A neural networks learning algorithm for minor component analysis and its convergence analysis

The eigenvector associated with the smallest eigenvalue of the autocorrelation matrix of input signals is called minor component. Minor component analysis (MCA) is a statistical approach for extracting minor component from input signals and has been applied in many fields of signal processing and data analysis. In this letter, we propose a neural networks learning algorithm for estimating adaptively minor component from input signals. Dynamics of the proposed algorithm are analyzed via a deterministic discrete time (DDT) method. Some sufficient conditions are obtained to guarantee convergence of the proposed algorithm.

On the discrete time dynamics of a self-stabilizing MCA learning algorithm

The stability of minor component analysis (MCA) learning algorithms is an important problem in many signal processing applications. In this paper, we propose an effective MCA learning algorithm that can offer better stability. The dynamics of the proposed algorithm are analyzed via a corresponding deterministic discrete time (DDT) system. It is proven that if the learning rate satisfies some mild conditions, almost all trajectories of the DDT system starting from points in an invariant set are bounded, and will converge to the minor component of the autocorrelation matrix of the input data. Simulation results will be furnished to illustrate the theoretical results achieved.

A stable MCA learning algorithm

Minor component analysis (MCA) is an important statistical tool for signal processing and data analysis. Neural networks can be used to extract online minor component from input data. Compared with traditional algebraic  approaches, a neural network method has a lower computational complexity. Stability of neural networks learning algorithms is crucial to practical applications. In this paper, we propose a stable MCA neural networks learning algorithm, which has a more satisfactory numerical stability than some existing MCA algorithms. Dynamical behaviors of the proposed algorithm are analyzed via deterministic discrete time (DDT) method and the conditions are obtained to guarantee convergence. Simulations are carried out to illustrate the theoretical results achieved.

Stability and convergence analysis of transform-domain LMS adaptive filters with second-order autoregressive process

In this paper, the stability and convergence properties of the class of transform-domain least mean square (LMS) adaptive filters with second-order autoregressive (AR) process are investigated. It is well known that this class of adaptive filters improve convergence property of the standard LMS adaptive filters by applying the fixed data-independent orthogonal transforms and power normalization. However, the convergence performance of this class of adaptive filters can be quite different for various input processes, and it has not been fully explored. In this paper, we first discuss the mean-square stability and steady-state performance of this class of adaptive filters. We then analyze the effects of the transforms and power normalization performed in the various adaptive filters for both first-order and second-order AR processes. We derive the input asymptotic eigenvalue distributions and make comparisons on their convergence performance. Finally, computer simulations on AR process as well as moving-average (MA) process and autoregressive-moving-average (ARMA) process are demonstrated for the support of the analytical results.

On performance of transform domain adaptive filters with Markov-2 inputs

In this paper, the analysis for the performance of the discrete Fourier transform LMS adaptive filter (DFT-LMS) and the discrete cosine transform LMS adaptive filter (DCT-LMS) for the Markov-2 inputs is presented. To improve the convergence property of the least mean squares (LMS) adaptive filter, the DFT-LMS and DCT-LMS preprocess the inputs with the fixed orthogonal transforms and power normalization. We derive the asymptotic results for the eigenvalues and eigenvalue distributions of the preprocessed input autocorrelation matrices with DFT-LMS and DCT-LMS for Markov-2 inputs. These results explicitly show the superior decorrelation property of DCT-LMS over that of DFT-LMS, and also provide the upper bounds for the eigenvalue spreads of the finite-length DFT-LMS and DCT-LMS adaptive filters. Simulation results are demonstrated to support the analytic results.

A unified learning algorithm to extract principal and minor components

Recently, many unified learning algorithms have been developed to solve the task of principal component analysis (PCA) and minor component analysis (MCA). These unified algorithms can be used to extract principal component and if altered simply by the sign, it can also serve as a minor component extractor. This is of practical significance in the implementations of algorithms. Convergence of the existing unified algorithms is guaranteed only under the condition that the learning rates of algorithms approach zero, which is impractical in many practical applications. In this paper, we propose a unified PCA & MCA algorithm with a constant learning rate, and derive the sufficient conditions to guarantee convergence via analyzing the discrete-time dynamics of the proposed algorithm. The achieved theoretical results lay a solid foundation for the applications of our proposed algorithm.

