1,091 research outputs found

    2023-2024 Catalog

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    The 2023-2024 Governors State University Undergraduate and Graduate Catalog is a comprehensive listing of current information regarding:Degree RequirementsCourse OfferingsUndergraduate and Graduate Rules and Regulation

    Hypercubes and Hamilton cycles of display sets of rooted phylogenetic networks

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    In the context of reconstructing phylogenetic networks from a collection of phylogenetic trees, several characterisations and subsequently algorithms have been established to reconstruct a phylogenetic network that collectively embeds all trees in the input in some minimum way. For many instances however, the resulting network also embeds additional phylogenetic trees that are not part of the input. However, little is known about these inferred trees. In this paper, we explore the relationships among all phylogenetic trees that are embedded in a given phylogenetic network. First, we investigate some combinatorial properties of the collection P of all rooted binary phylogenetic trees that are embedded in a rooted binary phylogenetic network N. To this end, we associated a particular graph G, which we call rSPR graph, with the elements in P and show that, if |P|=2^k, where k is the number of vertices with in-degree two in N, then G has a Hamiltonian cycle. Second, by exploiting rSPR graphs and properties of hypercubes, we turn to the well-studied class of rooted binary level-1 networks and give necessary and sufficient conditions for when a set of rooted binary phylogenetic trees can be embedded in a level-1 network without inferring any additional trees. Lastly, we show how these conditions translate into a polynomial-time algorithm to reconstruct such a network if it exists.Comment: final version of accepted manuscrip

    Geometric Data Analysis: Advancements of the Statistical Methodology and Applications

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    Data analysis has become fundamental to our society and comes in multiple facets and approaches. Nevertheless, in research and applications, the focus was primarily on data from Euclidean vector spaces. Consequently, the majority of methods that are applied today are not suited for more general data types. Driven by needs from fields like image processing, (medical) shape analysis, and network analysis, more and more attention has recently been given to data from non-Euclidean spaces–particularly (curved) manifolds. It has led to the field of geometric data analysis whose methods explicitly take the structure (for example, the topology and geometry) of the underlying space into account. This thesis contributes to the methodology of geometric data analysis by generalizing several fundamental notions from multivariate statistics to manifolds. We thereby focus on two different viewpoints. First, we use Riemannian structures to derive a novel regression scheme for general manifolds that relies on splines of generalized Bézier curves. It can accurately model non-geodesic relationships, for example, time-dependent trends with saturation effects or cyclic trends. Since Bézier curves can be evaluated with the constructive de Casteljau algorithm, working with data from manifolds of high dimensions (for example, a hundred thousand or more) is feasible. Relying on the regression, we further develop a hierarchical statistical model for an adequate analysis of longitudinal data in manifolds, and a method to control for confounding variables. We secondly focus on data that is not only manifold- but even Lie group-valued, which is frequently the case in applications. We can only achieve this by endowing the group with an affine connection structure that is generally not Riemannian. Utilizing it, we derive generalizations of several well-known dissimilarity measures between data distributions that can be used for various tasks, including hypothesis testing. Invariance under data translations is proven, and a connection to continuous distributions is given for one measure. A further central contribution of this thesis is that it shows use cases for all notions in real-world applications, particularly in problems from shape analysis in medical imaging and archaeology. We can replicate or further quantify several known findings for shape changes of the femur and the right hippocampus under osteoarthritis and Alzheimer's, respectively. Furthermore, in an archaeological application, we obtain new insights into the construction principles of ancient sundials. Last but not least, we use the geometric structure underlying human brain connectomes to predict cognitive scores. Utilizing a sample selection procedure, we obtain state-of-the-art results

    Northeastern Illinois University, Academic Catalog 2023-2024

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    https://neiudc.neiu.edu/catalogs/1064/thumbnail.jp

    A Structural Approach to the Design of Domain Specific Neural Network Architectures

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    This is a master's thesis concerning the theoretical ideas of geometric deep learning. Geometric deep learning aims to provide a structured characterization of neural network architectures, specifically focused on the ideas of invariance and equivariance of data with respect to given transformations. This thesis aims to provide a theoretical evaluation of geometric deep learning, compiling theoretical results that characterize the properties of invariant neural networks with respect to learning performance.Comment: 94 pages and 16 Figures Upload of my Master's thesis. Not peer reviewed and potentially contains error

    Robustness, scalability and interpretability of equivariant neural networks across different low-dimensional geometries

