Orthogonal and unitary tensor decomposition from an algebraic perspective

Boralevi, Ada; Draisma, Jan; Horobet, Emil; Robeva, Elina

doi:10.1007/s11856-017-1588-6

Cited by 26 publications

(51 citation statements)

References 17 publications

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“…(ii) We initiate a study of orthogonal decomposition over the field of complex numbers, which up to now had not been studied either from an algorithmic or structural point of view. In particular, we stress that all the results in [5] for the field of complex numbers are obtained for unitary rather than orthogonal decompositions.…”

Section: Results and Methodsmentioning

confidence: 99%

“…As mentioned earlier, over the field of complex numbers [5] studies unitary rather than orthogonal decompositions. It would be interesting to find out whether their results for unitary decompositions of symmetric and ordinary tensors, and for decompositions of (real and complex) alternating tensors can be recovered with our linear algebraic techniques.…”

Section: Open Problemsmentioning

confidence: 99%

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Orthogonal tensor decomposition and orbit closures from a linear algebraic perspective

Koiran

2019

Linear and Multilinear Algebra

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We study orthogonal decompositions of symmetric and ordinary tensors using methods from linear algebra. For the field of real numbers we show that the sets of decomposable tensors can be defined by equations of degree 2. This gives a new proof of some of the results of Robeva and Boralevi et al. Orthogonal decompositions over the field of complex numbers had not been studied previously; we give an explicit description of the set of decomposable tensors using polynomial equalities and inequalities, and we begin a study of their closures.The main open problem that arises from this work is to obtain a complete description of the closures. This question is akin to that of characterizing border rank of tensors in algebraic complexity. We give partial results using in particular a connection with approximate simultaneous diagonalization (the so-called ASD property).

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Section: Results and Methodsmentioning

confidence: 99%

Section: Open Problemsmentioning

confidence: 99%

Orthogonal tensor decomposition and orbit closures from a linear algebraic perspective

Koiran

2019

Linear and Multilinear Algebra

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“…Figure 2). In the algebraic geometry community, Robeva [2016] and Boralevi et al [2017] have completely characterized a family of algebraic varieties of orthogonally decomposable (odeco) tensors. We show how the octahedral variety is embedded in one of the odeco varieties, and we introduce a technique for optimization over the odeco frames.…”

Section: Alternative Frame Representationsmentioning

confidence: 99%

Algebraic Representations for Volumetric Frame Fields

2020

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Fig. 1. Octahedral fields generated with mMBO + RTR on various models.Field-guided parametrization methods have proven effective for quad meshing of surfaces; these methods compute smooth cross fields to guide the meshing process and then integrate the fields to construct a discrete mesh.A key challenge in extending these methods to three dimensions, however, is representation of field values. Whereas cross fields can be represented by tangent vector fields that form a linear space, the 3D analog-an octahedral frame field-takes values in a nonlinear manifold. In this work, we describe the space of octahedral frames in the language of differential and algebraic geometry. With this understanding, we develop geometry-aware tools for optimization of octahedral fields, namely geodesic stepping and exact projection via semidefinite relaxation. Our algebraic approach not only provides an elegant and mathematically-sound description of the space of octahedral frames but also suggests a generalization to frames whose three axes scale independently, better capturing the singular behavior we expect to see in volumetric frame fields. These new odeco frames, so-called as they are represented by orthogonally decomposable tensors, also admit a semidefinite program-based projection operator. Our description of the spaces of octahedral and odeco frames suggests computing frame fields via manifold-based optimization algorithms; we show that these algorithms efficiently produce high-quality fields while maintaining stability and smoothness.

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“…c = 0. In this case, we have two possibilities for our form: f is a sum of squares of 4 ternary cubic forms q 1 , q 2 , q 3 , q 4 or not.…”

Section: Examplesmentioning

confidence: 99%

The number of real eigenvectors of a real polynomial

Maccioni

2017

Boll Unione Mat Ital

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I investigate on the number t of real eigenvectors of a real symmetric tensor. In particular, given a homogeneous polynomial f of degree d in 3 variables, I prove that t is greater or equal than 2c + 1, if d is odd, and t is greater or equal than max(3, 2c + 1), if d is even, where c is the number of ovals in the zero locus of f . About binary forms, I prove that t is greater or equal than the number of real roots of f . Moreover, the above inequalities are sharp for binary forms of any degree and for cubic and quartic ternary forms.

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Orthogonal and unitary tensor decomposition from an algebraic perspective

Cited by 26 publications

References 17 publications

Orthogonal tensor decomposition and orbit closures from a linear algebraic perspective

Orthogonal tensor decomposition and orbit closures from a linear algebraic perspective

Algebraic Representations for Volumetric Frame Fields

The number of real eigenvectors of a real polynomial

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