Best rank-k approximations for tensors:generalizing Eckart–Young

\u3cp\u3eGiven 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\u3csub\u3ef\u3c/sub\u3e, 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\u3csub\u3ef\u3c/sub\u3e. 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\u3csub\u3ef\u3c/sub\u3e is spanned by the complex critical rank-one tensors. Since f itself belongs to H\u3csub\u3ef\u3c/sub\u3e, we deduce that also f itself is a linear combination of its critical rank-one tensors.\u3c/p\u3