# A consistent and flexible framework for deep matrix factorizations

arXiv: https://arxiv.org/abs/2206.10693 · HTML: https://mlsanity.com/paper/2206.10693/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 47 references that are in the map.
- First posted on arXiv 21 Jun 2022.

## Builds on (most cited first, 13 of 13)

- [Implicit Regularization in Deep Matrix Factorization](https://mlsanity.com/paper/1905.13655/) (2019)
- [Nonnegative Matrix Factorization for Signal and Data Analytics: Identifiability, Algorithms, and Applications](https://mlsanity.com/paper/1803.01257/) (2018)
- [A globally convergent algorithm for nonconvex optimization based on block coordinate update](https://mlsanity.com/paper/1410.1386/) (2014)
- [Robust Volume Minimization-Based Matrix Factorization for Remote Sensing and Document Clustering](https://mlsanity.com/paper/1608.04290/) (2016)
- [A deep matrix factorization method for learning attribute representations](https://mlsanity.com/paper/1509.03248/) (2015)
- [Simplex-Structured Matrix Factorization: Sparsity-based Identifiability and Provably Correct Algorithms](https://mlsanity.com/paper/2007.11446/) (2020)
- [Successive Nonnegative Projection Algorithm for Robust Nonnegative Blind Source Separation](https://mlsanity.com/paper/1310.7529/) (2013)
- [An Inertial Block Majorization Minimization Framework for Nonsmooth Nonconvex Optimization](https://mlsanity.com/paper/2010.12133/) (2020)
- [Hierarchical Clustering of Hyperspectral Images using Rank-Two Nonnegative Matrix Factorization](https://mlsanity.com/paper/1310.7441/) (2013)
- [Sparse and spurious: dictionary learning with noise and outliers](https://mlsanity.com/paper/1407.5155/) (2014)
- [Deep matrix factorizations](https://mlsanity.com/paper/2010.00380/) (2020)
- [Explicit Group Sparse Projection with Applications to Deep Learning and NMF](https://mlsanity.com/paper/1912.03896/) (2019)
- [Deep Self-representative Concept Factorization Network for Representation Learning](https://mlsanity.com/paper/1912.06444/) (2019)

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

- [Deep Nonnegative Matrix Factorization with Beta Divergences](https://mlsanity.com/paper/2309.08249/) (2023)
- [Hyperspectral Unmixing Hierarchies](https://mlsanity.com/paper/2604.16969/) (2026)
- [Supervised Deep Multimodal Matrix Factorization for Interpretable Brain Network Analysis](https://mlsanity.com/paper/2605.13312/) (2026)

## Abstract (arXiv)

Deep matrix factorizations (deep MFs) are recent unsupervised data mining techniques inspired by constrained low-rank approximations. They aim to extract complex hierarchies of features within high-dimensional datasets. Most of the loss functions proposed in the literature to evaluate the quality of deep MF models and the underlying optimization frameworks are not consistent because different losses are used at different layers. In this paper, we introduce two meaningful loss functions for deep MF and present a generic framework to solve the corresponding optimization problems. We illustrate the effectiveness of this approach through the integration of various constraints and regularizations, such as sparsity, nonnegativity and minimum-volume. The models are successfully applied on both synthetic and real data, namely for hyperspectral unmixing and extraction of facial features.