# Approximation Theory of Tree Tensor Networks: Tensorized Multivariate Functions

arXiv: https://arxiv.org/abs/2101.11932 · HTML: https://mlsanity.com/paper/2101.11932/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 36 references that are in the map.
- First posted on arXiv 28 Jan 2021.

## Builds on (most cited first, 9 of 9)

- [Error bounds for approximations with deep ReLU networks](https://mlsanity.com/paper/1610.01145/) (2016)
- [Adaptivity of deep ReLU network for learning in Besov and mixed smooth Besov spaces: optimal rate and curse of dimensionality](https://mlsanity.com/paper/1810.08033/) (2018)
- [Approximation spaces of deep neural networks](https://mlsanity.com/paper/1905.01208/) (2019)
- [Expressivity of Deep Neural Networks](https://mlsanity.com/paper/2007.04759/) (2020)
- [Expressive power of recurrent neural networks](https://mlsanity.com/paper/1711.00811/) (2017)
- [Approximation Theory of Tree Tensor Networks: Tensorized Univariate Functions -- Part I](https://mlsanity.com/paper/2007.00118/) (2020)
- [Approximation of Smoothness Classes by Deep Rectifier Networks](https://mlsanity.com/paper/2007.15645/) (2020)
- [Learning with tree tensor networks: complexity estimates and model selection](https://mlsanity.com/paper/2007.01165/) (2020)
- [Learning high-dimensional probability distributions using tree tensor networks](https://mlsanity.com/paper/1912.07913/) (2019)

## Built on by (most cited first, 3 of 3)

- [Generative Learning of Continuous Data by Tensor Networks](https://mlsanity.com/paper/2310.20498/) (2023)
- [SHAP Meets Tensor Networks: Provably Tractable Explanations with Parallelism](https://mlsanity.com/paper/2510.21599/) (2025)
- [Approximation and learning with compositional tensor trains](https://mlsanity.com/paper/2512.18059/) (2025)

## Abstract (arXiv)

We study the approximation of multivariate functions with tensor networks (TNs), providing some answers to the following two questions: ``what are the approximation capabilities of TNs for functions from classical smoothness classes?'' and ``what are the properties of the class of functions that can be approximated with TNs with a certain performance?'' As a partial answer to the former, we show that TNs can (near to) optimally replicate h-uniform and h-adaptive spline approximation, for any smoothness order of the target function. Tensor networks thus exhibit universal expressivity w.r.t. isotropic, anisotropic and mixed smoothness spaces that is comparable with more general neural networks families such as deep rectified linear unit (ReLU) networks. Put differently, TNs have the capacity to (near to) optimally approximate many function classes -- without being adapted to the particular class in question. As a partial answer to the latter, as a candidate model class we consider approximation classes of TNs and show that these are (quasi-)Banach spaces, that many types of classical smoothness spaces are continuously embedded into said approximation classes and that TNs approximation classes are themselves not embedded in any classical smoothness space. In other words, TNs can efficiently approximate functions that lie beyond classical smoothness spaces.