RT Conference Proceedings
T1 Best rank-k approximations for tensors: generalizing Eckart-Young
A1 Tocino-Sánchez, Alicia
K1 Matemáticas
K1 Tensores (Álgebra)
AB Joint work with Jan Draisma and Giorgio Ottaviani. Given a tensor f in a Euclidean tensor space, we are interested in the critical points of the distance function from f to the set of tensors of rank at most k, which we call the critical rank-at-most-k tensors for f. When f is a matrix, the critical rank-one matrices for f correspond to the singular pairs of f . The critical rank-one tensors for f lie in a linear subspace H_f, the critical space of f. Our main result is that, for any k, the critical rank-at-most-k tensors for a sufficiently general f also lie in the critical space H_f. This is the part of Eckart-Young Theorem that generalizes from matrices to tensors. Moreover, we show that when the tensor format satisfies the triangle inequalities, the critical space H_f is spanned by the complex critical rank-one tensors. Since f itself belongs to H_f, we deduce that also f itself is a linear combination of its critical rank-one tensors. For simplicity, we will focus on binary forms during the talk.
YR 2020
FD 2020-02-03
LK https://hdl.handle.net/10630/19237
UL https://hdl.handle.net/10630/19237
LA eng
NO Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech.
DS RIUMA. Repositorio Institucional de la Universidad de Málaga
RD 25 ene 2026