Geometrical methods for non-negative ICA: Manifolds, Lie groups and toral subalgebras

We explore the use of geometrical methods to tackle the non-negative independent component analysis (non-negative ICA) problem, without assuming the reader has an existing background in differential geometry. We concentrate on methods that achieve this by minimizing a cost function over the space of orthogonal matrices. We introduce the idea of the manifold and Lie group SO(n) of special orthogonal matrices that we wish to search over, and explain how this is related to the Lie algebra so(n) of skew-symmetric matrices. We describe how familiar optimization methods such as steepest-descent and conjugate gradients can be transformed into this Lie group setting, and how the Newton update step has an alternative Fourier version in SO(n). Finally we introduce the concept of a toral subgroup generated by a particular element of the Lie group or Lie algebra, and explore how this commutative subgroup might be used to simplify searches on our constraint surface. No proofs are presented in this article

Non-negative Independent Component Analysis Algorithm Based on 2D Givens Rotations and a Newton Optimization

ISBN 978-3-642-15994-7, SoftcoverInternational audienceIn this paper, we consider the Independent Component Analysis problem when the hidden sources are non-negative (Non-negative ICA). This problem is formulated as a non-linear cost function optimization over the special orthogonal matrix group SO(n). Using Givens rotations and Newton optimization, we developed an effective axis pair rotation method for Non-negative ICA. The performance of the proposed method is compared to those designed by Plumbley and simulations on synthetic data show the efficiency of the proposed algorithm

Semi-Supervised Sound Source Localization Based on Manifold Regularization

Conventional speaker localization algorithms, based merely on the received microphone signals, are often sensitive to adverse conditions, such as: high reverberation or low signal to noise ratio (SNR). In some scenarios, e.g. in meeting rooms or cars, it can be assumed that the source position is confined to a predefined area, and the acoustic parameters of the environment are approximately fixed. Such scenarios give rise to the assumption that the acoustic samples from the region of interest have a distinct geometrical structure. In this paper, we show that the high dimensional acoustic samples indeed lie on a low dimensional manifold and can be embedded into a low dimensional space. Motivated by this result, we propose a semi-supervised source localization algorithm which recovers the inverse mapping between the acoustic samples and their corresponding locations. The idea is to use an optimization framework based on manifold regularization, that involves smoothness constraints of possible solutions with respect to the manifold. The proposed algorithm, termed Manifold Regularization for Localization (MRL), is implemented in an adaptive manner. The initialization is conducted with only few labelled samples attached with their respective source locations, and then the system is gradually adapted as new unlabelled samples (with unknown source locations) are received. Experimental results show superior localization performance when compared with a recently presented algorithm based on a manifold learning approach and with the generalized cross-correlation (GCC) algorithm as a baseline

Spectral methods for multimodal data analysis

Spectral methods have proven themselves as an important and versatile tool in a wide range of problems in the fields of computer graphics, machine learning, pattern recognition, and computer vision, where many important problems boil down to constructing a Laplacian operator and finding a few of its eigenvalues and eigenfunctions. Classical examples include the computation of diffusion distances on manifolds in computer graphics, Laplacian eigenmaps, and spectral clustering in machine learning. In many cases, one has to deal with multiple data spaces simultaneously. For example, clustering multimedia data in machine learning applications involves various modalities or ``views'' (e.g., text and images), and finding correspondence between shapes in computer graphics problems is an operation performed between two or more modalities. In this thesis, we develop a generalization of spectral methods to deal with multiple data spaces and apply them to problems from the domains of computer graphics, machine learning, and image processing. Our main construction is based on simultaneous diagonalization of Laplacian operators. We present an efficient numerical technique for computing joint approximate eigenvectors of two or more Laplacians in challenging noisy scenarios, which also appears to be the first general non-smooth manifold optimization method. Finally, we use the relation between joint approximate diagonalizability and approximate commutativity of operators to define a structural similarity measure for images. We use this measure to perform structure-preserving color manipulations of a given image

Advanced optimization algorithms for sensor arrays and multi-antenna communications

Optimization problems arise frequently in sensor array and multi-channel signal processing applications. Often, optimization needs to be performed subject to a matrix constraint. In particular, unitary matrices play a crucial role in communications and sensor array signal processing. They are involved in almost all modern multi-antenna transceiver techniques, as well as sensor array applications in biomedicine, machine learning and vision, astronomy and radars. In this thesis, algorithms for optimization under unitary matrix constraint stemming from Riemannian geometry are developed. Steepest descent (SD) and conjugate gradient (CG) algorithms operating on the Lie group of unitary matrices are derived. They have the ability to find the optimal solution in a numerically efficient manner and satisfy the constraint accurately. Novel line search methods specially tailored for this type of optimization are also introduced. The proposed approaches exploit the geometrical properties of the constraint space in order to reduce the computational complexity. Array and multi-channel signal processing techniques are key technologies in wireless communication systems. High capacity and link reliability may be achieved by using multiple transmit and receive antennas. Combining multi-antenna techniques with multicarrier transmission leads to high the spectral efficiency and helps to cope with severe multipath propagation. The problem of channel equalization in MIMO-OFDM systems is also addressed in this thesis. A blind algorithm that optimizes of a combined criterion in order to be cancel both inter-symbol and co-channel interference is proposed. The algorithm local converge properties are established as well

Non-Negative Blind Source Separation Algorithm Based on Minimum Aperture Simplicial Cone

International audienceWe address the problem of Blind Source Separation (BSS) when the hidden sources are Nonnegative (N-BSS). In this case, the scatter plot of the mixed data is contained within the simplicial cone generated by the columns of the mixing matrix. The proposed method, termed SCSA-UNS for Simplicial Cone Shrinking Algorithm for Unmixing Non-negative Sources, aims at estimating the mixing matrix and the sources by fitting a Minimum Aperture Simplicial Cone (MASC) to the cloud of mixed data points. SCSA-UNS is evaluated on both independent and correlated synthetic data and compared to other N-BSS methods. Simulations are also performed on real Liquid Chromatography-Mass Spectrum (LC-MS) data for the metabolomic analysis of a chemical sample, and on real dynamic Positron Emission Tomography (PET) images, in order to study the pharmacokinetics of the [18F]-FDG (FluoroDeoxyGlucose) tracer in the brain