On the Relation Between Linear Autoencoders and Non-Negative Matrix Factorization for Mutational Signature Extraction

Since its introduction, non-negative matrix factorization (NMF) has been a popular tool for extracting interpretable, low-dimensional representations of high-dimensional data. However, several recent studies have proposed replacing NMF with autoencoders. The increasing popularity of autoencoders warrants an investigation on whether this replacement is in general valid and reasonable. Moreover, the exact relationship between non-negative autoencoders and NMF has not been thoroughly explored. Thus, a main aim of this study is to investigate in detail the relationship between autoencoders and NMF. We define a non-negative linear autoencoder, AE-NMF, which is mathematically equivalent with convex NMF, a constrained version of NMF. The performance of NMF and the non-negative linear autoencoder is compared within the context of mutational signature extraction from simulated and real-world cancer genomics data. We find that the reconstructions based on NMF are more accurate compared with AE-NMF, while the signatures extracted using both methods exhibit comparable consistency and performance when externally validated. These findings suggest that AE-NMF, the linear non-negative autoencoders investigated in this article, do not provide an improvement of NMF in the field of mutational signature extraction. Our study serves as a foundation for understanding the theoretical implication of replacing NMF with non-negative autoencoders.