Non-negative matrix factorization is a popular tool for decomposing data into feature and weight matrices under non-negativity constraints. It enjoys practical success but is poorly understood theoretically. This paper proposes an algorithm that alternates between decoding the weights and updating the features, and shows that assuming a generative model of the data, it provably recovers the ground-truth under fairly mild conditions. In particular, its only essential requirement on features is linear independence. Furthermore, the algorithm uses ReLU to exploit the non-negativity for decoding the weights, and thus can tolerate adversarial noise that can potentially be as large as the signal, and can tolerate unbiased noise much larger than the signal. The analysis relies on a carefully designed coupling between two potential functions, which we believe is of independent interest.

Non-negative matrix factorization is a popular tool for decomposing data into feature and weight matrices under non-negativity constraints.

PSD factorizations are fundamentally linked to understanding the expressiveness of semidefinite programs as well as the power and limitations of quantum resources in information theory.

GAMMA_FLOW is an open-source Python package for real-time analysis of spectral data. It supports classification, denoising, decomposition, and outlier detection of both single- and multi-component spectra. Instead of relying on large, computationally intensive models, it employs a supervised approach to non-negative matrix factorization (NMF) for dimensionality reduction. This ensures a fast, efficient, and adaptable analysis while reducing computational costs. gamma_flow achieves classification accuracies above 90% and enables reliable automated spectral interpretation. Originally developed for gamma-ray spectra, it is applicable to any type of one-dimensional spectral data. As an open and flexible alternative to proprietary software, it supports various applications in research and industry.

First provide a summary of the paper, and then address the following criteria: Quality, clarity, originality and significance. This paper describes a non-negative matrix factorization for tall and skinny matrices. This algorithm works in the bigdata scenario because it need only pass over the tall skinny matrix one time. This linear read of the tall skinny matrix does not fully utilize the distributed mapreduce framework. I wonder, is it possible to parallelize the reading of the matrix and combine the results from subsets of the data into one final result?

Abstract-- The movements of both animals and robots give rise to streams of high-dimensional motor and sensory information. Imagine the brain of a newborn or the controller of a baby humanoid robot trying to make sense of unprocessed sensorimotor time series. Here, we present a framework for studying the dynamic functional connectivity between the multimodal sensory signals of a robotic agent to uncover an underlying structure. Using instantaneous mutual information, we capture the time-varying functional connectivity (FC) between proprioceptive, tactile, and visual signals, revealing the sensorimotor relationships. Using an infinite relational model, we identified sensorimotor modules and their evolving connectivity. To further interpret these dynamic interactions, we employed non-negative matrix factorization, which decomposed the connectivity patterns into additive factors and their corresponding temporal coefficients. These factors can be considered the agent's motion primitives or movement synergies that the agent can use to make sense of its sensorimotor space and later for behavior selection. In the future, the method can be deployed in robot learning as well as in the analysis of human movement trajectories or brain signals.

The global prevalence of diabetes, particularly type 2 diabetes mellitus (T2DM), is rapidly increasing, posing significant health and economic challenges. T2DM not only disrupts blood glucose regulation but also damages vital organs such as the heart, kidneys, eyes, nerves, and blood vessels, leading to substantial morbidity and mortality. In the US alone, the economic burden of diagnosed diabetes exceeded \$400 billion in 2022. Early detection of individuals at risk is critical to mitigating these impacts. While machine learning approaches for T2DM prediction are increasingly adopted, many rely on supervised learning, which is often limited by the lack of confirmed negative cases. To address this limitation, we propose a novel unsupervised framework that integrates Non-negative Matrix Factorization (NMF) with statistical techniques to identify individuals at risk of developing T2DM. Our method identifies latent patterns of multimorbidity and polypharmacy among diagnosed T2DM patients and applies these patterns to estimate the T2DM risk in undiagnosed individuals. By leveraging data-driven insights from comorbidity and medication usage, our approach provides an interpretable and scalable solution that can assist healthcare providers in implementing timely interventions, ultimately improving patient outcomes and potentially reducing the future health and economic burden of T2DM.