Approximation Theory of Tree Tensor Networks: Tensorized Univariate Functions

Ali, Mazen; Nouy, Anthony

doi:10.1007/s00365-023-09620-w

Cited by 4 publications

(13 citation statements)

References 44 publications

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“…This corresponds to the algorithmic procedure in practical computations. Related results, obtained by other proof techniques, can be found in [1,3,18,29]. 6.1.…”

Section: Tensor Train Formatmentioning

confidence: 77%

“…Let Ω ∈ R n be a sufficiently smooth, bounded domain. 1 We consider a nested sequence of finite dimensional subspaces…”

Section: Approximation On the Subdomainsmentioning

confidence: 99%

“…Note that the tensor train (TT) format is a particular architecture of the hierarchical Tucker (HT) format [19]. 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].…”

Section: Introductionmentioning

confidence: 99%

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

Griebel¹,

Harbrecht²,

Schneider³

2023

Math. Comp.

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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.

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“…This corresponds to the algorithmic procedure in practical computations. Related results, obtained by other proof techniques, can be found in [1,3,18,29]. 6.1.…”

Section: Tensor Train Formatmentioning

confidence: 77%

“…Let Ω ∈ R n be a sufficiently smooth, bounded domain. 1 We consider a nested sequence of finite dimensional subspaces…”

Section: Approximation On the Subdomainsmentioning

confidence: 99%

See 1 more Smart Citation

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

Griebel¹,

Harbrecht²,

Schneider³

2023

Math. Comp.

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“…(iv) A general formulation of tensorized approximations on a function space level has been developed in Ali and Nouy (2023), where convergence rates for functions with Besov regularity are also obtained.…”

Section: Multilevel Tensorized Representationsmentioning

confidence: 99%

Low-rank tensor methods for partial differential equations

Bachmayr¹

2023

Acta Numerica

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Low-rank tensor representations can provide highly compressed approximations of functions. These concepts, which essentially amount to generalizations of classical techniques of separation of variables, have proved to be particularly fruitful for functions of many variables. We focus here on problems where the target function is given only implicitly as the solution of a partial differential equation. A first natural question is under which conditions we should expect such solutions to be efficiently approximated in low-rank form. Due to the highly nonlinear nature of the resulting low-rank approximations, a crucial second question is at what expense such approximations can be computed in practice. This article surveys basic construction principles of numerical methods based on low-rank representations as well as the analysis of their convergence and computational complexity.

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“…g (1) α (x 1 )g (2) α (x 2 ) • • • g (d) α (x d ), (1) then the task of approximating f is replaced by approximating the R • d univariate functions g ( ) α : [−1, 1] → R; see, e.g., [12]. It is often beneficial, especially when R is very large, to impose additional structure on (1), leading to the so-called functional tensor train (FTT) [13,38] and Tucker [28,49] formats. In practice, functions are rarely given directly in such a functional low-rank format, but they can sometimes be well approximated by it.…”

Section: Introductionmentioning

confidence: 99%

Approximation in the extended functional tensor train format

Strössner,

Sun,

Kressner

2024

Adv Comput Math

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This work proposes the extended functional tensor train (EFTT) format for compressing and working with multivariate functions on tensor product domains. Our compression algorithm combines tensorized Chebyshev interpolation with a low-rank approximation algorithm that is entirely based on function evaluations. Compared to existing methods based on the functional tensor train format, the adaptivity of our approach often results in reducing the required storage, sometimes considerably, while achieving the same accuracy. In particular, we reduce the number of function evaluations required to achieve a prescribed accuracy by up to over $$96\%$$ 96 % compared to the algorithm from Gorodetsky et al. (Comput. Methods Appl. Mech. Eng. 347, 59–84 2019).

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Approximation Theory of Tree Tensor Networks: Tensorized Univariate Functions

Cited by 4 publications

References 44 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 tensor methods for partial differential equations

Approximation in the extended functional tensor train format

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