# On solving symmetric multi-type orthogonal non-negative matrix tri-factorization problem

arXiv: https://arxiv.org/abs/2606.08291 · HTML: https://mlsanity.com/paper/2606.08291/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 46 references that are in the map.
- First posted on arXiv 6 Jun 2026.

## Builds on (most cited first, 8 of 8)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [node2vec: Scalable Feature Learning for Networks](https://mlsanity.com/paper/1607.00653/) (2016)
- [Revisiting Semi-Supervised Learning with Graph Embeddings](https://mlsanity.com/paper/1603.08861/) (2016)
- [Stochastic subgradient method converges on tame functions](https://mlsanity.com/paper/1804.07795/) (2018)
- [Network Representation Learning: From Preprocessing, Feature Extraction to Node Embedding](https://mlsanity.com/paper/2110.07582/) (2021)
- [A block inertial Bregman proximal algorithm for nonsmooth nonconvex problems with application to symmetric nonnegative matrix tri-factorization](https://mlsanity.com/paper/2003.03963/) (2020)
- [A Block Coordinate Descent-based Projected Gradient Algorithm for Orthogonal Non-negative Matrix Factorization](https://mlsanity.com/paper/2003.10269/) (2020)
- [Four algorithms to solve symmetric multi-type non-negative matrix tri-factorization problem](https://mlsanity.com/paper/2012.05963/) (2020)

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

- [Degree-Corrected Joint Matrix Factorization for Multilayer Community Detection](https://mlsanity.com/paper/2610.01361/) (2026)

## Abstract (arXiv)

We study the symmetric multi-type orthogonal non-negative matrix tri-factorization problem, where several symmetric non-negative matrices are simultaneously approximated by factors of the form GSiGtop, with a shared non-negative and orthogonal factor G. This model is motivated by clustering and network analysis, where non-negativity improves interpretability and orthogonality gives a natural assignment-type structure to the latent factor. Since the resulting optimization problem is highly non-convex, we develop two heuristic algorithms for computing high-quality local solutions. The first one is a fixed point method derived from the Karush-Kuhn-Tucker conditions after adding a penalty term for the orthogonality constraint. The second one is a three-stage ADAM-based method that combines non-negativity-preserving optimization, orthogonalization, and restricted ADAM refinement on the feasible set. We evaluate both methods on synthetic data, including noisy instances, and on citation network benchmarks. The synthetic experiments show that both algorithms recover factorizations close to the optimum and remain stable under noise. On real networks, the learned embeddings are competitive with or better than standard baselines such as SVD, node2vec, and classical link prediction heuristics in link prediction, node clustering, and node classification tasks.