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    In this thesis we develop neural networks that exploit the symmetries of four different low-dimensional geometries, namely 1D grids, 2D grids, 3D continuous spaces and graphs, through the consideration of translational, rotational, cylindrical and permutation symmetries. We apply these models to applications across a range of scientific disciplines demonstrating the predictive ability, robustness, scalability, and interpretability. We develop a neural network that exploits the translational symmetries on 1D grids to predict age and species of mosquitoes from high-dimensional mid-infrared spectra. We show that the model can learn to predict mosquito age and species with a higher accuracy than models that do not utilise any inductive bias. We also demonstrate that the model is sensitive to regions within the input spectra that are in agreement with regions identified by a domain expert. We present a transfer learning approach to overcome the challenge of working with small, real-world, wild collected data sets and demonstrate the benefit of the approach on a real-world application. We demonstrate the benefit of rotation equivariant neural networks on the task of segmenting deforestation regions from satellite images through exploiting the rotational symmetry present on 2D grids. We develop a novel physics-informed architecture, exploiting the cylindrical symmetries of the group SO+ (2, 1), which can invert the transmission effects of multi-mode optical fibres (MMFs). We develop a new connection between a physics understanding of MMFs and group equivariant neural networks. We show that this novel architecture requires fewer training samples to learn, better generalises to out-of-distribution data sets, scales to higher-resolution images, is more interpretable, and reduces the parameter count of the model. We demonstrate the capability of the model on real-world data and provide an adaption to the model to handle real-world deviations from theory. We also show that the model can scale to higher resolution images than was previously possible. We develop a novel architecture which provides a symmetry-preserving mapping between two different low-dimensional geometries and demonstrate its practical benefit for the application of 3D hand mesh generation from 2D images. This models exploits both the 2D rotational symmetries present in a 2D image and in a 3D hand mesh, and provides a mapping between the two data domains. We demonstrate that the model performs competitively on a range of benchmark data sets and justify the choice of inductive bias in the model. We develop an architecture which is equivariant to a novel choice of automorphism group through the use of a sub-graph selection policy. We demonstrate the benefit of the architecture, theoretically through proving the improved expressivity and improved scalability, and experimentally on a range of widely studied benchmark graph classification tasks. We present a method of comparison between models that had not been previously considered in this area of research, demonstrating recent SOTA methods are statistically indistinguishable

    LIPIcs, Volume 261, ICALP 2023, Complete Volume

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    LIPIcs, Volume 261, ICALP 2023, Complete Volum

    Data analysis with merge trees

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    Today’s data are increasingly complex and classical statistical techniques need growingly more refined mathematical tools to be able to model and investigate them. Paradigmatic situations are represented by data which need to be considered up to some kind of trans- formation and all those circumstances in which the analyst finds himself in the need of defining a general concept of shape. Topological Data Analysis (TDA) is a field which is fundamentally contributing to such challenges by extracting topological information from data with a plethora of interpretable and computationally accessible pipelines. We con- tribute to this field by developing a series of novel tools, techniques and applications to work with a particular topological summary called merge tree. To analyze sets of merge trees we introduce a novel metric structure along with an algorithm to compute it, define a framework to compare different functions defined on merge trees and investigate the metric space obtained with the aforementioned metric. Different geometric and topolog- ical properties of the space of merge trees are established, with the aim of obtaining a deeper understanding of such trees. To showcase the effectiveness of the proposed metric, we develop an application in the field of Functional Data Analysis, working with functions up to homeomorphic reparametrization, and in the field of radiomics, where each patient is represented via a clustering dendrogram

    Quantum-Inspired Machine Learning: a Survey

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    Quantum-inspired Machine Learning (QiML) is a burgeoning field, receiving global attention from researchers for its potential to leverage principles of quantum mechanics within classical computational frameworks. However, current review literature often presents a superficial exploration of QiML, focusing instead on the broader Quantum Machine Learning (QML) field. In response to this gap, this survey provides an integrated and comprehensive examination of QiML, exploring QiML's diverse research domains including tensor network simulations, dequantized algorithms, and others, showcasing recent advancements, practical applications, and illuminating potential future research avenues. Further, a concrete definition of QiML is established by analyzing various prior interpretations of the term and their inherent ambiguities. As QiML continues to evolve, we anticipate a wealth of future developments drawing from quantum mechanics, quantum computing, and classical machine learning, enriching the field further. This survey serves as a guide for researchers and practitioners alike, providing a holistic understanding of QiML's current landscape and future directions.Comment: 56 pages, 13 figures, 8 table
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