On Minimal Subspaces in Tensor Representations

In this paper we introduce and develop the notion of minimal subspaces in the framework of algebraic and topological tensor product spaces. This mathematical structure arises in a natural way in the study of tensor representations. We use minimal subspaces to prove the existence of a best approximation, for any element in a Banach tensor space, by means of a tensor given in a typical representation format (Tucker, hierarchical, or tensor train). We show that this result holds in a tensor Banach space with a norm stronger than the injective norm and in an intersection of finitely many Banach tensor spaces satisfying some additional conditions. Examples using topological tensor products of standard Sobolev spaces are given.

[1]  Narendra Ahuja,et al.  Compact representation of multidimensional data using tensor rank-one decomposition , 2004, Proceedings of the 17th International Conference on Pattern Recognition, 2004. ICPR 2004..

[2]  Vin de Silva,et al.  Tensor rank and the ill-posedness of the best low-rank approximation problem , 2006, math/0607647.

[3]  Francisco Chinesta,et al.  A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modeling of complex fluids , 2006 .

[4]  Tamara G. Kolda,et al.  Tensor Decompositions and Applications , 2009, SIAM Rev..

[5]  Jonathan M. Borwein.  Proximality and Chebyshev sets , 2007, Optim. Lett..

[6]  W. Hackbusch.  Tensor Spaces and Numerical Tensor Calculus , 2012, Springer Series in Computational Mathematics.

[7]  Angus E. Taylor.  Introduction to functional analysis , 1959 .

[8]  G. Vidal.  Efficient classical simulation of slightly entangled quantum computations. , 2003, Physical review letters.

[9]  George G. Lorentz,et al.  Constructive Approximation , 1993, Grundlehren der mathematischen Wissenschaften.

[10]  Will Light,et al.  Approximation Theory in Tensor Product Spaces , 1985 .

[11]  Antonio Falcó Montesinos,et al.  Algorithms and numerical methods for high dimensional financial market models , 2010 .

[12]  T. Mexia,et al.  Author ' s personal copy , 2009 .

[13]  Virginie Ehrlacher,et al.  Convergence of a greedy algorithm for high-dimensional convex nonlinear problems , 2010, 1004.0095.

[14]  A. Nouy.  Proper Generalized Decompositions and Separated Representations for the Numerical Solution of High Dimensional Stochastic Problems , 2010 .

[15]  W. Hackbusch,et al.  A New Scheme for the Tensor Representation , 2009 .

[16]  P. Holmes,et al.  The Proper Orthogonal Decomposition in the Analysis of Turbulent Flows , 1993 .

[17]  J. Chang,et al.  Analysis of individual differences in multidimensional scaling via an n-way generalization of “Eckart-Young” decomposition , 1970 .

[18]  W. Greub.  Linear Algebra , 1981 .

[19]  F. L. Hitchcock.  The Expression of a Tensor or a Polyadic as a Sum of Products , 1927 .

[20]  E. Davidson,et al.  Strategies for analyzing data from video fluorometric monitoring of liquid chromatographic effluents , 1981 .

[21]  André Uschmajew,et al.  Well-posedness of convex maximization problems on Stiefel manifolds and orthogonal tensor product approximations , 2010, Numerische Mathematik.

[22]  Antonio Falc,et al.  Algorithms and Numerical Methods for High Dimensional Financial Market Models , 2010 .

[23]  L. Lathauwer,et al.  Dimensionality reduction in higher-order signal processing and rank-(R1,R2,…,RN) reduction in multilinear algebra , 2004 .

[24]  DILEEP MENON,et al.  AN INTRODUCTION TO FUNCTIONAL ANALYSIS , 2010 .

[25]  E. Tyrtyshnikov,et al.  TT-cross approximation for multidimensional arrays , 2010 .

[26]  W. Dur,et al.  Concatenated tensor network states , 2009, 0904.1925.

[27]  BARRY SIMON,et al.  Uniform Crossnorms , 2004 .

[28]  M. Edelstein,et al.  Weekly proximinal sets , 1976 .

[29]  A. Nouy.  A generalized spectral decomposition technique to solve a class of linear stochastic partial differential equations , 2007 .

[30]  Gianluca Iaccarino,et al.  A least-squares approximation of partial differential equations with high-dimensional random inputs , 2009, J. Comput. Phys..

[31]  Demetri Terzopoulos,et al.  Multilinear Analysis of Image Ensembles: TensorFaces , 2002, ECCV.

[32]  L. Tucker,et al.  Some mathematical notes on three-mode factor analysis , 1966, Psychometrika.