Data augmentation with Mobius transformations

Data augmentation has led to substantial improvements in the performance and generalization of deep models, and remain a highly adaptable method to evolving model architectures and varying amounts of data---in particular, extremely scarce amounts of available training data. In this paper, we present a novel method of applying Mobius transformations to augment input images during training. Mobius transformations are bijective conformal maps that generalize image translation to operate over complex inversion in pixel space. As a result, Mobius transformations can operate on the sample level and preserve data labels. We show that the inclusion of Mobius transformations during training enables improved generalization over prior sample-level data augmentation techniques such as cutout and standard crop-and-flip transformations, most notably in low data regimes

Universal tools for analysing structures and interactions in geometry

This study examined symmetry and perspective in modern geometric transformations, treating them as functions that preserve specific properties while mapping one geometric figure to another. The purpose of this study was to investigate geometric transformations as a tool for analysis, to consider invariants as universal tools for studying geometry. Materials and Methods: The Erlangen ideas of F. I. Klein were used, which consider geometry as a theory of group invariants with respect to the transformation of the plane and space. Results and Discussion: Projective transformations and their extension to two-dimensional primitives were investigated. Two types of geometric correspondences, collinearity and correlation, and their properties were studied. The group of homotheties, including translations and parallel translations, and their role in the affine group were investigated. Homology with ideal line axes, such as stretching and centre stretching, was considered. Involutional homology and harmonic homology with the centre, axis, and homologous pairs of points were investigated. In this study unified geometry concepts, exploring how different geometric transformations relate and maintain properties across diverse geometric systems. Conclusions: It specifically examined Möbius transforms, including their matrix representation, trace, fixed points, and categorized them into identical transforms, nonlinear transforms, shifts, dilations, and inversions

Multi-relational Poincaré Graph Embeddings

Hyperbolic embeddings have recently gained attention in machine learning due to their ability to represent hierarchical data more accurately and succinctly than their Euclidean analogues. However, multi-relational knowledge graphs often exhibit multiple simultaneous hierarchies, which current hyperbolic models do not capture. To address this, we propose a model that embeds multi-relational graph data in the Poincar\'e ball model of hyperbolic space. Our Multi-Relational Poincar\'e model (MuRP) learns relation-specific parameters to transform entity embeddings by M\"obius matrix-vector multiplication and M\"obius addition. Experiments on the hierarchical WN18RR knowledge graph show that our Poincar\'e embeddings outperform their Euclidean counterpart and existing embedding methods on the link prediction task, particularly at lower dimensionality

Rethinking the compositionality of point clouds through regularization in the hyperbolic space

Point clouds of 3D objects exhibit an inherent compositional nature where simple parts can be assembled into progressively more complex shapes to form whole objects. Explicitly capturing such part-whole hierarchy is a long-sought objective in order to build effective models, but its tree-like nature has made the task elusive. In this paper, we propose to embed the features of a point cloud classifier into the hyperbolic space and explicitly regularize the space to account for the part-whole hierarchy. The hyperbolic space is the only space that can successfully embed the tree-like nature of the hierarchy. This leads to substantial improvements in the performance of state-of-art supervised models for point cloud classification

Characterising and classifying hypothalamus development

Understanding the development of the hypothalamus is important, due to its role as the central regulator of homeostasis. However, relative to development of other regions of the brain, characterisation and understanding of hypothalamus development is incomplete. Three important reasons for this are: (i) the hypothalamus is specified early compared to other brain regions, and then develops rapidly; (ii) the hypothalamus has a complex, anatomical structure even in the embryo; (iii) hypothalamic progenitor cells grow and migrate anisotropically. This non-linear growth makes it difficult to interpret downstream developmental events and molecular interactions that regulate early hypothalamus specification and regionalisation. One promising way to investigate development of the hypothalamus is through combining computational methods and traditional embryological approaches. To this end, I begin this thesis by developing a method of fine-grained classification of the Hamburger Hamilton (HH) stage 10 chick embryo. I was able to train an accurate classifier despite a limited dataset, by testing a variety of biologically motivated data augmentation techniques. I encouraged confidence in the staging system and subsequent classifications by analysing and visualising the output of the classifier. Using this classifier, I conducted a detailed morphological study of the developing hypothalamus at HH10 and surrounding stages, using both experimental embryology techniques and computational morphometric analyses. Using my increased understanding of the developing morphology, I characterised the expression of key hypothalamus morphogens: SHH, FGF10, and BMP2, as well as components of the SHH signalling pathway. I found that regionalisation between these morphogens occurred early and rapidly, with substantial heterogeneity in expression along both the anteroposterior and mediolateral axes. Finally, I tested to what extent this regionalisation is neuroepithelium intrinsic using ex vivo culture. I found both anteroposterior and mediolateral regionalisation in culture, which suggests that these processes are self-organising in the neuroepithelium. Overall, my thesis provides novel insights into early hypothalamic morphogenesis and molecular regionalisation, and shows through extension and use of the classifier how these complex processes may begin to be unpicked

