Thierry Chonavel

Mai-Quyen Pham, Jérémy Cohen, Thierry Chonavel

Nonnegative Matrix Factorization (NMF) is a fundamental tool in unsupervised learning, widely used for tasks such as dimensionality reduction, feature extraction, representation learning, and topic modeling. Many algorithms have been developed for NMF, including the well-known Multiplicative Updates (MU) algorithm, which belongs to a broader class of majorization-minimization techniques. In this work, we introduce a general second-order optimization framework for NMF under both quadratic and $β$-divergence loss functions. This approach, called Second-Order Majorant (SOM), constructs a local quadratic majorization of the loss function by majorizing its Hessian matrix. It includes MU as a special case, while enabling faster variants. In particular, we propose mSOM, a new algorithm within this class that leverages a tighter local approximation to accelerate convergence. We provide a convergence analysis, showing linear convergence for individual factor updates and global convergence to a stationary point for the alternating version, AmSOM algorithm. Numerical experiments on both synthetic and real data sets demonstrate that mSOM consistently outperforms state-of-the-art algorithms across multiple loss functions.

Maxime Mouchet, Sandrine Vaton, Thierry Chonavel et al.

Round-Trip Times are one of the most commonly collected performance metrics in computer networks. Measurement platforms such as RIPE Atlas provide researchers and network operators with an unprecedented amount of historical Internet delay measurements. It would be very useful to automate the processing of these measurements (statistical characterization of paths performance, change detection, recognition of recurring patterns, etc.). Humans are pretty good at finding patterns in network measurements but it can be difficult to automate this to enable many time series being processed at the same time. In this article we introduce a new model, the HDP-HMM or infinite hidden Markov model, whose performance in trace segmentation is very close to human cognition. This is obtained at the cost of a greater complexity and the ambition of this article is to make the theory accessible to network monitoring and management researchers. We demonstrate that this model provides very accurate results on a labeled dataset and on RIPE Atlas and CAIDA MANIC data. This method has been implemented in Atlas and we introduce the publicly accessible Web API.