Euclidean Distance Between Haar Orthogonal and Gaussian Matrices

In this work, we study a version of the general question of how well a Haar-distributed orthogonal matrix can be approximated by a random Gaussian matrix. Here, we consider a Gaussian random matrix (Formula presented.) of order n and apply to it the Gram–Schmidt orthonormalization procedure by columns to obtain a Haar-distributed orthogonal matrix (Formula presented.). If (Formula presented.) denotes the vector formed by the first m-coordinates of the ith row of (Formula presented.) and (Formula presented.), our main result shows that the Euclidean norm of (Formula presented.) converges exponentially fast to (Formula presented.), up to negligible terms. To show the extent of this result, we use it to study the convergence of the supremum norm (Formula presented.) and we find a coupling that improves by a factor (Formula presented.) the recently proved best known upper bound on (Formula presented.). Our main result also has applications in Quantum Information Theory.

Gender effects in orientation on primary students’ performance on items rich in graphics

This study investigated the longitudinal performance of 378 students who completed mathematics items rich in graphics. Specifically, this study explored student performance across axis (e.g., numbers lines), opposed-position (e.g., line and column graphs) and circular (e.g., pie charts) items over a three-year period (ages 9-11 years). The results of the study revealed significant performance differences in the favour of boys on graphics items that were represented in horizontal and vertical displays. There were no gender differences on items that were represented in a circular manner.

An old language for a new landscape

Because aesthetics can have a profound effect upon the human relationship to the non-human environment the importance of aesthetics to ecologically sustainable designed landscapes has been acknowledged. However, in recognition that the physical forms of designed landscapes are an expression of the social values of the time, some design professionals have called for a new aesthetic ― one that reflects these current ecological concerns. To address this, some authors have suggested various theoretical design frameworks upon which such an aesthetic could be based. Within these frameworks there is an underlying theme that the patterns and processes of natural systems have the potential to form a new aesthetic for landscape design —an aesthetic based on fractal rather than Euclidean geometry. Perry, Reeves and Sim (2008) have shown that it is possible to differentiate between different landscape forms by fractal analysis. However, this research also shows that individual scenes from within very different landscape forms can possess the same fractal properties. Early data, revealed by transforming landscape images from the spatial to the frequency domain, using the fast Fourier transform, suggest that fractal patterning can have a significant effect within the landscape. In fact, it may be argued that any landscape design that includes living processes will include some design element whose ultimate form can only be expressed through the mathematics of fractal geometry. This paper will present ongoing research into the potential role of fractal geometry as a basis for a new form language – a language that may articulate an aesthetic for landscape design that echoes our ecological awakening.

Learning the kernel matrix with semidefinite programming

Kernel-based learning algorithms work by embedding the data into a Euclidean space, and then searching for linear relations among the embedded data points. The embedding is performed implicitly, by specifying the inner products between each pair of points in the embedding space. This information is contained in the so-called kernel matrix, a symmetric and positive semidefinite matrix that encodes the relative positions of all points. Specifying this matrix amounts to specifying the geometry of the embedding space and inducing a notion of similarity in the input space - classical model selection problems in machine learning. In this paper we show how the kernel matrix can be learned from data via semidefinite programming (SDP) techniques. When applied to a kernel matrix associated with both training and test data this gives a powerful transductive algorithm -using the labeled part of the data one can learn an embedding also for the unlabeled part. The similarity between test points is inferred from training points and their labels. Importantly, these learning problems are convex, so we obtain a method for learning both the model class and the function without local minima. Furthermore, this approach leads directly to a convex method for learning the 2-norm soft margin parameter in support vector machines, solving an important open problem.

Learning the Kernel Matrix with Semi-Definite Programming

Kernel-based learning algorithms work by embedding the data into a Euclidean space, and then searching for linear relations among the embedded data points. The embedding is performed implicitly, by specifying the inner products between each pair of points in the embedding space. This information is contained in the so-called kernel matrix, a symmetric and positive definite matrix that encodes the relative positions of all points. Specifying this matrix amounts to specifying the geometry of the embedding space and inducing a notion of similarity in the input space -- classical model selection problems in machine learning. In this paper we show how the kernel matrix can be learned from data via semi-definite programming (SDP) techniques. When applied to a kernel matrix associated with both training and test data this gives a powerful transductive algorithm -- using the labelled part of the data one can learn an embedding also for the unlabelled part. The similarity between test points is inferred from training points and their labels. Importantly, these learning problems are convex, so we obtain a method for learning both the model class and the function without local minima. Furthermore, this approach leads directly to a convex method to learn the 2-norm soft margin parameter in support vector machines, solving another important open problem. Finally, the novel approach presented in the paper is supported by positive empirical results.

