Low rank approximation method for perturbed linear systems with applications to elliptic type stochastic PDEs

In this paper, we propose a low rank approximation method for efficiently solving stochastic partial differential equations. Specifically, our method utilizes a novel low rank approximation of the stiffness matrices, which can significantly reduce the computational load and storage requirements associated with matrix inversion without losing accuracy. To demonstrate the versatility and applicability of our method, we apply it to address two crucial uncertainty quantification problems: stochastic elliptic equations and optimal control problems governed by stochastic elliptic PDE constraints. Based on varying dimension reduction ratios, our algorithm exhibits the capability to yield a high precision numerical solution for stochastic partial differential equations, or provides a rough representation of the exact solutions as a pre-processing phase. Meanwhile, our algorithm for solving stochastic optimal control problems allows a diverse range of gradient-based unconstrained optimization methods, rendering it particularly appealing for computationally intensive large-scale problems. Numerical experiments are conducted and the results provide strong validation of the feasibility and effectiveness of our algorithm

Some statistical approaches to the analysis of matrix-valued data

In many modern applications, we encounter data sampled in the form of two-dimensional matrices. Simple vectorization of the matrix-valued observations would destroy the intrinsic row and column information embedded in such data. In this research, we study three statistical problems that are specific to matrix-valued data. The first one concerns dimension reduction for a group of high-dimensional matrix-valued data. We propose a novel dimension reduction approach that has nice approximation property, computes fast for high dimensionality, and also explicitly incorporates the intrinsic two-dimensional structure of the matrices. We discuss the connection of our proposal with existing approaches, and compare them both numerically and theoretically. We also obtain theoretical upper bounds on the approximation error of our method. The second one is a group independent component analysis approach. Motivated by analysis of groups of high-dimensional imaging data, we develop a framework in the frequency domain through Whittle log-likelihood maximization. Our method starts with efficient population value decomposition, and then models each temporally-dependent source signal via parametric linear processes. The superior performance of our approach is demonstrated through simulation studies and the ADHD200 data. The third one addresses the problem of regression with matrix-valued covariates. We consider the bilinear regression model, where two coefficient vectors are used to incorporate matrix covariates. We propose two maximum likelihood based estimators. Both estimators are shown to achieve the information lower bound and hence are theoretically optimal under the classical asymptotic framework. We further propose a bilinear ridge estimator and derive its convergence property. The superior performances of the proposed estimators are demonstrated both theoretically and numerically.Doctor of Philosoph

Connected Attribute Filtering Based on Contour Smoothness

A new attribute measuring the contour smoothness of 2-D objects is presented in the context of morphological attribute filtering. The attribute is based on the ratio of the circularity and non-compactness, and has a maximum of 1 for a perfect circle. It decreases as the object boundary becomes irregular. Computation on hierarchical image representation structures relies on five auxiliary data members and is rapid. Contour smoothness is a suitable descriptor for detecting and discriminating man-made structures from other image features. An example is demonstrated on a very-high-resolution satellite image using connected pattern spectra and the switchboard platform

State of the Art in Face Recognition

Notwithstanding the tremendous effort to solve the face recognition problem, it is not possible yet to design a face recognition system with a potential close to human performance. New computer vision and pattern recognition approaches need to be investigated. Even new knowledge and perspectives from different fields like, psychology and neuroscience must be incorporated into the current field of face recognition to design a robust face recognition system. Indeed, many more efforts are required to end up with a human like face recognition system. This book tries to make an effort to reduce the gap between the previous face recognition research state and the future state

Principal Component Analysis

This book is aimed at raising awareness of researchers, scientists and engineers on the benefits of Principal Component Analysis (PCA) in data analysis. In this book, the reader will find the applications of PCA in fields such as image processing, biometric, face recognition and speech processing. It also includes the core concepts and the state-of-the-art methods in data analysis and feature extraction