Canonical tensor scaling

Canonical Tensor Decompositions

The Singular Value Decomposition (SVD) may be extended to tensors at least in two very different ways. One is the High-Order SVD (HOSVD), and the other is the Canonical Decomposition (CanD). Only the latter is closely related to the tensor rank. Important basic questions are raised in this short paper, such as the maximal achievable rank of a tensor of given dimensions, or the computation of a ...

Finite-size scaling in the canonical ensemble

We investigate the critical scaling behavior of finite systems in the canonical ensemble. The essential difference with the grand canonical ensemble. i.e., the constraint on the number of particles, is already known to lead to the Fisher renormalization phenomenon that modifies the thermal critical singularities. We show that, in observables that are not Fisher renormalized, it also leads to a ...

Scaling solutions in scalar-tensor cosmologies

The possibility of a connection between dark energy and gravity through a direct coupling in the Lagrangian of the underlying theory has acquired an increasing interest due to the recently discovered capability of the extended quintessence model to encompass the fine-tuning problem of the cosmological constant. The gravity induced “R-boost” mechanism is indeed responsible for an early, enhanced...

Analysis and Approximation of the Canonical Polyadic Tensor Decomposition

We study the least-squares (LS) functional of the canonical polyadic (CP) tensor decomposition. Our approach is based on the elimination of one factor matrix which results in a reduced functional. The reduced functional is reformulated into a projection framework and into a Rayleigh quotient. An analysis of this functional leads to several conclusions: new sufficient conditions for the existenc...

Tensor product multiplicities, canonical bases and totally positive varieties