Multifractal characterisation and analysis of complex networks

Complex networks have been studied extensively due to their relevance to many real-world systems such as the world-wide web, the internet, biological and social systems. During the past two decades, studies of such networks in different fields have produced many significant results concerning their structures, topological properties, and dynamics. Three well-known properties of complex networks are scale-free degree distribution, small-world effect and self-similarity. The search for additional meaningful properties and the relationships among these properties is an active area of current research. This thesis investigates a newer aspect of complex networks, namely their multifractality, which is an extension of the concept of selfsimilarity. The first part of the thesis aims to confirm that the study of properties of complex networks can be expanded to a wider field including more complex weighted networks. Those real networks that have been shown to possess the self-similarity property in the existing literature are all unweighted networks. We use the proteinprotein interaction (PPI) networks as a key example to show that their weighted networks inherit the self-similarity from the original unweighted networks. Firstly, we confirm that the random sequential box-covering algorithm is an effective tool to compute the fractal dimension of complex networks. This is demonstrated on the Homo sapiens and E. coli PPI networks as well as their skeletons. Our results verify that the fractal dimension of the skeleton is smaller than that of the original network due to the shortest distance between nodes is larger in the skeleton, hence for a fixed box-size more boxes will be needed to cover the skeleton. Then we adopt the iterative scoring method to generate weighted PPI networks of five species, namely Homo sapiens, E. coli, yeast, C. elegans and Arabidopsis Thaliana. By using the random sequential box-covering algorithm, we calculate the fractal dimensions for both the original unweighted PPI networks and the generated weighted networks. The results show that self-similarity is still present in generated weighted PPI networks. This implication will be useful for our treatment of the networks in the third part of the thesis. The second part of the thesis aims to explore the multifractal behavior of different complex networks. Fractals such as the Cantor set, the Koch curve and the Sierspinski gasket are homogeneous since these fractals consist of a geometrical figure which repeats on an ever-reduced scale. Fractal analysis is a useful method for their study. However, real-world fractals are not homogeneous; there is rarely an identical motif repeated on all scales. Their singularity may vary on different subsets; implying that these objects are multifractal. Multifractal analysis is a useful way to systematically characterize the spatial heterogeneity of both theoretical and experimental fractal patterns. However, the tools for multifractal analysis of objects in Euclidean space are not suitable for complex networks. In this thesis, we propose a new box covering algorithm for multifractal analysis of complex networks. This algorithm is demonstrated in the computation of the generalized fractal dimensions of some theoretical networks, namely scale-free networks, small-world networks, random networks, and a kind of real networks, namely PPI networks of different species. Our main finding is the existence of multifractality in scale-free networks and PPI networks, while the multifractal behaviour is not confirmed for small-world networks and random networks. As another application, we generate gene interactions networks for patients and healthy people using the correlation coefficients between microarrays of different genes. Our results confirm the existence of multifractality in gene interactions networks. This multifractal analysis then provides a potentially useful tool for gene clustering and identification. The third part of the thesis aims to investigate the topological properties of networks constructed from time series. Characterizing complicated dynamics from time series is a fundamental problem of continuing interest in a wide variety of fields. Recent works indicate that complex network theory can be a powerful tool to analyse time series. Many existing methods for transforming time series into complex networks share a common feature: they define the connectivity of a complex network by the mutual proximity of different parts (e.g., individual states, state vectors, or cycles) of a single trajectory. In this thesis, we propose a new method to construct networks of time series: we define nodes by vectors of a certain length in the time series, and weight of edges between any two nodes by the Euclidean distance between the corresponding two vectors. We apply this method to build networks for fractional Brownian motions, whose long-range dependence is characterised by their Hurst exponent. We verify the validity of this method by showing that time series with stronger correlation, hence larger Hurst exponent, tend to have smaller fractal dimension, hence smoother sample paths. We then construct networks via the technique of horizontal visibility graph (HVG), which has been widely used recently. We confirm a known linear relationship between the Hurst exponent of fractional Brownian motion and the fractal dimension of the corresponding HVG network. In the first application, we apply our newly developed box-covering algorithm to calculate the generalized fractal dimensions of the HVG networks of fractional Brownian motions as well as those for binomial cascades and five bacterial genomes. The results confirm the monoscaling of fractional Brownian motion and the multifractality of the rest. As an additional application, we discuss the resilience of networks constructed from time series via two different approaches: visibility graph and horizontal visibility graph. Our finding is that the degree distribution of VG networks of fractional Brownian motions is scale-free (i.e., having a power law) meaning that one needs to destroy a large percentage of nodes before the network collapses into isolated parts; while for HVG networks of fractional Brownian motions, the degree distribution has exponential tails, implying that HVG networks would not survive the same kind of attack.

