# Interaction Decompositions for Tensor Network Regression

arXiv: https://arxiv.org/abs/2208.06029 · HTML: https://mlsanity.com/paper/2208.06029/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 38 references that are in the map.
- First posted on arXiv 11 Aug 2022.

## Builds on (most cited first, 11 of 11)

- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [Going Deeper with Convolutions](https://mlsanity.com/paper/1409.4842/) (2014)
- [Tensor Ring Decomposition](https://mlsanity.com/paper/1606.05535/) (2016)
- [Training Sparse Neural Networks](https://mlsanity.com/paper/1611.06694/) (2016)
- [Learning Relevant Features of Data with Multi-scale Tensor Networks](https://mlsanity.com/paper/1801.00315/) (2017)
- [Wide Compression: Tensor Ring Nets](https://mlsanity.com/paper/1802.09052/) (2018)
- [Tensor Ring Decomposition with Rank Minimization on Latent Space: An Efficient Approach for Tensor Completion](https://mlsanity.com/paper/1809.02288/) (2018)
- [Machine Learning by Unitary Tensor Network of Hierarchical Tree Structure](https://mlsanity.com/paper/1710.04833/) (2017)
- [Compressing Recurrent Neural Networks with Tensor Ring for Action Recognition](https://mlsanity.com/paper/1811.07503/) (2018)
- [TensorNetwork for Machine Learning](https://mlsanity.com/paper/1906.06329/) (2019)
- [Residual Matrix Product State for Machine Learning](https://mlsanity.com/paper/2012.11841/) (2020)

## Built on by (most cited first, 4 of 4)

- [Machine learning with tree tensor networks, CP rank constraints, and tensor dropout](https://mlsanity.com/paper/2305.19440/) (2023)
- [Generative Learning of Continuous Data by Tensor Networks](https://mlsanity.com/paper/2310.20498/) (2023)
- [Tractable Shapley Values and Interactions via Tensor Networks](https://mlsanity.com/paper/2510.22138/) (2025)
- [Tensor Ring Optimized Quantum-Enhanced Tensor Neural Networks](https://mlsanity.com/paper/2310.01515/) (2023)

## Abstract (arXiv)

It is well known that tensor network regression models operate on an exponentially large feature space, but questions remain as to how effectively they are able to utilize this space. Using a polynomial featurization, we propose the interaction decomposition as a tool that can assess the relative importance of different regressors as a function of their polynomial degree. We apply this decomposition to tensor ring and tree tensor network models trained on the MNIST and Fashion MNIST datasets, and find that up to 75% of interaction degrees are contributing meaningfully to these models. We also introduce a new type of tensor network model that is explicitly trained on only a small subset of interaction degrees, and find that these models are able to match or even outperform the full models using only a fraction of the exponential feature space. This suggests that standard tensor network models utilize their polynomial regressors in an inefficient manner, with the lower degree terms being vastly under-utilized.