SoTT: greedy approximation of a tensor as a sum of Tensor Trains

Virginie Ehrlacher (1, 2) , Maria Fuente-Ruiz (3) , Damiano Lombardi (3)

1 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique
2 MATHERIALS - MATHematics for MatERIALS
3 COMMEDIA - COmputational Mathematics for bio-MEDIcal Applications

In the present work, a method is proposed in order to compute an approximation of a given tensor as a sum of Tensor Trains (TTs), where the order of the variates and the values of the ranks can vary from one term to the other in an adaptive way. The numerical scheme is based on a greedy algorithm and an adaptation of the TT-SVD method. The proposed approach can also be used in order to compute an approximation of a tensor in a Canonical Polyadic format (CP), as an alternative to standard algorithms like Alternating Linear Squares (ALS) or Alternating Singular Value Decomposition (ASVD) methods. Some numerical experiments are proposed, in which the proposed method is compared to ALS and ASVD methods for the construction of a CP approximation of a given tensor and performs particularly well for high-order tensors. The interest of approximating a tensor as a sum of Tensor Trains is illustrated in several numerical test cases.

Mots clés: Tensor methods; Canonical Polyadic; Tensor Train AMS subject classifications. 65F99; 65D15

Domaines: Modélisation et simulation; Analyse numérique [math.NA]

Virginie Ehrlacher, Maria Fuente-Ruiz, Damiano Lombardi. SoTT: greedy approximation of a tensor as a sum of Tensor Trains. SIAM Journal on Scientific Computing, 2022, 44 (2), ⟨10.1137/20M1381472⟩. ⟨hal-03018646v3⟩