The unfinished landscape fractal geometry and the aesthetics of ecological design

During the late 20th century it was proposed that a design aesthetic reflecting current ecological concerns was required within the overall domain of the built environment and specifically within landscape design. To address this, some authors suggested various theoretical frameworks upon which such an aesthetic could be based. Within these frameworks there was an underlying theme that the patterns and processes of Nature may have the potential to form this aesthetic — an aesthetic based on fractal rather than Euclidean geometry. In order to understand how fractal geometry, described as the geometry of Nature, could become the referent for a design aesthetic, this research examines the mathematical concepts of fractal Geometry, and the underlying philosophical concepts behind the terms ‘Nature’ and ‘aesthetics’. The findings of this initial research meant that a new definition of Nature was required in order to overcome the barrier presented by the western philosophical Nature¯culture duality. This new definition of Nature is based on the type and use of energy. Similarly, it became clear that current usage of the term aesthetics has more in common with the term ‘style’ than with its correct philosophical meaning. The aesthetic philosophy of both art and the environment recognises different aesthetic criteria related to either the subject or the object, such as: aesthetic experience; aesthetic attitude; aesthetic value; aesthetic object; and aesthetic properties. Given these criteria, and the fact that the concept of aesthetics is still an active and ongoing philosophical discussion, this work focuses on the criteria of aesthetic properties and the aesthetic experience or response they engender. The examination of fractal geometry revealed that it is a geometry based on scale rather than on the location of a point within a three-dimensional space. This enables fractal geometry to describe the complex forms and patterns created through the processes of Wild Nature. Although fractal geometry has been used to analyse the patterns of built environments from a plan perspective, it became clear from the initial review of the literature that there was a total knowledge vacuum about the fractal properties of environments experienced every day by people as they move through them. To overcome this, 21 different landscapes that ranged from highly developed city centres to relatively untouched landscapes of Wild Nature have been analysed. Although this work shows that the fractal dimension can be used to differentiate between overall landscape forms, it also shows that by itself it cannot differentiate between all images analysed. To overcome this two further parameters based on the underlying structural geometry embedded within the landscape are discussed. These parameters are the Power Spectrum Median Amplitude and the Level of Isotropy within the Fourier Power Spectrum. Based on the detailed analysis of these parameters a greater understanding of the structural properties of landscapes has been gained. With this understanding, this research has moved the field of landscape design a step close to being able to articulate a new aesthetic for ecological design.

Population structure of Bactrocera dorsalis s.s., B. papayae and B. philippinensis (Diptera: Tephritidae) in southeast Asia: evidence for a single species hypothesis using mitochondrial DNA and wingshape data

