# Fast Rank-1 NMF for Missing Data with KL Divergence

arXiv: https://arxiv.org/abs/2110.12595 · HTML: https://mlsanity.com/paper/2110.12595/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 3 of its 38 references that are in the map.
- First posted on arXiv 25 Oct 2021.

## Builds on (most cited first, 3 of 3)

- [Tensor Balancing on Statistical Manifold](https://mlsanity.com/paper/1702.08142/) (2017)
- [Fast Tucker Rank Reduction for Non-Negative Tensors Using Mean-Field Approximation](https://mlsanity.com/paper/2103.02898/) (2021)
- [A Symmetric Rank-one Quasi Newton Method for Non-negative Matrix Factorization](https://mlsanity.com/paper/1305.5829/) (2013)

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

- [Many-body Approximation for Non-negative Tensors](https://mlsanity.com/paper/2209.15338/) (2022)

## Abstract (arXiv)

We propose a fast non-gradient-based method of rank-1 non-negative matrix factorization (NMF) for missing data, called A1GM, that minimizes the KL divergence from an input matrix to the reconstructed rank-1 matrix. Our method is based on our new finding of an analytical closed-formula of the best rank-1 non-negative multiple matrix factorization (NMMF), a variety of NMF. NMMF is known to exactly solve NMF for missing data if positions of missing values satisfy a certain condition, and A1GM transforms a given matrix so that the analytical solution to NMMF can be applied. We empirically show that A1GM is more efficient than a gradient method with competitive reconstruction errors.