# Cons-training Tensor Networks: Embedding and Optimization Over Discrete Linear Constraints

arXiv: https://arxiv.org/abs/2405.09005 · HTML: https://mlsanity.com/paper/2405.09005/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 33 references that are in the map.
- First posted on arXiv 15 May 2024.

## Builds on (most cited first, 12 of 12)

- [Unsupervised Generative Modeling Using Matrix Product States](https://mlsanity.com/paper/1709.01662/) (2017)
- [Learning Relevant Features of Data with Multi-scale Tensor Networks](https://mlsanity.com/paper/1801.00315/) (2017)
- [Variational Neural Annealing](https://mlsanity.com/paper/2101.10154/) (2021)
- [Machine Learning by Unitary Tensor Network of Hierarchical Tree Structure](https://mlsanity.com/paper/1710.04833/) (2017)
- [JuMP 1.0: Recent improvements to a modeling language for mathematical optimization](https://mlsanity.com/paper/2206.03866/) (2022)
- [Supervised Learning with Projected Entangled Pair States](https://mlsanity.com/paper/2009.09932/) (2020)
- [Anomaly Detection with Tensor Networks](https://mlsanity.com/paper/2006.02516/) (2020)
- [Privacy-preserving machine learning with tensor networks](https://mlsanity.com/paper/2202.12319/) (2022)
- [A quantum-inspired tensor network method for constrained combinatorial optimization problems](https://mlsanity.com/paper/2203.15246/) (2022)
- [Global Optimization of Mixed-Integer Nonlinear Programs with SCIP 8](https://mlsanity.com/paper/2301.00587/) (2023)
- [Tensor Network Contractions for #SAT](https://mlsanity.com/paper/1405.7375/) (2014)
- [Fast counting with tensor networks](https://mlsanity.com/paper/1805.00475/) (2018)

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

- [Tensor Network Estimation of Distribution Algorithms](https://mlsanity.com/paper/2412.19780/) (2024)

## Abstract (arXiv)

In this study, we introduce a novel family of tensor networks, termed constrained matrix product states (MPS), designed to incorporate exactly arbitrary discrete linear constraints, including inequalities, into sparse block structures. These tensor networks are particularly tailored for modeling distributions with support strictly over the feasible space, offering benefits such as reducing the search space in optimization problems, alleviating overfitting, improving training efficiency, and decreasing model size. Central to our approach is the concept of a quantum region, an extension of quantum numbers traditionally used in U(1) symmetric tensor networks, adapted to capture any linear constraint, including the unconstrained scenario. We further develop a novel canonical form for these new MPS, which allow for the merging and factorization of tensor blocks according to quantum region fusion rules and permit optimal truncation schemes. Utilizing this canonical form, we apply an unsupervised training strategy to optimize arbitrary objective functions subject to discrete linear constraints. Our method's efficacy is demonstrated by solving the quadratic knapsack problem, achieving superior performance compared to a leading nonlinear integer programming solver. Additionally, we analyze the complexity and scalability of our approach, demonstrating its potential in addressing complex constrained combinatorial optimization problems.