# Log-based Sparse Nonnegative Matrix Factorization for Data Representation

arXiv: https://arxiv.org/abs/2204.10647 · HTML: https://mlsanity.com/paper/2204.10647/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 50 references that are in the map.
- First posted on arXiv 22 Apr 2022.

## Builds on (most cited first, 5 of 5)

- [Discriminative Ridge Machine: A Classifier for High-Dimensional Data or Imbalanced Data](https://mlsanity.com/paper/1904.07496/) (2019)
- [Kernel Two-Dimensional Ridge Regression for Subspace Clustering](https://mlsanity.com/paper/2011.01477/) (2020)
- [Nonnegative Matrix Factorization with Local Similarity Learning](https://mlsanity.com/paper/1907.04150/) (2019)
- [Two-Dimensional Semi-Nonnegative Matrix Factorization for Clustering](https://mlsanity.com/paper/2005.09229/) (2020)
- [A Unified Framework for Sparse Non-Negative Least Squares using Multiplicative Updates and the Non-Negative Matrix Factorization Problem](https://mlsanity.com/paper/1604.02181/) (2016)

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

- [Nonnegative Matrix Factorization in Dimensionality Reduction: A Survey](https://mlsanity.com/paper/2405.03615/) (2024)
- [Majorization-minimization for Sparse Nonnegative Matrix Factorization with the β-divergence](https://mlsanity.com/paper/2207.06316/) (2022)
- [Adaptively Topological Tensor Network for Multi-view Subspace Clustering](https://mlsanity.com/paper/2305.00716/) (2023)

## Abstract (arXiv)

Nonnegative matrix factorization (NMF) has been widely studied in recent years due to its effectiveness in representing nonnegative data with parts-based representations. For NMF, a sparser solution implies better parts-based representation.However, current NMF methods do not always generate sparse solutions.In this paper, we propose a new NMF method with log-norm imposed on the factor matrices to enhance the sparseness.Moreover, we propose a novel column-wisely sparse norm, named ℓ2,log-(pseudo) norm to enhance the robustness of the proposed method.The ℓ2,log-(pseudo) norm is invariant, continuous, and differentiable.For the ℓ2,log regularized shrinkage problem, we derive a closed-form solution, which can be used for other general problems.Efficient multiplicative updating rules are developed for the optimization, which theoretically guarantees the convergence of the objective value sequence.Extensive experimental results confirm the effectiveness of the proposed method, as well as the enhanced sparseness and robustness.