# (MPO)2: Multivariate Polynomial Optimization based on Matrix Product Operators

arXiv: https://arxiv.org/abs/2607.15916 · HTML: https://mlsanity.com/paper/2607.15916/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 18 of its 48 references that are in the map.
- First posted on arXiv 17 Jul 2026.

## Builds on (most cited first, 18 of 18)

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- [Deep Polynomial Neural Networks](https://mlsanity.com/paper/2006.13026/) (2020)
- [Tensor Methods in Computer Vision and Deep Learning](https://mlsanity.com/paper/2107.03436/) (2021)
- [Supervised Learning with Quantum-Inspired Tensor Networks](https://mlsanity.com/paper/1605.05775/) (2016)
- [The Variational Gaussian Process](https://mlsanity.com/paper/1511.06499/) (2015)
- [TensorNetwork for Machine Learning](https://mlsanity.com/paper/1906.06329/) (2019)
- [Augmenting Deep Classifiers with Polynomial Neural Networks](https://mlsanity.com/paper/2104.07916/) (2021)
- [Discovering interpretable elastoplasticity models via the neural polynomial method enabled symbolic regressions](https://mlsanity.com/paper/2307.13149/) (2023)
- [Exploring the Approximation Capabilities of Multiplicative Neural Networks for Smooth Functions](https://mlsanity.com/paper/2301.04605/) (2023)
- [A block-sparse Tensor Train Format for sample-efficient high-dimensional Polynomial Regression](https://mlsanity.com/paper/2104.14255/) (2021)
- [Interpretable Bayesian Tensor Network Kernel Machines with Automatic Rank and Feature Selection](https://mlsanity.com/paper/2507.11136/) (2025)
- [Tensor machines for learning target-specific polynomial features](https://mlsanity.com/paper/1504.01697/) (2015)

## Abstract (arXiv)

Central to machine learning and signal processing is the ability to perform universal function approximation and learn complex input-output relationships from limited numbers of observations. Multivariate polynomial models offer a natural way to express such relationships through multiplicative feature interactions, but their coefficient tensors grow exponentially in size with the polynomial degree. Existing tensorized polynomial models reduce this cost, yet canonical polyadic decompositions have rank-limited expressivity, and tensor train formulations are feature order dependent. We introduce Multivariate Polynomial Optimization based on Matrix Product Operators (MPO)2, a framework that combines learned MPO feature embeddings with compact polynomial weight tensors. This yields feature order independent polynomial representations that can incorporate structured operators such as projections, convolutions, and masks for weight tensor symmetries. Across regression and classification benchmarks, (MPO)2 improves over existing tensor decomposition based polynomial models and provides a flexible alternative for efficient polynomial function approximation.