# Fast Tucker Rank Reduction for Non-Negative Tensors Using Mean-Field Approximation

arXiv: https://arxiv.org/abs/2103.02898 · HTML: https://mlsanity.com/paper/2103.02898/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 46 references that are in the map.
- First posted on arXiv 4 Mar 2021.

## Builds on (most cited first, 6 of 6)

- [Tensor Robust Principal Component Analysis with A New Tensor Nuclear Norm](https://mlsanity.com/paper/1804.03728/) (2018)
- [TensorLy: Tensor Learning in Python](https://mlsanity.com/paper/1610.09555/) (2016)
- [Tensor-based algorithms for image classification](https://mlsanity.com/paper/1910.02150/) (2019)
- [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)

## Built on by (most cited first, 2 of 2)

- [Many-body Approximation for Non-negative Tensors](https://mlsanity.com/paper/2209.15338/) (2022)
- [Fast Rank-1 NMF for Missing Data with KL Divergence](https://mlsanity.com/paper/2110.12595/) (2021)

## Abstract (arXiv)

We present an efficient low-rank approximation algorithm for non-negative tensors. The algorithm is derived from our two findings: First, we show that rank-1 approximation for tensors can be viewed as a mean-field approximation by treating each tensor as a probability distribution. Second, we theoretically provide a sufficient condition for distribution parameters to reduce Tucker ranks of tensors; interestingly, this sufficient condition can be achieved by iterative application of the mean-field approximation. Since the mean-field approximation is always given as a closed formula, our findings lead to a fast low-rank approximation algorithm without using a gradient method. We empirically demonstrate that our algorithm is faster than the existing non-negative Tucker rank reduction methods and achieves competitive or better approximation of given tensors.