Nonlinear algebra and applications

Breiding, Paul; Çelik, Türkü Özlüm; Duff, Timothy; Heaton, Alexander; Maraj, Aida; Sattelberger, Anna-Laura; Venturello, Lorenzo; Yürük, Oğuzhan

doi:10.3934/naco.2021045

Cited by 4 publications

(5 citation statements)

References 208 publications

(215 reference statements)

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“…The reader might also take a look at the previous books by Sturmfels [20,21]. We have not included references to the recent papers in the area because there are simply too many, but the reader could find in the article [3] over two hundred references on applications of nonlinear algebra to polynomial optimization, partial differential equations, algebraic statistics, integrable systems, configuration spaces of frameworks, biochemical reaction networks, algebraic vision, and tensor decompositions. Further references can be found in the book [5].…”

Section: The Structure Of the Bookmentioning

confidence: 99%

Book Review: Invitation to nonlinear algebra

Dickenstein,

Ottaviani

2023

Bull. Amer. Math. Soc.

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Section: The Structure Of the Bookmentioning

confidence: 99%

Book Review: Invitation to nonlinear algebra

Dickenstein,

Ottaviani

2023

Bull. Amer. Math. Soc.

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“…Due to the sum structure, also the gradient properties generalize to the tensor case. 3 Due to the distinguishable roles of J ∈ K and the map J K γ , we here remain faithful to prior literature as for both the letter J has been used before.…”

Section: 1mentioning

confidence: 99%

“…While real tensors of at most rank r (rs) do not form varieties, complex ones with at most this border rank do, here with a dimension of dim(V ≤r (rs) ,C ) = r (rs) (d(n − 1) + 1) = 51 (cf. [3,6,33]). Though we assume this dimension to be lower than the one for K max , we take this smaller value as reference.…”

Section: Details Of Comparisonmentioning

confidence: 99%

Asymptotic Log-Det Sum-of-Ranks Minimization via Tensor (Alternating) Iteratively Reweighted Least Squares

Krämer

2021

Preprint

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Affine sum-of-ranks minimization (ASRM) generalizes the affine rank minimization (ARM) problem from matrices to tensors. Here, the interest lies in the ranks of a family K of different matricizations. Transferring our priorly discussed results on asymptotic log-det rank minimization, we show that iteratively reweighted least squares with weight strength p = 0 remains a, theoretically and practically, particularly viable method denoted as IRLS-0K. As in the matrix case, we prove global convergence of asymptotic minimizers of the log-det sum-of-ranks function to desired solutions. Further, we show local convergence of IRLS-0K in dependence of the rate of decline of the therein appearing regularization parameter γ 0. For hierarchical families K, we show how an alternating version (AIRLS-0K, related to prior work under the name SALSA) can be evaluated solely through tensor tree network based operations. The method can thereby be applied to high dimensions through the avoidance of exponential computational complexity. Further, the otherwise crucial rank adaption process becomes essentially superfluous even for completion problems. In numerical experiments, we show that the therefor required subspace restrictions and relaxation of the affine constraint cause only a marginal loss of approximation quality. On the other hand, we demonstrate that IRLS-0K allows to observe the theoretical phase transition also for generic tensor recoverability in practice. Concludingly, we apply AIRLS-0K to larger scale problems.

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“…The following examples demonstrate that our assumptions on X are not very restrictive and the results of this work can be applied in a number of different settings. For an overview of polynomial models in the sciences see also [11].…”

Section: Introductionmentioning

confidence: 99%