Identifiability of an X-Rank Decomposition of Polynomial Maps

In this paper, we study a polynomial decomposition model that arises in problems of system identification, signal processing, and machine learning. We show that this decomposition is a special case of the X-rank decomposition---a powerful novel concept in algebraic geometry that generalizes the tensor CP decomposition. We prove new results on generic/maximal rank and on identifiability of a particular polynomial decomposition model. We try to make the results and basic tools accessible to a general audience (i.e., assuming no knowledge of algebraic geometry or its prerequisites).

[1]  Gian-Carlo Rota,et al.  Apolarity and Canonical Forms for Homogeneous Polynomials , 1993, Eur. J. Comb..

[2]  E. Bai,et al.  Block Oriented Nonlinear System Identification , 2010 .

[3]  Konstantin Usevich,et al.  Identification of a block-structured model with several sources of nonlinearity , 2014, 2014 European Control Conference (ECC).

[4]  Yves Rolain,et al.  System identification in a real world , 2014, 2014 IEEE 13th International Workshop on Advanced Motion Control (AMC).

[5]  L. Chiantini,et al.  Weakly defective varieties , 2001 .

[6]  V. Strassen.  Rank and optimal computation of generic tensors , 1983 .

[7]  L. Chiantini,et al.  On generic identifiability of symmetric tensors of subgeneric rank , 2015, 1504.00547.

[8]  A. Iarrobino,et al.  Power Sums, Gorenstein Algebras, and Determinantal Loci , 2000 .

[9]  Johan Schoukens,et al.  Decoupling Multivariate Polynomials Using First-Order Information , 2014, ArXiv.

[10]  F. Sottile.  Real Algebraic Geometry for Geometric Constraints , 2016, 1606.03127.

[11]  Meera Sitharam,et al.  Real Algebraic Geometry for Geometric Constraints , 2018 .

[12]  Giorgio Ottaviani,et al.  On the number of Waring decompositions for a generic polynomial vector , 2016, 1601.01869.

[13]  Kurt Hornik,et al.  Multilayer feedforward networks are universal approximators , 1989, Neural Networks.

[14]  Pierre Comon,et al.  SEMIALGEBRAIC GEOMETRY OF NONNEGATIVE , 2016 .

[15]  Pierre Comon,et al.  Handbook of Blind Source Separation: Independent Component Analysis and Applications , 2010 .

[16]  Andrzej Schinzel.  On a decomposition of polynomials in several variables , 2002 .

[17]  Giorgio Ottaviani,et al.  An Algorithm For Generic and Low-Rank Specific Identifiability of Complex Tensors , 2014, SIAM J. Matrix Anal. Appl..

[18]  C. D. Boor,et al.  Polynomial interpolation in several variables , 1994 .

[19]  E. Ballico.  On the weak non-defectivity of veronese embeddings of projective spaces , 2005 .

[20]  Luca Chiantini,et al.  On the Concept of k‐Secant Order of a Variety , 2006 .

[21]  M. Mella.  Singularities of linear systems and the Waring problem , 2004, math/0406288.

[22]  Enrico Carlini,et al.  The solution to the Waring problem for monomials and the sum of coprime monomials , 2012 .

[23]  Lieven De Lathauwer,et al.  A Link between the Canonical Decomposition in Multilinear Algebra and Simultaneous Matrix Diagonalization , 2006, SIAM J. Matrix Anal. Appl..

[24]  Tomas Sauer,et al.  Polynomial interpolation in several variables , 2000, Adv. Comput. Math..

[25]  Michael L. Catalano-Johnson.  The possible dimensions of the higher secant varieties , 1996 .

[26]  Pierre Comon,et al.  A Polynomial Formulation for Joint Decomposition of Symmetric Tensors of Different Orders , 2015, LVA/ICA.

[27]  B. Logan,et al.  Optimal reconstruction of a function from its projections , 1975 .

[28]  T. Willmore.  Algebraic Geometry , 1973, Nature.

[29]  Joe Harris,et al.  Representation Theory: A First Course , 1991 .

[30]  K. Usevich.  Decomposing multivariate polynomials with structured low-rank matrix completion , 2014 .

[31]  C. Bocci,et al.  Refined methods for the identifiability of tensors , 2013, 1303.6915.

[32]  Zach Teitler,et al.  On maximum, typical and generic ranks , 2014, ArXiv.

[33]  P. Comon,et al.  Generic and typical ranks of multi-way arrays , 2009 .

[34]  Michael A. Saunders,et al.  Atomic Decomposition by Basis Pursuit , 1998, SIAM J. Sci. Comput..

[35]  J. Landsberg.  Tensors: Geometry and Applications , 2011 .

[36]  Joydeep Ghosh,et al.  Ridge polynomial networks , 1995, IEEE Trans. Neural Networks.

[37]  Donal O'Shea,et al.  Ideals, varieties, and algorithms - an introduction to computational algebraic geometry and commutative algebra (2. ed.) , 1997, Undergraduate texts in mathematics.

[38]  F. Zak.  Determinants of Projective Varieties and their Degrees , 2004 .

[39]  Andrzej Schinzel.  On a decomposition of polynomials in several variables, II , 2002 .

[40]  LimLek-Heng,et al.  Symmetric Tensors and Symmetric Tensor Rank , 2008 .

[41]  J. M. Landsberg,et al.  On the Ideals of Secant Varieties of Segre Varieties , 2004, Found. Comput. Math..

[42]  David Eisenbud,et al.  LINEAR SECTIONS OF DETERMINANTAL VARIETIES , 1988 .

[43]  P. McCullagh.  Tensor Methods in Statistics , 1987 .