Approximation by tree tensor networks in high dimensions: Sobolev and compositional functions

Bachmayr, Markus; Nouy, Anthony; Schneider, Reinhold

doi:10.48550/arxiv.2112.01474

Cited by 5 publications

(9 citation statements)

References 18 publications

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“…We basically proceed as in [14] and successively apply the singular value decomposition as studied in the previous section. Related results can be found in [1,3,28].…”

Section: Tensor Train Formatmentioning

confidence: 92%

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Low-rank approximation of continuous functions in Sobolev spaces with dominating mixed smoothness

Griebel¹,

Harbrecht²,

Schneider³

2022

Preprint

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Let Ω i ⊂ R ni , i = 1, . . . , m, be given domains. In this article, we study the low-rank approximation with respect to L 2 (Ω 1 × • • • × Ω m ) of functions from Sobolev spaces with dominating mixed smoothness. To this end, we first estimate the rank of a bivariate approximation, i.e., the rank of the continuous singular value decomposition. In comparison to the case of functions from Sobolev spaces with isotropic smoothness, compare [12,13], we obtain improved results due to the additional mixed smoothness. This convergence result is then used to study the tensor train decomposition as a method to construct multivariate low-rank approximations of functions from Sobolev spaces with dominating mixed smoothness. We show that this approach is able to beat the curse of dimension.

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“…We basically proceed as in [14] and successively apply the singular value decomposition as studied in the previous section. Related results can be found in [1,3,28].…”

Section: Tensor Train Formatmentioning

confidence: 92%

“…Such tensor representations are known in computational quantum physics and quantum information theory as tensor networks or more precisely, as tree-based tensor networks [1]. All these names are used in the literature [3]. The present investigations can be extended to the hierarchical Tucker format by using the same or similar ideas, cf.…”

Section: Introductionmentioning

confidence: 99%

Low-rank approximation of continuous functions in Sobolev spaces with dominating mixed smoothness

Griebel¹,

Harbrecht²,

Schneider³

2022

Preprint

Self Cite

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show abstract

“…Opposite to the GAN formulation, we do not use the dual characterization and also do not require a discriminator approximation or Lipschitz constraints. The used explicit polynomial chaos model class corresponds to the generator only within the GAN min-max formulation (5). This leads to a single model that needs to be trained.…”

Section: Relation To Generative Adversarial Neural Network (Gan)mentioning

confidence: 99%

“…To get a bigger picture, it should be noted that the graph structure of tensor formats in some way resembles the topology (and expressiveness) of neural networks (NN) [18,19,1,2,5]. In fact, tensor networks can be seen as a subset of NNs with somewhat lesser expressivity 2 on the one hand, but much richer mathematical structure on the other.…”

mentioning

confidence: 99%

Low-rank Wasserstein polynomial chaos expansions in the framework of optimal transport

Gruhlke¹,

Eigel²

2022

Preprint

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A unsupervised learning approach for the computation of an explicit functional representation of a random vector Y is presented, which only relies on a finite set of samples with unknown distribution. Motivated by recent advances with computational optimal transport for estimating Wasserstein distances, we develop a new Wasserstein multi-element polynomial chaos expansion (WPCE). It relies on the minimization of a regularized empirical Wasserstein metric known as debiased Sinkhorn divergence.As a requirement for an efficient polynomial basis expansion, a suitable (minimal) stochastic coordinate system X has to be determined with the aim to identify ideally independent random variables. This approach generalizes representations through diffeomorphic transport maps to the case of noncontinuous and non-injective model classes M with different input and output dimension, yielding the relation Y = M(X) in distribution. Moreover, since the used PCE grows exponentially in the number of random coordinates of X, we introduce an appropriate low-rank format given as stacks of tensor trains, which alleviates the curse of dimensionality, leading to only linear dependence on the input dimension. By the choice of the model class M and the smooth loss function, higher order optimization schemes become possible. It is shown that the relaxation to a discontinuous model class is necessary to explain multimodal distributions. Moreover, the proposed framework is applied to a numerical upscaling task, considering a computationally challenging microscopic random nonperiodic composite material. This leads to tractable effective macroscopic random field in adopted stochastic coordinates.

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Section: Introductionmentioning

confidence: 99%

Low-rank approximation of continuous functions in Sobolev spaces with dominating mixed smoothness

Griebel¹,

Harbrecht²,

Schneider³

2023

Math. Comp.

View full text Add to dashboard Cite

Let Ω i ⊂ R n i \Omega _i\subset \mathbb {R}^{n_i} , i = 1 , … , m i=1,\ldots ,m , be given domains. In this article, we study the low-rank approximation with respect to L 2 ( Ω 1 × ⋯ × Ω m ) L^2(\Omega _1\times \dots \times \Omega _m) of functions from Sobolev spaces with dominating mixed smoothness. To this end, we first estimate the rank of a bivariate approximation, i.e., the rank of the continuous singular value decomposition. In comparison to the case of functions from Sobolev spaces with isotropic smoothness, compare Griebel and Harbrecht [IMA J. Numer. Anal. 34 (2014), pp. 28–54] and Griebel and Harbrecht [IMA J. Numer. Anal. 39 (2019), pp. 1652–1671], we obtain improved results due to the additional mixed smoothness. This convergence result is then used to study the tensor train decomposition as a method to construct multivariate low-rank approximations of functions from Sobolev spaces with dominating mixed smoothness. We show that this approach is able to beat the curse of dimension.

show abstract

Approximation by tree tensor networks in high dimensions: Sobolev and compositional functions

Cited by 5 publications

References 18 publications

Low-rank approximation of continuous functions in Sobolev spaces with dominating mixed smoothness

Low-rank approximation of continuous functions in Sobolev spaces with dominating mixed smoothness

Low-rank Wasserstein polynomial chaos expansions in the framework of optimal transport

Low-rank approximation of continuous functions in Sobolev spaces with dominating mixed smoothness

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