Background Bactrocera dorsalis s.s. is a pestiferous tephritid fruit fly distributed from Pakistan to the Pacific, with the Thai/Malay peninsula its southern limit. Sister pest taxa, B. papayae and B. philippinensis, occur in the southeast Asian archipelago and the Philippines, respectively. The relationship among these species is unclear due to their high molecular and morphological similarity. This study analysed population structure of these three species within a southeast Asian biogeographical context to assess potential dispersal patterns and the validity of their current taxonomic status. Results Geometric morphometric results generated from 15 landmarks for wings of 169 flies revealed significant differences in wing shape between almost all sites following canonical variate analysis. For the combined data set there was a greater isolation-by-distance (IBD) effect under a ‘non-Euclidean’ scenario which used geographical distances within a biogeographical ‘Sundaland context’ (r2 = 0.772, P < 0.0001) as compared to a ‘Euclidean’ scenario for which direct geographic distances between sample sites was used (r2 = 0.217, P < 0.01). COI sequence data were obtained for 156 individuals and yielded 83 unique haplotypes with no correlation to current taxonomic designations via a minimum spanning network. BEAST analysis provided a root age and location of 540kya in northern Thailand, with migration of B. dorsalis s.l. into Malaysia 470kya and Sumatra 270kya. Two migration events into the Philippines are inferred. Sequence data revealed a weak but significant IBD effect under the ‘non-Euclidean’ scenario (r2 = 0.110, P < 0.05), with no historical migration evident between Taiwan and the Philippines. Results are consistent with those expected at the intra-specific level. Conclusions Bactrocera dorsalis s.s., B. papayae and B. philippinensis likely represent one species structured around the South China Sea, having migrated from northern Thailand into the southeast Asian archipelago and across into the Philippines. No migration is apparent between the Philippines and Taiwan. This information has implications for quarantine, trade and pest management.

Chronological categorization and decomposition of customer loads

The majority of distribution utilities do not have accurate information on the constituents of their loads. This information is very useful in managing and planning the network, adequately and economically. Customer loads are normally categorized in three main sectors: 1) residential; 2) industrial; and 3) commercial. In this paper, penalized least-squares regression and Euclidean distance methods are developed for this application to identify and quantify the makeup of a feeder load with unknown sectors/subsectors. This process is done on a monthly basis to account for seasonal and other load changes. The error between the actual and estimated load profiles are used as a benchmark of accuracy. This approach has shown to be accurate in identifying customer types in unknown load profiles, and is used in cross-validation of the results and initial assumptions.

Kernel analysis on Grassmann manifolds for action recognition

Modelling video sequences by subspaces has recently shown promise for recognising human actions. Subspaces are able to accommodate the effects of various image variations and can capture the dynamic properties of actions. Subspaces form a non-Euclidean and curved Riemannian manifold known as a Grassmann manifold. Inference on manifold spaces usually is achieved by embedding the manifolds in higher dimensional Euclidean spaces. In this paper, we instead propose to embed the Grassmann manifolds into reproducing kernel Hilbert spaces and then tackle the problem of discriminant analysis on such manifolds. To achieve efficient machinery, we propose graph-based local discriminant analysis that utilises within-class and between-class similarity graphs to characterise intra-class compactness and inter-class separability, respectively. Experiments on KTH, UCF Sports, and Ballet datasets show that the proposed approach obtains marked improvements in discrimination accuracy in comparison to several state-of-the-art methods, such as the kernel version of affine hull image-set distance, tensor canonical correlation analysis, spatial-temporal words and hierarchy of discriminative space-time neighbourhood features.

Pre-crash and non-crash traffic flow trends analysis on motorways

Crashes that occur on motorways contribute to a significant proportion (40-50%) of non-recurrent motorway congestions. Hence, reducing the frequency of crashes assists in addressing congestion issues (Meyer, 2008). Crash likelihood estimation studies commonly focus on traffic conditions in a short time window around the time of a crash while longer-term pre-crash traffic flow trends are neglected. In this paper we will show, through data mining techniques that a relationship between pre-crash traffic flow patterns and crash occurrence on motorways exists. We will compare them with normal traffic trends and show this knowledge has the potential to improve the accuracy of existing models and opens the path for new development approaches. The data for the analysis was extracted from records collected between 2007 and 2009 on the Shibuya and Shinjuku lines of the Tokyo Metropolitan Expressway in Japan. The dataset includes a total of 824 rear-end and sideswipe crashes that have been matched with crashes corresponding to traffic flow data using an incident detection algorithm. Traffic trends (traffic speed time series) revealed that crashes can be clustered with regards to the dominant traffic patterns prior to the crash. Using the K-Means clustering method with Euclidean distance function allowed the crashes to be clustered. Then, normal situation data was extracted based on the time distribution of crashes and were clustered to compare with the “high risk” clusters. Five major trends have been found in the clustering results for both high risk and normal conditions. The study discovered traffic regimes had differences in the speed trends. Based on these findings, crash likelihood estimation models can be fine-tuned based on the monitored traffic conditions with a sliding window of 30 minutes to increase accuracy of the results and minimize false alarms.

