An Alternating Direction Method of Multipliers for Constrained Joint Diagonalization by Congruence (Invited Paper)

International audienceIn this paper, we address the problem of joint diagonalization by congruence (i.e. the canonical polyadic decomposition of semi-symmetric 3rd order tensors) subject to arbitrary convex constraints. Sufficient conditions for the existence of a solution are given. An efficient algorithm based on the Alternating Direction Method of Multipliers (ADMM) is then designed. ADMM provides an elegant approach for handling the additional constraint terms, while taking advantage of the structure of the objective function. Numerical tests on simulated matrices show the benefits of the proposed method for low signal to noise ratios. Simulations in the context of nuclear magnetic resonance spectroscopy are also provided

Nonnegative Compression for Semi-Nonnegative Independent Component Analysis

International audienceIn many Independent Component Analysis (ICA) problems the mixing matrix is nonnegative while the sources are unconstrained, giving rise to what we call hereafter the Semi-Nonnegative ICA (SN-ICA) problems. Exploiting the nonnegativity property can improve the ICA result. Besides, in some practical applications, the dimension of the observation space must be reduced. However, the classical dimension compression procedure, such as prewhitening, breaks the nonnegativity property of the compressed mixing matrix. In this paper, we introduce a new nonnegative compression method, which guarantees the nonnegativity of the compressed mixing matrix. Simulation results show its fast convergence property. An illustration of Blind Source Separation (BSS) of Magnetic Resonance Spectroscopy (MRS) data confirms the validity of the proposed method

Audio source separation for music in low-latency and high-latency scenarios

Aquesta tesi proposa mètodes per tractar les limitacions de les tècniques existents de separació de fonts musicals en condicions de baixa i alta latència. En primer lloc, ens centrem en els mètodes amb un baix cost computacional i baixa latència. Proposem l'ús de la regularització de Tikhonov com a mètode de descomposició de l'espectre en el context de baixa latència. El comparem amb les tècniques existents en tasques d'estimació i seguiment dels tons, que són passos crucials en molts mètodes de separació. A continuació utilitzem i avaluem el mètode de descomposició de l'espectre en tasques de separació de veu cantada, baix i percussió. En segon lloc, proposem diversos mètodes d'alta latència que milloren la separació de la veu cantada, gràcies al modelatge de components específics, com la respiració i les consonants. Finalment, explorem l'ús de correlacions temporals i anotacions manuals per millorar la separació dels instruments de percussió i dels senyals musicals polifònics complexes.Esta tesis propone métodos para tratar las limitaciones de las técnicas existentes de separación de fuentes musicales en condiciones de baja y alta latencia. En primer lugar, nos centramos en los métodos con un bajo coste computacional y baja latencia. Proponemos el uso de la regularización de Tikhonov como método de descomposición del espectro en el contexto de baja latencia. Lo comparamos con las técnicas existentes en tareas de estimación y seguimiento de los tonos, que son pasos cruciales en muchos métodos de separación. A continuación utilizamos y evaluamos el método de descomposición del espectro en tareas de separación de voz cantada, bajo y percusión. En segundo lugar, proponemos varios métodos de alta latencia que mejoran la separación de la voz cantada, gracias al modelado de componentes que a menudo no se toman en cuenta, como la respiración y las consonantes. Finalmente, exploramos el uso de correlaciones temporales y anotaciones manuales para mejorar la separación de los instrumentos de percusión y señales musicales polifónicas complejas.This thesis proposes specific methods to address the limitations of current music source separation methods in low-latency and high-latency scenarios. First, we focus on methods with low computational cost and low latency. We propose the use of Tikhonov regularization as a method for spectrum decomposition in the low-latency context. We compare it to existing techniques in pitch estimation and tracking tasks, crucial steps in many separation methods. We then use the proposed spectrum decomposition method in low-latency separation tasks targeting singing voice, bass and drums. Second, we propose several high-latency methods that improve the separation of singing voice by modeling components that are often not accounted for, such as breathiness and consonants. Finally, we explore using temporal correlations and human annotations to enhance the separation of drums and complex polyphonic music signals

Gradient Flow Based Matrix Joint Diagonalization for Independent Componenet Analysis

In this thesis, employing the theory of matrix Lie groups, we develop gradient based flows for the problem of Simultaneous or Joint Diagonalization (JD) of a set of symmetric matrices. This problem has applications in many fields especially in the field of Independent Component Analysis (ICA). We consider both orthogonal and non-orthogonal JD. We view the JD problem as minimization of a common quadric cost function on a matrix group. We derive gradient based flows together with suitable discretizations for minimization of this cost function on the Riemannian manifolds of O(n) and GL(n).\\ We use the developed JD methods to introduce a new class of ICA algorithms that sphere the data, however do not restrict the subsequent search for the un-mixing matrix to orthogonal matrices. These methods provide robust ICA algorithms in Gaussian noise by making effective use of both second and higher order statistics

Semi-nonnegative joint diagonalization by congruence and semi-nonnegative ICA

International audienceIn this paper, we focus on the Joint Diagonalization by Congruence (JDC) decomposition of a set of matrices, while imposing nonnegative constraints on the joint diagonalizer. The latter will be referred to the semi-nonnegative JDC fitting problem. This problem appears in semi-nonnegative Independent Component Analysis (ICA), say ICA involving nonnegative static mixtures, such as those encountered for instance in image processing and in magnetic resonance spectroscopy. In order to achieve the semi-nonnegative JDC decomposition, we propose two novel algorithms called ELS-ALSexp and CGexp, which optimize an unconstrained problem obtained by means of an exponential change of variable. The proposed methods are based on the line search strategy for which an analytic global plane search procedure has been considered. All derivatives have been jointly calculated in matrix form using the algebraic basis for matrix calculus and product operator properties. Our algorithms have been tested on synthetic arrays and the semi-nonnegative ICA problem is illustrated through simulations in magnetic resonance spectroscopy and in image processing. The numerical results show the benefit of using a priori information, such as nonnegativity