Face image super-resolution using 2D CCA

In this paper a face super-resolution method using two-dimensional canonical correlation analysis (2D CCA) is presented. A detail compensation step is followed to add high-frequency components to the reconstructed high-resolution face. Unlike most of the previous researches on face super-resolution algorithms that first transform the images into vectors, in our approach the relationship between the high-resolution and the low-resolution face image are maintained in their original 2D representation. In addition, rather than approximating the entire face, different parts of a face image are super-resolved separately to better preserve the local structure. The proposed method is compared with various state-of-the-art super-resolution algorithms using multiple evaluation criteria including face recognition performance. Results on publicly available datasets show that the proposed method super-resolves high quality face images which are very close to the ground-truth and performance gain is not dataset dependent. The method is very efficient in both the training and testing phases compared to the other approaches. © 2013 Elsevier B.V

Copula-based high dimensional dependence modelling

University of Technology Sydney. Faculty of Engineering and Information Technology.Big data applications increasingly involve high-dimensional and sophisticated dependence structures in complex data. Modelling high-dimensional dependence, that is, the dependence between a set of high-dimensional variables, is a critical but challenging issue in many applications including social media analysis and financial markets. A typical example concerns the interplay of financial variables involved in driving complex market movements. A particular problem is understanding the dependence between high-dimensional variables with tail dependence and asymmetric characteristics which appear widely in financial markets. Typically, existing methods, such as the Bayesian logic program, relational dependency networks and relational Markov networks, build a graph to represent the conditional dependence structure between random variables. These models aim at high-dimensional domains, and have the advantage of learning latent relationships from data. However, they tend to force the local quantitative part of the model to take a simple form such as the discretized form of the data when multivariate Gaussian or its mixtures cannot capture the data in the real world. The complex dependencies between high-dimensional variables are difficult to capture. In statistics and finance, the copula has been shown to be a powerful tool for modelling high-dimensional dependencies. The copula splits the multivariate marginal distributions from dependence structures, so that the specification of dependence structures can be investigated independently of the marginal distributions. It can provide a flexible mechanism for modelling real world distributions that cannot be handled well by graphical models. Thus, researchers have tried to combine copula and probability graphical models, such as the tree-structured copula model and copula Bayesian networks. These copula-based models aim to resolve the limitations of discretizing data, but they impose assumptions and restrictions on the dependence structure. These assumptions and restrictions are not appropriate for dependence modelling among financial variables. In order to address these research limitations and challenges, this thesis proposes the use of the truncated partial correlation-based canonical vine copula, partial correlation-based regular vine copula and truncated partial correlation-based regular vine copula to model the dependence of high-dimensional variables. Chapter 3 introduces a new partial correlation-based canonical vine to identify the asymmetric and non-linear dependence structures of asset returns without any prior dependence assumptions. To simplify the model while maintaining its merit, a partial correlation-based truncation method is proposed to truncate the canonical vine. The truncated partial correlation-based canonical vine copula is then applied to construct and analyse the dependence structures of European stocks as a case study. Chapter 4 introduces the truncated partial correlation-based regular vine copula to explore the relations in multiple variables. Very often, strong restrictions are applied on a dependence structure by existing high-dimensional dependence models. These restrictions disabled the detection of sophisticated structures such as the upper and lower tail dependence between multiple variables. A partial correlation-based regular vine copula model may relax these restrictions. The partial correlation-based regular vine copula model employs a partial correlation to construct the regular vine structure, which is algebraically independent. This model is able to capture the asymmetric characteristics among multiple variables by using a two-parametric copula with flexible lower and upper tail dependence. The method is tested on a cross-country stock market data set to analyse the asymmetry and tail dependence in the dynamic period. Chapter 5 proposes a novel truncated partial correlation-based regular vine copula model which can capture more flexible dependence structures without making pre-assumptions about the data. Specifically, the model employs a new partial correlation to build the dependence structures via a bottom-up strategy. It can identify important dependencies and information among high-dimensional variables, truncating the irrelevant information to significantly reduce the parameter estimate time. The in-sample and out-of-sample performance of the model are examined by using the data in currency markets over a period of 17 years. Chapter 6 discusses how to resolve the high-dimensional asset allocation problem through a partial correlation-based canonical vine. Typically, the mean-variance criteria which is widely used in asset allocation, is actually not the optimal solution for asset allocation as the joint distribution of asset returns are distributed in asymmetric ways rather than in the assumed normal distribution. The partial correlation-based canonical vine can resolve the issue by producing the asymmetric joint distribution of asset returns in the utility function. Then, the utility function is then used for determining the optimal allocation of the assets. The performance of the model is examined by using data in both European and United State stock markets. In summary, this thesis proposes three dependence models, including one canonical vine and two regular vines. The three dependence models, which do not impose any dependence assumption on the dependence structure, can be used for modelling different high-dimensional dependencies, such as asymmetry or tail dependencies. All of these models are examined by the datasets in the real world, such as stock or currency markets. In addition, the partial correlation-based canonical vine is used to resolve optimisation allocation of assets in stock markets. This thesis works to show that there is great potential in applying copula to model complex dependence, particular in modelling time-varying parameters, or in developing efficient vine copula simplification methods

