MM-algorithms for traditional and convex NMF with Tweedie and Negative Binomial cost functions and empirical evaluation

Non-negative matrix factorisation (NMF) is a widely used tool for unsupervised learning and feature extraction, with applications ranging from genomics to text analysis and signal processing. Standard formulations of NMF are typically derived under Gaussian or Poisson noise assumptions, which may be inadequate for data exhibiting overdispersion or other complex mean-variance relationships. In this paper, we develop a unified framework for both traditional and convex NMF under a broad class of distributional assumptions, including Negative Binomial and Tweedie models, where the connection between the Tweedie and the $β$-divergence is also highlighted. Using a Majorize-Minimisation approach, we derive multiplicative update rules for all considered models, and novel updates for convex NMF with Poisson and Negative Binomial cost functions. We provide a unified implementation of all considered models, including the first implementations of several convex NMF models. Empirical evaluations on mutational and word count data demonstrate that the choice of noise model critically affects model fit and feature recovery, and that convex NMF can provide an efficient and robust alternative to traditional NMF in scenarios where the number of classes is large. The code for our proposed updates is available in the R package nmfgenr and can be found at https://github.com/MartaPelizzola/nmfgenr.

Advanced posterior analyses of hidden Markov models: finite Markov chain imbedding and hybrid decoding

Two major tasks in applications of hidden Markov models are to (i) compute distributions of summary statistics of the hidden state sequence, and (ii) decode the hidden state sequence. We describe finite Markov chain imbedding (FMCI) and hybrid decoding to solve each of these two tasks. In the first part of our paper we use FMCI to compute posterior distributions of summary statistics such as the number of visits to a hidden state, the total time spent in a hidden state, the dwell time in a hidden state, and the longest run length. We use simulations from the hidden state sequence, conditional on the observed sequence, to establish the FMCI framework. In the second part of our paper we apply hybrid segmentation for improved decoding of a HMM. We demonstrate that hybrid decoding shows increased performance compared to Viterbi or Posterior decoding (often also referred to as global or local decoding), and we introduce a novel procedure for choosing the tuning parameter in the hybrid procedure. Furthermore, we provide an alternative derivation of the hybrid loss function based on weighted geometric means. We demonstrate and apply FMCI and hybrid decoding on various classical data sets, and supply accompanying code for reproducibility.

On the Relation Between Autoencoders and Non-negative Matrix Factorization, and Their Application for Mutational Signature Extraction

The aim of this study is to provide a foundation to understand the relationship between non-negative matrix factorization (NMF) and non-negative autoencoders enabling proper interpretation and understanding of autoencoder-based alternatives to NMF. Since its introduction, NMF has been a popular tool for extracting interpretable, low-dimensional representations of high-dimensional data. However, recently, several studies have proposed to replace NMF with autoencoders. This 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 non-negative autoencoders and NMF. We find that the connection between the two models can be established through convex NMF, which is a restricted case of NMF. In particular, convex NMF is a special case of an autoencoder. The performance of NMF and autoencoders is compared within the context of extraction of mutational signatures from cancer genomics data. We find that the reconstructions based on NMF are more accurate compared to autoencoders, while the signatures extracted using both methods show comparable consistencies and values when externally validated. These findings suggest that the non-negative autoencoders investigated in this article do not provide an improvement of NMF in the field of mutational signature extraction.