# Representation Theorem for Matrix Product States

arXiv: https://arxiv.org/abs/2103.08277 · HTML: https://mlsanity.com/paper/2103.08277/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 39 references that are in the map.
- First posted on arXiv 15 Mar 2021.

## Builds on (most cited first, 7 of 7)

- [Unsupervised Generative Modeling Using Matrix Product States](https://mlsanity.com/paper/1709.01662/) (2017)
- [Tensor Networks for Dimensionality Reduction and Large-Scale Optimizations. Part 2 Applications and Future Perspectives](https://mlsanity.com/paper/1708.09165/) (2017)
- [TensorNetwork for Machine Learning](https://mlsanity.com/paper/1906.06329/) (2019)
- [Equivalence of restricted Boltzmann machines and tensor network states](https://mlsanity.com/paper/1701.04831/) (2017)
- [Language Design as Information Renormalization](https://mlsanity.com/paper/1708.01525/) (2017)
- [Accelerating Block Coordinate Descent for Nonnegative Tensor Factorization](https://mlsanity.com/paper/2001.04321/) (2020)
- [Infinitely Wide Tensor Networks as Gaussian Process](https://mlsanity.com/paper/2101.02333/) (2021)

## Built on by (most cited first, 1 of 1)

- [Neural Tangent Kernel of Matrix Product States: Convergence and Applications](https://mlsanity.com/paper/2111.14046/) (2021)

## Abstract (arXiv)

In this work, we investigate the universal representation capacity of the Matrix Product States (MPS) from the perspective of boolean functions and continuous functions. We show that MPS can accurately realize arbitrary boolean functions by providing a construction method of the corresponding MPS structure for an arbitrarily given boolean gate. Moreover, we prove that the function space of MPS with the scale-invariant sigmoidal activation is dense in the space of continuous functions defined on a compact subspace of the n-dimensional real coordinate space mathbbRn. We study the relation between MPS and neural networks and show that the MPS with a scale-invariant sigmoidal function is equivalent to a one-hidden-layer neural network equipped with a kernel function. We construct the equivalent neural networks for several specific MPS models and show that non-linear kernels such as the polynomial kernel which introduces the couplings between different components of the input into the model appear naturally in the equivalent neural networks. At last, we discuss the realization of the Gaussian Process (GP) with infinitely wide MPS by studying their equivalent neural networks.