A Comparison of Relaxations of Multiset Cannonical Correlation Analysis and Applications

Canonical correlation analysis is a statistical technique that is used to find relations between two sets of variables. An important extension in pattern analysis is to consider more than two sets of variables. This problem can be expressed as a quadratically constrained quadratic program (QCQP), commonly referred to Multi-set Canonical Correlation Analysis (MCCA). This is a non-convex problem and so greedy algorithms converge to local optima without any guarantees on global optimality. In this paper, we show that despite being highly structured, finding the optimal solution is NP-Hard. This motivates our relaxation of the QCQP to a semidefinite program (SDP). The SDP is convex, can be solved reasonably efficiently and comes with both absolute and output-sensitive approximation quality. In addition to theoretical guarantees, we do an extensive comparison of the QCQP method and the SDP relaxation on a variety of synthetic and real world data. Finally, we present two useful extensions: we incorporate kernel methods and computing multiple sets of canonical vectors

Microcanonical finite-size scaling in specific heat diverging 2nd order phase transitions

A Microcanonical Finite Site Ansatz in terms of quantities measurable in a Finite Lattice allows to extend phenomenological renormalization (the so called quotients method) to the microcanonical ensemble. The Ansatz is tested numerically in two models where the canonical specific-heat diverges at criticality, thus implying Fisher-renormalization of the critical exponents: the 3D ferromagnetic Ising model and the 2D four-states Potts model (where large logarithmic corrections are known to occur in the canonical ensemble). A recently proposed microcanonical cluster method allows to simulate systems as large as L=1024 (Potts) or L=128 (Ising). The quotients method provides extremely accurate determinations of the anomalous dimension and of the (Fisher-renormalized) thermal $\nu$ exponent. While in the Ising model the numerical agreement with our theoretical expectations is impressive, in the Potts case we need to carefully incorporate logarithmic corrections to the microcanonical Ansatz in order to rationalize our data.Comment: 13 pages, 8 figure

Random fields of multivariate test statistics, with applications to shape analysis

Our data are random fields of multivariate Gaussian observations, and we fit a multivariate linear model with common design matrix at each point. We are interested in detecting those points where some of the coefficients are nonzero using classical multivariate statistics evaluated at each point. The problem is to find the $P$-value of the maximum of such a random field of test statistics. We approximate this by the expected Euler characteristic of the excursion set. Our main result is a very simple method for calculating this, which not only gives us the previous result of Cao and Worsley [Ann. Statist. 27 (1999) 925--942] for Hotelling's $T^2$, but also random fields of Roy's maximum root, maximum canonical correlations [Ann. Appl. Probab. 9 (1999) 1021--1057], multilinear forms [Ann. Statist. 29 (2001) 328--371], $\bar{\chi}^2$ [Statist. Probab. Lett 32 (1997) 367--376, Ann. Statist. 25 (1997) 2368--2387] and $\chi^2$ scale space [Adv. in Appl. Probab. 33 (2001) 773--793]. The trick involves approaching the problem from the point of view of Roy's union-intersection principle. The results are applied to a problem in shape analysis where we look for brain damage due to nonmissile trauma.Comment: Published in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org