Approximate Joint Diagonalization Algorithm Research Articles

An alternating least-squares algorithm for approximate joint diagonalization of Hermitian matrices

Approximate joint diagonalization (AJD) of a set of target matrices is one of the approaches for blind identification, blind source separation (BSS), and blind source extraction. This approach is useful for applications with measurement noise and/or estimation errors. This paper presents an alternating least-squares (ALS) algorithm for solving AJD problem. The ALS algorithm consists of two processing phases. The first processing phase works for finding diagonal matrices by minimizing the indirect least-squares criterion. A mixing matrix is estimated by minimizing the constrained direct least-squares criterion in the second processing phase. Our ALS algorithm can circumvent numerically unstable behavior by improving the condition number of the estimated mixing matrix in the second processing phase. The proposed algorithm is shown through experimental results to have better estimation performance and faster convergence than existing ALS algorithms with lower computational requirement in the rectangular mixing case.

Approximate Joint Singular Value Decomposition Algorithm Based on Givens-Like Rotation

An approximate joint singular value decomposition algorithm is proposed for a set of $K (K\geq 2)$ complex matrices. It can be seen as an orthogonal non-Hermitian approximate joint diagonalization algorithm. We exploit a Givens-like rotation method based on a special parameterization of the updating matrices and a reasonable approximation. The main points consist of the presentation of the new parameter structure, analytical derivation of the elementary updating matrices. High accuracy and fast convergence rate are obtained. Numerical simulations illustrate the overall good performance of the proposed algorithm.

Measurement-induced nonlocality in arbitrary dimensions in terms of the inverse approximate joint diagonalization

Here we focus on the measurement induced nonlocality and present a redefinition in terms of the skew information subject to a broken observable. It is shown that the obtained quantity possesses an obvious operational meaning, can tackle the noncontractivity of the measurement induced nonlocality and has analytic expressions for many quantum states. Most importantly, an inverse approximate joint diagonalization algorithm, due to its simplicity, high efficiency, stability, and state independence, is presented to provide almost analytic expressions for any quantum state, which can also shed light on other aspects in physics.

Approximate joint diagonalization and geometric mean of symmetric positive definite matrices.

We explore the connection between two problems that have arisen independently in the signal processing and related fields: the estimation of the geometric mean of a set of symmetric positive definite (SPD) matrices and their approximate joint diagonalization (AJD). Today there is a considerable interest in estimating the geometric mean of a SPD matrix set in the manifold of SPD matrices endowed with the Fisher information metric. The resulting mean has several important invariance properties and has proven very useful in diverse engineering applications such as biomedical and image data processing. While for two SPD matrices the mean has an algebraic closed form solution, for a set of more than two SPD matrices it can only be estimated by iterative algorithms. However, none of the existing iterative algorithms feature at the same time fast convergence, low computational complexity per iteration and guarantee of convergence. For this reason, recently other definitions of geometric mean based on symmetric divergence measures, such as the Bhattacharyya divergence, have been considered. The resulting means, although possibly useful in practice, do not satisfy all desirable invariance properties. In this paper we consider geometric means of covariance matrices estimated on high-dimensional time-series, assuming that the data is generated according to an instantaneous mixing model, which is very common in signal processing. We show that in these circumstances we can approximate the Fisher information geometric mean by employing an efficient AJD algorithm. Our approximation is in general much closer to the Fisher information geometric mean as compared to its competitors and verifies many invariance properties. Furthermore, convergence is guaranteed, the computational complexity is low and the convergence rate is quadratic. The accuracy of this new geometric mean approximation is demonstrated by means of simulations.

A Simple Nonorthogonal Approximate Joint Diagonalization Algorithm and Its Application for Blind Source Separation

A Simple Nonorthogonal Approximate Joint Diagonalization Algorithm and Its Application for Blind Source Separation

Coupled quasi‐harmonic bases

AbstractThe use of Laplacian eigenbases has been shown to be fruitful in many computer graphics applications. Today, state‐of‐the‐art approaches to shape analysis, synthesis, and correspondence rely on these natural harmonic bases that allow using classical tools from harmonic analysis on manifolds. However, many applications involving multiple shapes are obstacled by the fact that Laplacian eigenbases computed independently on different shapes are often incompatible with each other. In this paper, we propose the construction of common approximate eigenbases for multiple shapes using approximate joint diagonalization algorithms, taking as input a set of corresponding functions (e.g. indicator functions of stable regions) on the two shapes. We illustrate the benefits of the proposed approach on tasks from shape editing, pose transfer, correspondence, and similarity.