Underdetermined blind source separation based on relaxed sparsity condition of sources

Recently, Aissa-El-Bey et al. have proposed two subspacebased methods for underdetermined blind source separation (UBSS) in time-frequency (TF) domain. These methods allow multiple active sources at TF points so long as the number of active sources at any TF point is strictly less than the number of sensors, and the column vectors of the mixing matrix are pairwise linearly independent. In this correspondence, we first show that the subspace-based methods must also satisfy the condition that any M × M submatrix of the mixing matrix is of full rank. Then we present a new UBSS approach which only requires that the number of active sources at any TF point does not exceed that of sensors. An algorithm is proposed to perform the UBSS.

A globally convergent MC algorithm with an adaptive learning rate

This brief deals with the problem of minor component analysis (MCA). Artificial neural networks can be exploited to achieve the task of MCA. Recent research works show that convergence of neural networks based MCA algorithms can be guaranteed if the learning rates are less than certain thresholds. However, the computation of these thresholds needs information about the eigenvalues of the autocorrelation matrix of data set, which is unavailable in online extraction of minor component from input data stream. In this correspondence, we introduce an adaptive learning rate into the OJAn MCA algorithm, such that its convergence condition does not depend on any unobtainable information, and can be easily satisfied in practical applications.

Investigating nanoparticle-substrate interaction in LSPR biosensing using the image-charge theory

Localized surface plasmon resonance (LSPR) has been used to develop optical biosensors. Tuning the resonance wavelength to detect target biomolecules with a particular dipolar resonance is essential when designing LSPR biosensors. In this paper, the interaction of nanoparticles (NPs) with glass substrate (SiO2) for LSPR wavelength is investigated using the concept of the image-charge theory. Using the FDTD method, it is shown how the NP and substrate size change the plasmon wavelength. Next, this phenomenon is interpreted using the analytical electrostatic eigenvalue method.

Face set classification using maximally probable mutual modes

In this paper we consider face recognition from sets of face images and, in particular, recognition invariance to illumination. The main contribution is an algorithm based on the novel concept of maximally probable mutual modes (MMPM). Specifically: (i) we discuss and derive a local manifold illumination invariant and (ii) show how the invariant naturally leads to a formulation of "common modes" of two face appearance distributions. Recognition is then performed by finding the most probable mode, which is shown to be an eigenvalue problem. The effectiveness of the proposed method is demonstrated empirically on a challenging database containing the total of 700 video sequences of 100 individuals

Impact of high wind penetration on the voltage profile of distribution systems

In this paper, simulation results showing the effect of lower and higher penetration of distributed wind generation on the voltage profile in distribution systems have been presented. The analysis is carried out over two distribution test systems. The detailed mathematical modeling of the system is also presented. It also investigates the small-signal stability of distribution systems using eigenvalue approach. The analyses show that voltage variation problems occur in different nodes of the distribution networks with an increase of penetration level. However, proper selection of dispersion level can improve the voltage profile of the distribution systems

JLMC: A clustering method based on Jordan-Form of Laplacian-Matrix

Among the current clustering algorithms of complex networks, Laplacian-based spectral clustering algorithms have the advantage of rigorous mathematical basis and high accuracy. However, their applications are limited due to their dependence on prior knowledge, such as the number of clusters. For most of application scenarios, it is hard to obtain the number of clusters beforehand. To address this problem, we propose a novel clustering algorithm - Jordan-Form of Laplacian-Matrix based Clustering algorithm (JLMC). In JLMC, we propose a model to calculate the number (n) of clusters in a complex network based on the Jordan-Form of its corresponding Laplacian matrix. JLMC clusters the network into n clusters by using our proposed modularity density function (P function). We conduct extensive experiments over real and synthetic data, and the experimental results reveal that JLMC can accurately obtain the number of clusters in a complex network, and outperforms Fast-Newman algorithm and Girvan-Newman algorithm in terms of clustering accuracy and time complexity.