# An Advance on Variable Elimination with Applications to Tensor-Based Computation

arXiv: https://arxiv.org/abs/2002.09320 · HTML: https://mlsanity.com/paper/2002.09320/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 21 Feb 2020.

## Built on by (most cited first, 10 of 10)

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- [Dolphin: A Programmable Framework for Scalable Neurosymbolic Learning](https://mlsanity.com/paper/2410.03348/) (2024)
- [Probabilistic Inference in the Era of Tensor Networks and Differential Programming](https://mlsanity.com/paper/2405.14060/) (2024)
- [Identifying Causal Effects Under Functional Dependencies](https://mlsanity.com/paper/2403.04919/) (2024)
- [An Algorithm and Complexity Results for Causal Unit Selection](https://mlsanity.com/paper/2302.14412/) (2023)
- [Constrained Identifiability of Causal Effects](https://mlsanity.com/paper/2412.02869/) (2024)
- [Combining Local Symmetry Exploitation and Reinforcement Learning for Optimised Probabilistic Inference -- A Work In Progress](https://mlsanity.com/paper/2503.08786/) (2025)
- [Causal Neural Probabilistic Circuits](https://mlsanity.com/paper/2603.01372/) (2026)

## Abstract (arXiv)

We present new results on the classical algorithm of variable elimination, which underlies many algorithms including for probabilistic inference. The results relate to exploiting functional dependencies, allowing one to perform inference and learning efficiently on models that have very large treewidth. The highlight of the advance is that it works with standard (dense) factors, without the need for sparse factors or techniques based on knowledge compilation that are commonly utilized. This is significant as it permits a direct implementation of the improved variable elimination algorithm using tensors and their operations, leading to extremely efficient implementations especially when learning model parameters. Moreover, the proposed technique does not require knowledge of the specific functional dependencies, only that they exist, so can be used when learning these dependencies. We illustrate the efficacy of our proposed algorithm by compiling Bayesian network queries into tensor graphs and then learning their parameters from labeled data using a standard tool for tensor computation.