Best rank k approximation for binary forms

Ottaviani, Giorgio; Tocino, Alicia

doi:10.1007/s13348-017-0206-6

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“…Remark 2.10. In the case of binary forms (dim V = 2), H f is the hyperplane orthogonal to D(f ) [16]. In the case of ordinary tensors, H f was first defined in [15] where it was called singular space, but in view of the results in this paper we feel that critical space is a better name.…”

Section: The Critical Space Of a Tensormentioning

confidence: 93%

“…Remark 2.1. In the case of binary forms (dim V = 2 and arbitrary d), the pairing [f |g] coincides (up to scalar multiples) with (f |D(g)), where D(g) = g x y − g y x; see [16]. Note the skew-symmetry property (f |D(g)) = − (g|D(f )).…”

Section: The Critical Space Of a Tensormentioning

confidence: 99%

“…In the case of symmetric tensors, these critical rank-one tensors correspond to the so-called eigenvectors of f [11], while in the case of ordinary tensors, they correspond to singular vector tuples [10]. In the case n = 1 of binary forms, Corollary 1.3 was proved in [16]. The two corollaries above can be regarded as generalizations of the Eckart-Young Theorem and the Spectral Theorem from matrices to tensors.…”

Section: Introductionmentioning

confidence: 99%

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

2018

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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.2000 Mathematics Subject Classification. 15A69, 15A18, 14M17, 14P05.

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Section: The Critical Space Of a Tensormentioning

confidence: 93%

Section: The Critical Space Of a Tensormentioning

confidence: 99%