A Direct Algorithm for Nonorthogonal Approximate Joint Diagonalization

While a pair of N × N matrices can almost always be exactly and simultaneously diagonalized by a generalized eigendecomposition, no exact solution exists in the case of a set with more than two matrices. This problem, termed approximate joint diagonalization (AJD), is instrumental in blind signal processing. When the set of matrices to be jointly diagonalized includes at least N linearly independent matrices, we propose a suboptimal but closed-form solution for AJD in the direct least-squares sense. The corresponding non-iterative algorithm is given the acronym DIEM (DIagonalization using Equivalent Matrices). Extensive numerical simulations show that DIEM is both fast and accurate compared to the state-of-the-art iterative AJD algorithms.

A New Approximate Joint Diagonalization Algorithm Based on the Fused Complex-Valued Cost Function

A new algorithm to solve the fused complex-valued cost function for approximate joint diagonalization (AJD), named CVAJD (Complex-Valued Approximate Joint Diagonalization), is presented. The CVAJD algorithm adopts an iterative scheme to update the demixing matrix through the strictly diagonally-dominant residual mixing matrix obtained in each of iterations. Due to the relaxation of several constraints on the target matrices, it has more general utilizations. Besides, it is also easy to implement. A numerical simulation illustrates fast convergence and good performance of the CVAJD.

QML-Based Joint Diagonalization of Positive-Definite Hermitian Matrices

In this paper, a new algorithm for approximate joint diagonalization (AJD) of positive-definite Hermitian matrices is presented. The AJD matrix, which is assumed to be square non-unitary, is derived via minimization of a quasi-maximum likelihood (QML) objective function. This objective function coincides asymptotically with the maximum likelihood (ML) objective function, hence enabling the proposed algorithm to asymptotically approach the ML estimation performance. In the proposed method, the rows of the AJD matrix are obtained independently, in an iterative manner. This feature enables direct estimation of full row-rank rectangular AJD sub-matrices. Under some mild assumptions, convergence of the proposed algorithm is asymptotically guarantied, such that the error norm corresponding to each row of the AJD matrix reduces significantly after the first iteration, and the convergence is almost Q-super linear. This property results rapid convergence, which leads to low computational load in the proposed method. The performance of the proposed algorithm is evaluated and compared to other state-of-the-art algorithms for AJD and its practical use is demonstrated in the blind source separation and blind source extraction problems. The results imply that under the assumptions of high signal-to-noise ratio and large amount of matrices, the proposed algorithm is computationally efficient with performance similar to state-of-the-art algorithms for AJD.

Gradient Search Non-Orthogonal Approximate Joint Diagonalization Algorithm

The problem of approximate joint diagonalization of a set of matrices is instrumental in numerous statistical signal processing applications. This paper describes a relative gradient non-orthogonal approximate joint diagonalization (AJD) algorithm based on a non-least squares AJD criterion and a special AJD using a non-square diagonalizing matrix and an AJD method for ill-conditioned matrices. Simulation results demonstrate the better performance of the relative gradient AJD algorithm compared with the conventional least squares (LS) criteria based gradient-type AJD algorithms. The algorithm is attractive for practical applications since it is simple and efficient.

A Comparative Study of Approximate Joint Diagonalization Algorithms for Blind Source Separation in Presence of Additive Noise

A comparative study of approximate joint diagonalization algorithms of a set of matrices is presented. Using a weighted least-squares criterion, without the orthogonality constraint, an algorithm is compared with an analogous one for blind source separation (BSS). The criterion of the present algorithm is on the separating matrix while the other is on the mixing matrix. The convergence of the algorithm is proved under some mild assumptions. The performances of the two algorithms are compared with usual standard algorithms using BSS simulations results. We show that the improvement in estimating the separating matrix, resulting from the relaxation of the orthogonality restriction, can be achieved in presence of additive noise when the length of observed sequences is sufficiently large and when the mixing matrix is not close to an orthogonal matrix