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

Abstract

In 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-ALS exp and CG exp , 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.

An Alternating Direction Method of Multipliers for constrained joint diagonalization by congruence

“…The indicator of a set S of a Hilbert space H is defined as (∀x ∈ H) ι S (x) = 0 if x ∈ S, and +∞ otherwise. It is noteworthy that the minimization problem (8) can be rewritten similarly to (5) where J = 5 and the matrices (Q T j Q j ) 1≤j≤5 correspond to isomorphisms. The augmented Lagrangian function associated with (8) is given by…”

“…Then, the nonnegativity constraint can be viewed as a limit case when A min = 0 and A max → +∞. Its use in the context of Nonnegative JDC (NJDC) was investigated through derivativebased [4], [5], alternating least squares [5], [19] and Jacobilike [20] optimization algorithms. Regarding the choice of D, because of the imposed diagonal structure for the matrices (D (k) ) 1≤k≤K , this set must have the form…”

Nonnegative compression for Semi-Nonnegative Independent Component Analysis

“…The Semi-Nonnegative Independent Component Analysis (SeNICA) problem is defined as follows [1]- [3]: Problem 1. Given M realizations of a real N -dimensional random vector x, find an (N ×P ) full column rank mixing matrix A and the corresponding M realizations of P -dimensional source random vector s, such that:…”

“…However, such a compression method can not guarantee the nonnegativity of the compressed mixing matrix A = W T A, because generally the unitary matrix W is not nonnegative. Now, in some practical situations, exploiting the nonnegativity property of the mixing matrix can improve the ICA result [1]- [3], [6]- [8]. Therefore, the objective of this paper is to propose a nonnegative compression method that guarantees the nonnegativity of the compressed mixing matrix A.…”

“…Indeed, the (P × P ) compressed mixing matrix A = W T A is yet nonnegative. Then A and the M realizations of s can be estimated using a SeNICA method such as one of those described in [1]- [3], [6], [7]. Therefore it is not necessary to estimate the original mixing matrix A.…”