Random projections on manifolds of symmetric positive definite matrices for image classification

Recent advances suggest that encoding images through Symmetric Positive Definite (SPD) matrices and then interpreting such matrices as points on Riemannian manifolds can lead to increased classification performance. Taking into account manifold geometry is typically done via (1) embedding the manifolds in tangent spaces, or (2) embedding into Reproducing Kernel Hilbert Spaces (RKHS). While embedding into tangent spaces allows the use of existing Euclidean-based learning algorithms, manifold shape is only approximated which can cause loss of discriminatory information. The RKHS approach retains more of the manifold structure, but may require non-trivial effort to kernelise Euclidean-based learning algorithms. In contrast to the above approaches, in this paper we offer a novel solution that allows SPD matrices to be used with unmodified Euclidean-based learning algorithms, with the true manifold shape well-preserved. Specifically, we propose to project SPD matrices using a set of random projection hyperplanes over RKHS into a random projection space, which leads to representing each matrix as a vector of projection coefficients. Experiments on face recognition, person re-identification and texture classification show that the proposed approach outperforms several recent methods, such as Tensor Sparse Coding, Histogram Plus Epitome, Riemannian Locality Preserving Projection and Relational Divergence Classification.

Dictionary learning and sparse coding on Grassmann manifolds : an extrinsic solution

Recent advances in computer vision and machine learning suggest that a wide range of problems can be addressed more appropriately by considering non-Euclidean geometry. In this paper we explore sparse dictionary learning over the space of linear subspaces, which form Riemannian structures known as Grassmann manifolds. To this end, we propose to embed Grassmann manifolds into the space of symmetric matrices by an isometric mapping, which enables us to devise a closed-form solution for updating a Grassmann dictionary, atom by atom. Furthermore, to handle non-linearity in data, we propose a kernelised version of the dictionary learning algorithm. Experiments on several classification tasks (face recognition, action recognition, dynamic texture classification) show that the proposed approach achieves considerable improvements in discrimination accuracy, in comparison to state-of-the-art methods such as kernelised Affine Hull Method and graph-embedding Grassmann discriminant analysis.

Multi-shot person re-identification via relational stein divergence

Person re-identification is particularly challenging due to significant appearance changes across separate camera views. In order to re-identify people, a representative human signature should effectively handle differences in illumination, pose and camera parameters. While general appearance-based methods are modelled in Euclidean spaces, it has been argued that some applications in image and video analysis are better modelled via non-Euclidean manifold geometry. To this end, recent approaches represent images as covariance matrices, and interpret such matrices as points on Riemannian manifolds. As direct classification on such manifolds can be difficult, in this paper we propose to represent each manifold point as a vector of similarities to class representers, via a recently introduced form of Bregman matrix divergence known as the Stein divergence. This is followed by using a discriminative mapping of similarity vectors for final classification. The use of similarity vectors is in contrast to the traditional approach of embedding manifolds into tangent spaces, which can suffer from representing the manifold structure inaccurately. Comparative evaluations on benchmark ETHZ and iLIDS datasets for the person re-identification task show that the proposed approach obtains better performance than recent techniques such as Histogram Plus Epitome, Partial Least Squares, and Symmetry-Driven Accumulation of Local Features.

Comparison of block and event-related experimental designs in diffusion-weighted functional MRI

PURPOSE To compare diffusion-weighted functional magnetic resonance imaging (DfMRI), a novel alternative to the blood oxygenation level-dependent (BOLD) contrast, in a functional MRI experiment. MATERIALS AND METHODS Nine participants viewed contrast reversing (7.5 Hz) black-and-white checkerboard stimuli using block and event-related paradigms. DfMRI (b = 1800 mm/s2 ) and BOLD sequences were acquired. Four parameters describing the observed signal were assessed: percent signal change, spatial extent of the activation, the Euclidean distance between peak voxel locations, and the time-to-peak of the best fitting impulse response for different paradigms and sequences. RESULTS The BOLD conditions showed a higher percent signal change relative to DfMRI; however, event-related DfMRI showed the strongest group activation (t = 21.23, P < 0.0005). Activation was more diffuse and spatially closer to the BOLD response for DfMRI when the block design was used. DfMRIevent showed the shortest TTP (4.4 +/- 0.88 sec). CONCLUSION The hemodynamic contribution to DfMRI may increase with the use of block designs.