Dictionary-based Tensor Canonical Polyadic Decomposition

To ensure interpretability of extracted sources in tensor decomposition, we introduce in this paper a dictionary-based tensor canonical polyadic decomposition which enforces one factor to belong exactly to a known dictionary. A new formulation of sparse coding is proposed which enables high dimensional tensors dictionary-based canonical polyadic decomposition. The benefits of using a dictionary in tensor decomposition models are explored both in terms of parameter identifiability and estimation accuracy. Performances of the proposed algorithms are evaluated on the decomposition of simulated data and the unmixing of hyperspectral images

Using separable non-negative matrix factorization techniques for the analysis of time-resolved Raman spectra

The key challenge of time-resolved Raman spectroscopy is the identification of the constituent species and the analysis of the kinetics of the underlying reaction network. In this work we present an integral approach that allows for determining both the component spectra and the rate constants simultaneously from a series of vibrational spectra. It is based on an algorithm for non-negative matrix factorization which is applied to the experimental data set following a few pre-processing steps. As a prerequisite for physically unambiguous solutions, each component spectrum must include one vibrational band that does not significantly interfere with vibrational bands of other species. The approach is applied to synthetic "experimental" spectra derived from model systems comprising a set of species with component spectra differing with respect to their degree of spectral interferences and signal-to-noise ratios. In each case, the species involved are connected via monomolecular reaction pathways. The potential and limitations of the approach for recovering the respective rate constants and component spectra are discussed

A review on initialization methods for nonnegative matrix factorization: Towards omics data experiments

Nonnegative Matrix Factorization (NMF) has acquired a relevant role in the panorama of knowledge extraction, thanks to the peculiarity that non-negativity applies to both bases and weights, which allows meaningful interpretations and is consistent with the natural human part-based learning process. Nevertheless, most NMF algorithms are iterative, so initialization methods affect convergence behaviour, the quality of the final solution, and NMF performance in terms of the residual of the cost function. Studies on the impact of NMF initialization techniques have been conducted for text or image datasets, but very few considerations can be found in the literature when biological datasets are studied, even though NMFs have largely demonstrated their usefulness in better understanding biological mechanisms with omic datasets. This paper aims to present the state-of-the-art on NMF initialization schemes along with some initial considerations on the impact of initialization methods when microarrays (a simple instance of omic data) are evaluated with NMF mechanisms. Using a series of measures to qualitatively examine the biological information extracted by a given NMF scheme, it preliminary appears that some information (e.g., represented by genes) can be extracted regardless of the initialization scheme used

Random mixtures of polycyclic aromatic hydrocarbon spectra match interstellar infrared emission

The mid-infrared (IR; 5-15~$\mu$m) spectrum of a wide variety of astronomical objects exhibits a set of broad emission features at 6.2, 7.7, 8.6, 11.3 and 12.7 $\mu$m. About 30 years ago it was proposed that these signatures are due to emission from a family of UV heated nanometer-sized carbonaceous molecules known as polycyclic aromatic hydrocarbons (PAHs), causing them to be referred to as aromatic IR bands (AIBs). Today, the acceptance of the PAH model is far from settled, as the identification of a single PAH in space has not yet been successful and physically relevant theoretical models involving ``true'' PAH cross sections do not reproduce the AIBs in detail. In this paper, we use the NASA Ames PAH IR Spectroscopic Database, which contains over 500 quantum-computed spectra, in conjunction with a simple emission model, to show that the spectrum produced by any random mixture of at least 30 PAHs converges to the same 'kernel'-spectrum. This kernel-spectrum captures the essence of the PAH emission spectrum and is highly correlated with observations of AIBs, strongly supporting PAHs as their source. Also, the fact that a large number of molecules are required implies that spectroscopic signatures of the individual PAHs contributing to the AIBs spanning the visible, near-infrared, and far infrared spectral regions are weak, explaining why they have not yet been detected. An improved effort, joining laboratory, theoretical, and observational studies of the PAH emission process, will support the use of PAH features as a probe of physical and chemical conditions in the nearby and distant Universe

Nonnegative Matrix Factorization for Signal and Data Analytics: Identifiability, Algorithms, and Applications

Nonnegative matrix factorization (NMF) has become a workhorse for signal and data analytics, triggered by its model parsimony and interpretability. Perhaps a bit surprisingly, the understanding to its model identifiability---the major reason behind the interpretability in many applications such as topic mining and hyperspectral imaging---had been rather limited until recent years. Beginning from the 2010s, the identifiability research of NMF has progressed considerably: Many interesting and important results have been discovered by the signal processing (SP) and machine learning (ML) communities. NMF identifiability has a great impact on many aspects in practice, such as ill-posed formulation avoidance and performance-guaranteed algorithm design. On the other hand, there is no tutorial paper that introduces NMF from an identifiability viewpoint. In this paper, we aim at filling this gap by offering a comprehensive and deep tutorial on model identifiability of NMF as well as the connections to algorithms and applications. This tutorial will help researchers and graduate students grasp the essence and insights of NMF, thereby avoiding typical `pitfalls' that are often times due to unidentifiable NMF formulations. This paper will also help practitioners pick/design suitable factorization tools for their own problems.Comment: accepted version, IEEE Signal Processing Magazine; supplementary materials added. Some minor revisions implemente