# Efficient Nonnegative Tucker Decompositions: Algorithms and Uniqueness

arXiv: https://arxiv.org/abs/1404.4412 · HTML: https://mlsanity.com/paper/1404.4412/

## Key facts

- Cited by 17 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 Apr 2014.

## Built on by (most cited first, 17 of 17)

- [Uncovering audio patterns in music with Nonnegative Tucker Decomposition for structural segmentation](https://mlsanity.com/paper/2104.08580/) (2021)
- [Faster Robust Tensor Power Method for Arbitrary Order](https://mlsanity.com/paper/2306.00406/) (2023)
- [Unsupervised EHR-based Phenotyping via Matrix and Tensor Decompositions](https://mlsanity.com/paper/2209.00322/) (2022)
- [D-Tracker: Modeling Interest Diffusion in Social Activity Tensor Data Streams](https://mlsanity.com/paper/2505.00242/) (2025)
- [Bayesian Robust Tensor Ring Model for Incomplete Multiway Data](https://mlsanity.com/paper/2202.13321/) (2022)
- [Spatiotemporal Regularized Tucker Decomposition Approach for Traffic Data Imputation](https://mlsanity.com/paper/2305.06563/) (2023)
- [Efficient Algorithms for Regularized Nonnegative Scale-invariant Low-rank Approximation Models](https://mlsanity.com/paper/2403.18517/) (2024)
- [Identifiability of Nonnegative Tucker Decompositions -- Part I: Theory](https://mlsanity.com/paper/2505.12713/) (2025)
- [Near-Universal Multiplicative Updates for Nonnegative Einsum Factorization](https://mlsanity.com/paper/2602.02759/) (2026)
- [Nonnegative Tucker Decomposition with Beta-divergence for Music Structure Analysis of Audio Signals](https://mlsanity.com/paper/2110.14434/) (2021)
- [Doubly Non-Central Beta Matrix Factorization for Stable Dimensionality Reduction of Bounded Support Matrix Data](https://mlsanity.com/paper/2410.18425/) (2024)
- [Robust Low-Rank Tensor Completion based on M-product with Weighted Correlated Total Variation and Sparse Regularization](https://mlsanity.com/paper/2604.13525/) (2026)
- [Fast Hypergraph Regularized Nonnegative Tensor Ring Factorization Based on Low-Rank Approximation](https://mlsanity.com/paper/2109.02314/) (2021)
- [Optical Coherence Tomography Image Enhancement via Block Hankelization and Low Rank Tensor Network Approximation](https://mlsanity.com/paper/2306.11750/) (2023)
- [Tensor State Space-based Dynamic Multilayer Network Modeling](https://mlsanity.com/paper/2506.02413/) (2025)
- [Score-Based Model for Low-Rank Tensor Recovery](https://mlsanity.com/paper/2506.22295/) (2025)
- [Data Understanding Survey: Pursuing Improved Dataset Characterization Via Tensor-based Methods](https://mlsanity.com/paper/2510.14161/) (2025)

## Abstract (arXiv)

Nonnegative Tucker decomposition (NTD) is a powerful tool for the extraction of nonnegative parts-based and physically meaningful latent components from high-dimensional tensor data while preserving the natural multilinear structure of data. However, as the data tensor often has multiple modes and is large-scale, existing NTD algorithms suffer from a very high computational complexity in terms of both storage and computation time, which has been one major obstacle for practical applications of NTD. To overcome these disadvantages, we show how low (multilinear) rank approximation (LRA) of tensors is able to significantly simplify the computation of the gradients of the cost function, upon which a family of efficient first-order NTD algorithms are developed. Besides dramatically reducing the storage complexity and running time, the new algorithms are quite flexible and robust to noise because any well-established LRA approaches can be applied. We also show how nonnegativity incorporating sparsity substantially improves the uniqueness property and partially alleviates the curse of dimensionality of the Tucker decompositions. Simulation results on synthetic and real-world data justify the validity and high efficiency of the proposed NTD algorithms.