A Unified Framework for Sparse Non-Negative Least Squares using Multiplicative Updates and the Non-Negative Matrix Factorization Problem

We study the sparse non-negative least squares (S-NNLS) problem. S-NNLS occurs naturally in a wide variety of applications where an unknown, non-negative quantity must be recovered from linear measurements. We present a unified framework for S-NNLS based on a rectified power exponential scale mixture prior on the sparse codes. We show that the proposed framework encompasses a large class of S-NNLS algorithms and provide a computationally efficient inference procedure based on multiplicative update rules. Such update rules are convenient for solving large sets of S-NNLS problems simultaneously, which is required in contexts like sparse non-negative matrix factorization (S-NMF). We provide theoretical justification for the proposed approach by showing that the local minima of the objective function being optimized are sparse and the S-NNLS algorithms presented are guaranteed to converge to a set of stationary points of the objective function. We then extend our framework to S-NMF, showing that our framework leads to many well known S-NMF algorithms under specific choices of prior and providing a guarantee that a popular subclass of the proposed algorithms converges to a set of stationary points of the objective function. Finally, we study the performance of the proposed approaches on synthetic and real-world data.Comment: To appear in Signal Processin

Single-Channel Signal Separation Using Spectral Basis Correlation with Sparse Nonnegative Tensor Factorization

A novel approach for solving the single-channel signal separation is presented the proposed sparse nonnegative tensor factorization under the framework of maximum a posteriori probability and adaptively fine-tuned using the hierarchical Bayesian approach with a new mixing mixture model. The mixing mixture is an analogy of a stereo signal concept given by one real and the other virtual microphones. An “imitated-stereo” mixture model is thus developed by weighting and time-shifting the original single-channel mixture. This leads to an artificial mixing system of dual channels which gives rise to a new form of spectral basis correlation diversity of the sources. Underlying all factorization algorithms is the principal difficulty in estimating the adequate number of latent components for each signal. This paper addresses these issues by developing a framework for pruning unnecessary components and incorporating a modified multivariate rectified Gaussian prior information into the spectral basis features. The parameters of the imitated-stereo model are estimated via the proposed sparse nonnegative tensor factorization with Itakura–Saito divergence. In addition, the separability conditions of the proposed mixture model are derived and demonstrated that the proposed method can separate real-time captured mixtures. Experimental testing on real audio sources has been conducted to verify the capability of the proposed method

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