Methods · Comparison with Alternative Feature Extraction Methods
Non-Negative Matrix Factorisation (NMF)
Survey sections: §6.9.1 Non-Negative Matrix Factorisation
Non-negative matrix factorisation (NMF; Lee & Seung, 1999) decomposes a non-negative activation matrix as , where both the feature dictionary (m is the preset number of features) and the coefficient matrix are constrained to have non-negative elements. Because all components are non-negative, each feature contributes positively to the final representation, making it easier to interpret.
Assumptions and requirements
Additionally, classical NMF’s optimisation process uses simple multiplicative update rules that preserve non-negativity. Moreover, under assumptions like separability (each latent feature appears in at least one sufficiently “pure” example), NMF guarantees that the recovered decomposition is unique (Donoho & Stodden, 2003). However, classical NMF is not suitable for neural network activations because they can have negative elements, requiring additional preprocessing steps that may distort the activation-space structure.
Evaluation considerations
Shafran et al. (2026) showed that Semi-NMF can recover features competitive with SAEs on small-scale evaluations, but systematic comparisons at large scale remain limited.
Limitations
NMF does not explicitly encourage feature sparsity, and scaling NMF to large dictionaries is computationally expensive.
Related methods
Key papers
Cited in §6.9.1.
- Daniel D. Lee and H. Sebastian Seung. Learning the parts of objects by non-negative matrix factorization. Nature, 401 (6755): 788–791, 1999. .
- David Donoho and Victoria Stodden. When does non-negative matrix factorization give a correct decomposition into parts? In S. Thrun, L. Saul, and B. Schölkopf (eds.), Advances in Neural Information Processing Systems, volume 16. MIT Press, 2003. URL https://proceedings.neurips.cc/paper_files/paper/2003/file/1843e35d41ccf6e63273495ba42df3c1-Paper.pdf.
- Chris H.Q. Ding, Tao Li, and Michael I. Jordan. Convex and semi-nonnegative matrix factorizations. IEEE Transactions on Pattern Analysis and Machine Intelligence, 32 (1): 45–55, 2010. .
- Or David Shafran, Atticus Geiger, and Mor Geva. Constructing interpretable features from compositional neuron groups. In Maria Liakata, Viviane P. Moreira, Jiajun Zhang, and David Jurgens (eds.), Proceedings of the 64th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers), pp. 42326–42348, San Diego, California, United States, July 2026. Association for Computational Linguistics. ISBN 979-8-89176-390-6. . URL https://aclanthology.org/2026.acl-long.1959/.
- PreprintZeyu Yun, Yubei Chen, Bruno Olshausen, and Yann LeCun. Transformer visualization via dictionary learning: contextualized embedding as a linear superposition of transformer factors. In Eneko Agirre, Marianna Apidianaki, and Ivan Vulić (eds.), Proceedings of Deep Learning Inside Out (DeeLIO): The 2nd Workshop on Knowledge Extraction and Integration for Deep Learning Architectures, pp. 1–10, Online, June 2021. Association for Computational Linguistics. . URL https://aclanthology.org/2021.deelio-1.1/.
- Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova. BERT: Pre-training of deep bidirectional transformers for language understanding. In Jill Burstein, Christy Doran, and Thamar Solorio (eds.), Proceedings of the 2019 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Volume 1 (Long and Short Papers), pp. 4171–4186, Minneapolis, Minnesota, June 2019. Association for Computational Linguistics. . URL https://aclanthology.org/N19-1423/.
Other methods in Comparison with Alternative Feature Extraction Methods
Principal Component Analysis (PCA) · Independent Component Analysis (ICA) · Supervised Feature Extraction Methods (DAS, SMDS, probes) · Dictionary Learning Baselines (K-SVD)