# Heterogeneous Tensor Decomposition for Clustering via Manifold Optimization

arXiv: https://arxiv.org/abs/1504.01777 · HTML: https://mlsanity.com/paper/1504.01777/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Apr 2015.

## Built on by (most cited first, 10 of 10)

- [Manifold optimization for non-linear optimal transport problems](https://mlsanity.com/paper/2103.00902/) (2021)
- [Riemannian block SPD coupling manifold and its application to optimal transport](https://mlsanity.com/paper/2201.12933/) (2022)
- [Nonnegative Matrix Factorization in Dimensionality Reduction: A Survey](https://mlsanity.com/paper/2405.03615/) (2024)
- [HyperNTF: A Hypergraph Regularized Nonnegative Tensor Factorization for Dimensionality Reduction](https://mlsanity.com/paper/2101.06827/) (2021)
- [Riemannian Optimization on Relaxed Indicator Matrix Manifold](https://mlsanity.com/paper/2503.20505/) (2025)
- [HFN: Heterogeneous Feature Network for Multivariate Time Series Anomaly Detection](https://mlsanity.com/paper/2211.00277/) (2022)
- [Multiway clustering of 3-order tensor via affinity matrix](https://mlsanity.com/paper/2303.07757/) (2023)
- [A Riemannian Approach to Low-Rank Optimal Transport](https://mlsanity.com/paper/2606.12120/) (2026)
- [Riemannian Manifold Optimization for Discriminant Subspace Learning](https://mlsanity.com/paper/2101.08032/) (2021)
- [Robust Manifold Nonnegative Tucker Factorization for Tensor Data Representation](https://mlsanity.com/paper/2211.03934/) (2022)

## Abstract (arXiv)

Tensors or multiarray data are generalizations of matrices. Tensor clustering has become a very important research topic due to the intrinsically rich structures in real-world multiarray datasets. Subspace clustering based on vectorizing multiarray data has been extensively researched. However, vectorization of tensorial data does not exploit complete structure information. In this paper, we propose a subspace clustering algorithm without adopting any vectorization process. Our approach is based on a novel heterogeneous Tucker decomposition model. In contrast to existing techniques, we propose a new clustering algorithm that alternates between different modes of the proposed heterogeneous tensor model. All but the last mode have closed-form updates. Updating the last mode reduces to optimizing over the so-called multinomial manifold, for which we investigate second order Riemannian geometry and propose a trust-region algorithm. Numerical experiments show that our proposed algorithm compete effectively with state-of-the-art clustering algorithms that are based on tensor factorization.