# Fast and Structured Block-Term Tensor Decomposition For Hyperspectral Unmixing

arXiv: https://arxiv.org/abs/2205.03798 · HTML: https://mlsanity.com/paper/2205.03798/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 64 references that are in the map.
- First posted on arXiv 8 May 2022.

## Builds on (most cited first, 19 of 19)

- [Tensor Decomposition for Signal Processing and Machine Learning](https://mlsanity.com/paper/1607.01668/) (2016)
- [Nonnegative Matrix Factorization for Signal and Data Analytics: Identifiability, Algorithms, and Applications](https://mlsanity.com/paper/1803.01257/) (2018)
- [On Identifiability of Nonnegative Matrix Factorization](https://mlsanity.com/paper/1709.00614/) (2017)
- [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)
- [Nonlinear unmixing of hyperspectral images: models and algorithms](https://mlsanity.com/paper/1304.1875/) (2013)
- [Identifiability of the Simplex Volume Minimization Criterion for Blind Hyperspectral Unmixing: The No Pure-Pixel Case](https://mlsanity.com/paper/1406.5273/) (2014)
- [Block stochastic gradient iteration for convex and nonconvex optimization](https://mlsanity.com/paper/1408.2597/) (2014)
- [Successive Nonnegative Projection Algorithm for Robust Nonnegative Blind Source Separation](https://mlsanity.com/paper/1310.7529/) (2013)
- [Spectrum Cartography via Coupled Block-Term Tensor Decomposition](https://mlsanity.com/paper/1911.12468/) (2019)
- [Semiblind Hyperspectral Unmixing in the Presence of Spectral Library Mismatches](https://mlsanity.com/paper/1507.01661/) (2015)
- [Block-Randomized Stochastic Proximal Gradient for Low-Rank Tensor Factorization](https://mlsanity.com/paper/1901.05529/) (2019)
- [Self-Dictionary Sparse Regression for Hyperspectral Unmixing: Greedy Pursuit and Pure Pixel Search are Related](https://mlsanity.com/paper/1409.4320/) (2014)
- [Hyperspectral Super-Resolution via Interpretable Block-Term Tensor Modeling](https://mlsanity.com/paper/2006.10248/) (2020)
- [On the Convergence of Projected-Gradient Methods with Low-Rank Projections for Smooth Convex Minimization over Trace-Norm Balls and Related Problems](https://mlsanity.com/paper/1902.01644/) (2019)
- [Identifiability-Guaranteed Simplex-Structured Post-Nonlinear Mixture Learning via Autoencoder](https://mlsanity.com/paper/2106.09070/) (2021)
- [A Framework for One-Bit and Constant-Envelope Precoding over Multiuser Massive MISO Channels](https://mlsanity.com/paper/1810.03159/) (2018)
- [Memory-Efficient Convex Optimization for Self-Dictionary Separable Nonnegative Matrix Factorization: A Frank-Wolfe Approach](https://mlsanity.com/paper/2109.11135/) (2021)
- [On Hyperspectral Unmixing](https://mlsanity.com/paper/2106.14177/) (2021)

## Built on by (most cited first, 6 of 6)

- [Quantized Radio Map Estimation Using Tensor and Deep Generative Models](https://mlsanity.com/paper/2303.01770/) (2023)
- [Hyperspectral Anomaly Detection Fused Unified Nonconvex Tensor Ring Factors Regularization](https://mlsanity.com/paper/2505.17881/) (2025)
- [Rethinking Coupled Tensor Analysis for Hyperspectral Super-Resolution: Recoverable Modeling Under Endmember Variability](https://mlsanity.com/paper/2512.19489/) (2025)
- [Unregistered Spectral Image Fusion: Unmixing, Adversarial Learning, and Recoverability](https://mlsanity.com/paper/2603.21510/) (2026)
- [Hyperspectral Unmixing Under Endmember Variability: A Variational Inference Framework](https://mlsanity.com/paper/2407.14899/) (2024)
- [Variational Low-rank Tensor Decomposition for Multisubject Spatiotemporal Data Analysis](https://mlsanity.com/paper/2607.22262/) (2026)

## Abstract (arXiv)

The block-term tensor decomposition model with multilinear rank-(Lr,Lr,1) terms (or, the "LL1 tensor decomposition" in short) offers a valuable alternative for hyperspectral unmixing (HU) under the linear mixture model. Particularly, the LL1 decomposition ensures the endmember/abundance identifiability in scenarios where such guarantees are not supported by the classic matrix factorization (MF) approaches. However, existing LL1-based HU algorithms use a three-factor parameterization of the tensor (i.e., the hyperspectral image cube), which leads to a number of challenges including high per-iteration complexity, slow convergence, and difficulties in incorporating structural prior information. This work puts forth an LL1 tensor decomposition-based HU algorithm that uses a constrained two-factor re-parameterization of the tensor data. As a consequence, a two-block alternating gradient projection (GP)-based LL1 algorithm is proposed for HU. With carefully designed projection solvers, the GP algorithm enjoys a relatively low per-iteration complexity. Like in MF-based HU, the factors under our parameterization correspond to the endmembers and abundances. Thus, the proposed framework is natural to incorporate physics-motivated priors that arise in HU. The proposed algorithm often attains orders-of-magnitude speedup and substantial HU performance gains compared to the existing three-factor parameterization-based HU algorithms.