Real identifiability vs. complex identifiability

Angelini, Elena; Bocci, Cristiano; Chiantini, Luca

doi:10.1080/03081087.2017.1347137

Cited by 19 publications

(28 citation statements)

References 39 publications

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“…Perhaps the most interesting among these labels are those arising in open Euclidean subsets of the ambient real projective space P r (R), the typical labels; see §3 and §4. We highlight differences and similarities that we find between typical labels and typical ranks; the latter ones have been intensively studied in recent years; see for instance [1,4,5,7,10,21,22,23].…”

Section: Introductionmentioning

confidence: 58%

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Labels of real projective varieties

Ballico

Ventura

2020

Boll Unione Mat Ital

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Let X be a complex projective variety defined over R. Recently, Bernardi and the first author introduced the notion of admissible rank with respect to X. This rank takes into account only decompositions that are stable under complex conjugation. Such a decomposition carries a label, i.e., a pair of integers recording the cardinality of its totally real part. We study basic properties of admissible ranks for varieties, along with special examples of curves; for instance, for rational normal curves admissible and complex ranks coincide. Along the way, we introduce the scheme theoretic version of admissible rank. Finally, analogously to the situation of real ranks, we analyze typical labels, i.e., those arising as labels of a full-dimensional Euclidean open set. We highlight similarities and differences with typical ranks.Let X be a projective variety defined over an arbitrary field K. For a subfield L ⊆ K, the set of L-points of X is denoted X(L). The set of smooth L-points of X is X reg (L) and the singular locus X sing (L). In the following, K = C and L = R.Let X(C) ⊂ P r (C) be an integral and nondegenerate complex projective variety. The pair consisting of X(C) and the given embedding X(C) ֒→ P r (C) is defined over R if and

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Section: Introductionmentioning

confidence: 58%

“…The definition of typical rank is a key notion for real ranks of (real) algebraic varieties. Recently this topic has witnessed a tremendous amount of results; see for instance [1,4,7,5,10,21,22,23]. We extend some of them here to the setting of typical labels which will be defined in a moment.…”

Section: Typical Labelsmentioning

confidence: 94%

Labels of real projective varieties

Ballico

Ventura

2020

Boll Unione Mat Ital

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“…In this case, f is a single polynomial f (u) = f (u), and (1) becomes (2) f (u) = g 1 (v 1 u) + · · · + g r (v r u), since we can assume that w k = [1]. An example of (2) is shown in Figure 1.…”

Section: Model and Examplesmentioning

confidence: 99%

“…Hence, if the polynomial f admits a decomposition (2), then all the homogeneous parts f (d) can be decomposed as…”

Section: Lemma 18 (Terracini)mentioning

confidence: 99%

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Identifiability of an X-Rank Decomposition of Polynomial Maps

Comon¹,

Yang²,

Usevich³

2017

SIAM J. Appl. Algebra Geometry

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Abstract. In this paper, we study a polynomial decomposition model that arises in problems of system identification, signal processing, and machine learning. We show that this decomposition is a special case of the X-rank decomposition-a powerful novel concept in algebraic geometry that generalizes the tensor CP decomposition. We prove new results on generic/maximal rank and on identifiability of a particular polynomial decomposition model. We try to make the results and basic tools accessible to a general audience (i.e., assuming no knowledge of algebraic geometry or its prerequisites).

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A Note on Nonclosed Tensor Formats

Hackbusch

2019

Vietnam J. Math.

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Various tensor formats exist which allow a data-sparse representation of tensors. Some of these formats are not closed. The consequences are (i) possible non-existence of best approximations and (ii) divergence of the representing parameters when a tensor within the format tends to a border tensor outside. The paper tries to describe the nature of this divergence. A particular question is whether the divergence is uniform for all border tensors.

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Real identifiability vs. complex identifiability

Abstract: We report about the state of the art on complex and real generic identifiability of tensors, we describe some of our recent results obtained in [6] and we present perspectives on the subject.

Cited by 19 publications

References 39 publications

Labels of real projective varieties

Labels of real projective varieties

Identifiability of an X-Rank Decomposition of Polynomial Maps

A Note on Nonclosed Tensor Formats

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