# American Institute of Mathematical Sciences

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:

show all references

##### References:
 [1] H. Bercovici, V. Niţică. Cohomology of higher rank abelian Anosov actions for Banach algebra valued cocycles. Conference Publications, 2001, 2001 (Special) : 50-55. doi: 10.3934/proc.2001.2001.50 [2] Yangyang Xu, Ruru Hao, Wotao Yin, Zhixun Su. Parallel matrix factorization for low-rank tensor completion. Inverse Problems & Imaging, 2015, 9 (2) : 601-624. doi: 10.3934/ipi.2015.9.601 [3] Zhouchen Lin. A review on low-rank models in data analysis. Big Data & Information Analytics, 2016, 1 (2&3) : 139-161. doi: 10.3934/bdia.2016001 [4] Gabriele Link, Jean-Claude Picaud. Ergodic geometry for non-elementary rank one manifolds. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 6257-6284. doi: 10.3934/dcds.2016072 [5] Felipe Hernandez. A decomposition for the Schrödinger equation with applications to bilinear and multilinear estimates. Communications on Pure & Applied Analysis, 2018, 17 (2) : 627-646. doi: 10.3934/cpaa.2018034 [6] Sikhar Patranabis, Debdeep Mukhopadhyay. Identity-based key aggregate cryptosystem from multilinear maps. Advances in Mathematics of Communications, 2019, 13 (4) : 759-778. doi: 10.3934/amc.2019044 [7] Umberto Martínez-Peñas. Rank equivalent and rank degenerate skew cyclic codes. Advances in Mathematics of Communications, 2017, 11 (2) : 267-282. doi: 10.3934/amc.2017018 [8] Relinde Jurrius, Ruud Pellikaan. On defining generalized rank weights. Advances in Mathematics of Communications, 2017, 11 (1) : 225-235. doi: 10.3934/amc.2017014 [9] Tomasz Downarowicz, Yonatan Gutman, Dawid Huczek. Rank as a function of measure. Discrete & Continuous Dynamical Systems - A, 2014, 34 (7) : 2741-2750. doi: 10.3934/dcds.2014.34.2741 [10] Mostafa Karimi, Noor Akma Ibrahim, Mohd Rizam Abu Bakar, Jayanthi Arasan. Rank-based inference for the accelerated failure time model in the presence of interval censored data. Numerical Algebra, Control & Optimization, 2017, 7 (1) : 107-112. doi: 10.3934/naco.2017007 [11] Nick Cercone, F'IEEE. What's the big deal about big data?. Big Data & Information Analytics, 2016, 1 (1) : 31-79. doi: 10.3934/bdia.2016.1.31 [12] Mariantonia Cotronei, Tomas Sauer. Full rank filters and polynomial reproduction. Communications on Pure & Applied Analysis, 2007, 6 (3) : 667-687. doi: 10.3934/cpaa.2007.6.667 [13] Roman VodiČka, Vladislav MantiČ. An energy based formulation of a quasi-static interface damage model with a multilinear cohesive law. Discrete & Continuous Dynamical Systems - S, 2017, 10 (6) : 1539-1561. doi: 10.3934/dcdss.2017079 [14] Richard Boire. Understanding AI in a world of big data. Big Data & Information Analytics, 2017, 2 (5) : 22-42. doi: 10.3934/bdia.2018001 [15] Alex L Castro, Wyatt Howard, Corey Shanbrom. Bridges between subriemannian geometry and algebraic geometry: Now and then. Conference Publications, 2015, 2015 (special) : 239-247. doi: 10.3934/proc.2015.0239 [16] Frank Blume. Minimal rates of entropy convergence for rank one systems. Discrete & Continuous Dynamical Systems - A, 2000, 6 (4) : 773-796. doi: 10.3934/dcds.2000.6.773 [17] Michael Blank. Finite rank approximations of expanding maps with neutral singularities. Discrete & Continuous Dynamical Systems - A, 2008, 21 (3) : 749-762. doi: 10.3934/dcds.2008.21.749 [18] John Sheekey. A new family of linear maximum rank distance codes. Advances in Mathematics of Communications, 2016, 10 (3) : 475-488. doi: 10.3934/amc.2016019 [19] Keith Burns, Katrin Gelfert. Lyapunov spectrum for geodesic flows of rank 1 surfaces. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 1841-1872. doi: 10.3934/dcds.2014.34.1841 [20] Gábor Székelyhidi, Ben Weinkove. On a constant rank theorem for nonlinear elliptic PDEs. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 6523-6532. doi: 10.3934/dcds.2016081

Impact Factor: