounded-rank tensors are defined in bounded degree, Duke Mathematical Journal, 163(2014), 35-63.
    [12] [ J. D. Hauenstein and A. J. Sommese, Witness sets of projections, Applied Mathematics and Computation, Elsevier, 217(2010), 3349-3354.
    [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.
    [14] [ R. Ke, W. Li and M. Xiao, Characterization of extreme points of multi-stochastic tensors, Computational Methods in Applied Mathematics, 16(2016), 459-474.
    [15] [ T. G. Kolda and B. W. Bader, Tensor decompositions and applications, SIAM Review, 51(2009), 455-500.
    [16] [ J. M. Landsberg, Tensors:Geometry and Applications, AMS, Providence, Rhode Island, USA, 2012.
    [17] [ M. W. Mahoney, L.-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.
    [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 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.
  • Reader Comments
  • © 2016 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索
Big Data and Information Analytics

Metrics

Article views(2869) PDF downloads(518) Cited by(0)

Preview PDF
Download XML
Export Citation
Article outline

Other Articles By Authors

Related pages

Tools

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog