Primal-Dual Algorithms for Non-negative Matrix Factorization with the Kullback-Leibler Divergence

Non-negative matrix factorization (NMF) approximates a given matrix as a product of two non-negative matrices. Multiplicative algorithms deliver reliable results, but they show slow convergence for high-dimensional data and may be stuck away from local minima. Gradient descent methods have better behavior, but only apply to smooth losses such as the least-squares loss. In this article, we propose a first-order primal-dual algorithm for non-negative decomposition problems (where one factor is fixed) with the KL divergence, based on the Chambolle-Pock algorithm. All required computations may be obtained in closed form and we provide an efficient heuristic way to select step-sizes. By using alternating optimization, our algorithm readily extends to NMF and, on synthetic examples, face recognition or music source separation datasets, it is either faster than existing algorithms, or leads to improved local optima, or both

State-dependent brainstem ensemble dynamics and their interactions with hippocampus across sleep states

The brainstem plays a crucial role in sleep-wake regulation. However, the ensemble dynamics underlying sleep regulation remain poorly understood. Here, we show slow, state-predictive brainstem ensemble dynamics and state-dependent interactions between the brainstem and the hippocampus in mice. On a timescale of seconds to minutes, brainstem populations can predict pupil dilation and vigilance states and exhibit longer prediction power than hippocampal CA1 neurons. On a timescale of sub-seconds, pontine waves (P-waves) are accompanied by synchronous firing of brainstem neurons during both rapid eye movement (REM) and non-REM (NREM) sleep. Crucially, P-waves functionally interact with CA1 activity in a state-dependent manner: during NREM sleep, hippocampal sharp wave-ripples (SWRs) precede P-waves. On the other hand, P-waves during REM sleep are phase-locked with ongoing theta oscillations and are followed by burst firing of CA1 neurons. This state-dependent global coordination between the brainstem and hippocampus implicates distinct functional roles of sleep

On the separation of T Tauri star spectra using non-negative matrix factorization and Bayesian positive source separation

The objective of this study is to compare and evaluate Bayesian and deterministic methods of positive source separation of young star spectra. In the Bayesian approach, the proposed Bayesian Positive Source Separation (BPSS) method uses Gamma priors to enforce non-negativity in the source signals and mixing coefficients and a Markov Chain Monte Carlo (MCMC) algorithm, modified by suggesting simpler proposal distributions and randomly initializing the MCMC to correctly separate spectra. In the deterministic approach, two Non-negative Matrix Factorization (NNMF) algorithms, the multiplicative update rule algorithm and an alternating least squares algorithm, are used to separate the star spectra into sources. The BPSS and NNMF algorithms are applied to the field of Astrophysics by applying the source separation techniques to T Tauri star spectra, resulting in a successful decomposition of the spectra into their sources. These methods are for the first time being applied and evaluated in optical spectroscopy. The results show that, while both methods perform well, BPSS outperforms NNMF. The NNMF and BPSS algorithms improve upon the current methodology used in Astrophysics iu two important ways. First, they permit the identification of additional components of the spectra in addition to the photosphere and boundary layer which can be modeled with current methods. Second, by applying a statistical algorithm, the modeling of T Tauri stars becomes less subjective. These methods may be further extrapolated to model spectra from other types of stars or astrophysical phenomena

Spatial Mass Spectral Data Analysis Using Factor and Correlation Models

ToF-SIMS is a powerful and information rich tool with high resolution and sensitivity compared to conventional mass spectrometers. Recently, its application has been extended to metabolic profiling analysis. However, there are only a few algorithms currently available to handle such output data from metabolite samples. Therefore some novel and innovative algorithms are undoubtedly in need to provide new insights into the application of ToF-SIMS for metabolic profiling analysis. In this thesis, we develop novel multivariate analysis techniques that can be used in processing ToF-SIMS data extracted from metabolite samples. Firstly, several traditional multivariate analysis methodologies that have previously been suggested for ToF-SIMS data analysis are discussed, including Clustering, Principal Components Analysis (PCA), Maximum Autocorrelation Factor (MAF), and Multivariate Curve Resolution (MCR). In particular, PCA is selected as an example to show the performance of traditional multivariate analysis techniques in dealing with large ToF-SIMS data extracted from metabolite samples. In order to provide more realistic and meaningful interpretation of the results, Non-negative Matrix Factorisation (NMF) is presented. This algorithm is combined with the Bayesian Framework to improve the reliability of the results and the convergence of the algorithm. However, the iterative process involved leads to considerable computational complexity in the estimation procedure. Another novel algorithm is also proposed which is an optimised MCR algorithm within alternating non-negativity constrained least squares (ANLS) framework. It provides a more simple approximation procedure by implementing a dimensionality reduction based on a basis function decomposition approach. The novel and main feature of the proposed algorithm is that it incorporates a spatially continuous representation of ToF-SIMS data which decouples the computational complexity of the estimation procedure from the image resolution. The proposed algorithm can be used as an efficient tool in processing ToF-SIMS data obtained from metabolite samples