# Many-body Approximation for Non-negative Tensors

arXiv: https://arxiv.org/abs/2209.15338 · HTML: https://mlsanity.com/paper/2209.15338/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 54 references that are in the map.
- First posted on arXiv 30 Sep 2022.

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

- [Tensor Ring Decomposition](https://mlsanity.com/paper/1606.05535/) (2016)
- [TensorLy: Tensor Learning in Python](https://mlsanity.com/paper/1610.09555/) (2016)
- [Tree Tensor Networks for Generative Modeling](https://mlsanity.com/paper/1901.02217/) (2019)
- [Efficient tensor completion for color image and video recovery: Low-rank tensor train](https://mlsanity.com/paper/1606.01500/) (2016)
- [Tensor Methods in Computer Vision and Deep Learning](https://mlsanity.com/paper/2107.03436/) (2021)
- [Supervised Learning with Projected Entangled Pair States](https://mlsanity.com/paper/2009.09932/) (2020)
- [Tensor Balancing on Statistical Manifold](https://mlsanity.com/paper/1702.08142/) (2017)
- [Legendre Decomposition for Tensors](https://mlsanity.com/paper/1802.04502/) (2018)
- [Kullback-Leibler Principal Component for Tensors is not NP-hard](https://mlsanity.com/paper/1711.07925/) (2017)
- [Graph Regularized Nonnegative Tensor Ring Decomposition for Multiway Representation Learning](https://mlsanity.com/paper/2010.05657/) (2020)
- [Fast Tucker Rank Reduction for Non-Negative Tensors Using Mean-Field Approximation](https://mlsanity.com/paper/2103.02898/) (2021)
- [Fast Rank-1 NMF for Missing Data with KL Divergence](https://mlsanity.com/paper/2110.12595/) (2021)

## Built on by (most cited first, 3 of 3)

- [A Complete Decomposition of KL Error using Refined Information and Mode Interaction Selection](https://mlsanity.com/paper/2410.11964/) (2024)
- [Near-Universal Multiplicative Updates for Nonnegative Einsum Factorization](https://mlsanity.com/paper/2602.02759/) (2026)
- [Pseudo-Nonlinear Data Augmentation: A Constrained Energy Minimization Viewpoint](https://mlsanity.com/paper/2410.00718/) (2024)

## Abstract (arXiv)

We present an alternative approach to decompose non-negative tensors, called many-body approximation. Traditional decomposition methods assume low-rankness in the representation, resulting in difficulties in global optimization and target rank selection. We avoid these problems by energy-based modeling of tensors, where a tensor and its mode correspond to a probability distribution and a random variable, respectively. Our model can be globally optimized in terms of the KL divergence minimization by taking the interaction between variables (that is, modes), into account that can be tuned more intuitively than ranks. Furthermore, we visualize interactions between modes as tensor networks and reveal a nontrivial relationship between many-body approximation and low-rank approximation. We demonstrate the effectiveness of our approach in tensor completion and approximation.