Nelikulmion modulin numeerinen laskenta

The module of a quadrilateral is a positive real number which divides quadrilaterals into conformal equivalence classes. This is an introductory text to the module of a quadrilateral with some historical background and some numerical aspects. This work discusses the following topics: 1. Preliminaries 2. The module of a quadrilateral 3. The Schwarz-Christoffel Mapping 4. Symmetry properties of the module 5. Computational results 6. Other numerical methods Appendices include: Numerical evaluation of the elliptic integrals of the first kind. Matlab programs and scripts and possible topics for future research. Numerical results section covers additive quadrilaterals and the module of a quadrilateral under the movement of one of its vertex.Nelikulmion moduli on positiivinen reaaliluku, joka jakaa nelikulmiot konformisiin ekvivalenssi luokkiin. Tämä on johdanto teksti nelikulmion moduliin ja sen numeeriseen laskentaan. Lisäksi työssä on näiden alojen historiaa. Työssä käsitellään mm. seuraavia asioita: 1. Esitiedot 2. Nelikulmion modulin määritelmä 3. Schwarz-Christoffel kuvaus 4. Nelikulmion modulin symmetriaominaisuuksia 5. Laskennallisia tuloksia 6. Muita numeerisia menetelmiä Liitteet sisältävät: Elliptisten, ensimmäisen luokan, integraalien numeerinen laskeminen. Matlab ohjelmia, joita on käytetty työssä ja ehdotuksia tutkimuskohteiksi. Laskennallisissa tuloksissa osiossa tutkitaan summautuvia nelikulmioita ja nelikulmion modulia. Lisäksi tutkitaan miten nelikulmion moduli muuttuu kun yksi sen kärkipiste liikkuu

Accurate staging of chick embryonic tissues via deep learning of salient features

Recent work shows that the developmental potential of progenitor cells in the HH10 chick brain changes rapidly, accompanied by subtle changes in morphology. This demands increased temporal resolution for studies of the brain at this stage, necessitating precise and unbiased staging. Here, we investigated whether we could train a deep convolutional neural network to sub-stage HH10 chick brains using a small dataset of 151 expertly labelled images. By augmenting our images with biologically informed transformations and data-driven preprocessing steps, we successfully trained a classifier to sub-stage HH10 brains to 87.1% test accuracy. To determine whether our classifier could be generally applied, we re-trained it using images (269) of randomised control and experimental chick wings, and obtained similarly high test accuracy (86.1%). Saliency analyses revealed that biologically relevant features are used for classification. Our strategy enables training of image classifiers for various applications in developmental biology with limited microscopy data

Cross Connections: Online Activism, Real World Outcomes

This research paper examines the role of the Internet as it relates to the development of social movements and political protest in the ‘physical’ world. It also analyses the role of independent media and reporting methodologies used by activists and net-artists. The emergence of online activism and an emphasis on collaboration, information sharing and open source tools also had a significant impact on new media arts discussions and aesthetics. The refugee activist movement in Australia is a key case study in this thesis, as it is an excellent example of how activists have used the Internet and WWW to garner support within the community and to engage people to come to protests. In addition, activists at the protests have reported these events on the WWW and this subject has also had a resounding impact within the context of contemporary and media arts. The implications of identity online is a major factor in constructing the arguments in this thesis, as the relationship between ‘real’ and ‘virtual’ space is explored in detail as it relates to personal identity and online community

Novel Perspectives in String Phenomenology

String theory is the leading contemporary framework to explore the synthesis of quantum mechanics with gravity. String phenomenology aims to study string theory while maintaining contact with observational data. The fermionic $Z_2\times Z_2$ orbifold provides a case study that yielded a rich space of phenomenological models. String theory in ten dimensions gives rise to non--supersymmetric tachyonic vacua that may serve as good starting points for the construction of phenomenologically viable models. I discuss an example of such a three generation standard--like model in which all the moduli, aside from the dilaton, are frozen. The M\"obius symmetry may turn out to play a central role in the synthesis of quantum mechanics and gravity. In a local version it plays a central role in string theory. In a global version it underlies the Equivalence Postulate of Quantum Mechanics (EPOQM) formalism, which implies that spatial space is compact. It was recently proposed that evidence that the universe is closed exists in the Cosmic Microwave Background Radiation \cite{DiVMS}

Coordinate Independent Convolutional Networks -- Isometry and Gauge Equivariant Convolutions on Riemannian Manifolds

Motivated by the vast success of deep convolutional networks, there is a great interest in generalizing convolutions to non-Euclidean manifolds. A major complication in comparison to flat spaces is that it is unclear in which alignment a convolution kernel should be applied on a manifold. The underlying reason for this ambiguity is that general manifolds do not come with a canonical choice of reference frames (gauge). Kernels and features therefore have to be expressed relative to arbitrary coordinates. We argue that the particular choice of coordinatization should not affect a network's inference -- it should be coordinate independent. A simultaneous demand for coordinate independence and weight sharing is shown to result in a requirement on the network to be equivariant under local gauge transformations (changes of local reference frames). The ambiguity of reference frames depends thereby on the G-structure of the manifold, such that the necessary level of gauge equivariance is prescribed by the corresponding structure group G. Coordinate independent convolutions are proven to be equivariant w.r.t. those isometries that are symmetries of the G-structure. The resulting theory is formulated in a coordinate free fashion in terms of fiber bundles. To exemplify the design of coordinate independent convolutions, we implement a convolutional network on the M\"obius strip. The generality of our differential geometric formulation of convolutional networks is demonstrated by an extensive literature review which explains a large number of Euclidean CNNs, spherical CNNs and CNNs on general surfaces as specific instances of coordinate independent convolutions.Comment: The implementation of orientation independent M\"obius convolutions is publicly available at https://github.com/mauriceweiler/MobiusCNN