In this thesis, we study the polynomial equations that describe the highest weight orbit of an irreducible finite dimensional highest weight module under a semisimple Lie group. We also study the connection of the convex hull of this orbit and the Carathéodory orbitope.

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The user perspective of image search remains poorly understood. the purpose of this study is to identify and investigate the key issues relevant to a user’s interaction with images and the user’s approach to image search. a deeper understanding of these issues will serve to inform the design of image retrieval systems and in turn better serve the user. Previous research explores areas of information seeking behavior, representation in information science, query formulation, and image retrieval. the theoretical framework for this study includes an articulation of image search scenarios as adapted from Yoon and O’Connor’s taxonomy of image query types, Copeland’s Engineering Design Approach for rigorous qualitative research, and Anderson’s Functional Ontology Construction Model for building robust models of human behavior. a series of semi-structured interviews were conducted with expert-level image users. Interviewees discussed their motivations for image search, types of image searches they pursue, and varied approaches to image search, as well as how they decide that an information need has been met and which factors influence their experience of search. a content analysis revealed themes repeated across responses, including a collection of 23 emergent concepts and 6 emergent categories. a functional analysis revealed further insight into these themes. Results from both analyses may be used as a framework for future exploration of this topic. Implications are discussed and future research directions are indicated. Among possibilities for future research are investigations into collaborative search and ubiquitous image search.

Let A be a ring with involution *. The group Sl*(2,A), defined by Pantoja and Soto-Andrade, is a noncommutative version of Sl(2,F) where F is a field. In the case of A being artinian, they determined when Sl*(2,A) admitted a Bruhat presentation, and with Gutiérrez, constructed a representation for Sl*(2,A) from its generators. In particular, if A=Mn(F) and * is transposition, then Sl*(2,A) = Sp(2n,F). In this paper, we are interested in the representation theory of G=Sp4(O/p2) where A=M2(O/p2) and O is a local ring with prime ideal p. It has a normal, abelian subgroup K, and by Clifford's theorem we can find distinct irreducible representations of G starting with one-dimensional representations of K. The outline of our strategy will be demonstrated in the example of finding irreducible representations of SL2,(O/p2).

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Boston University. University Professors Program Senior theses.

Many ontologies, in particular in the biomedical domain, are based on the Description Logic EL++. Several efforts have been made to interpret and exploit EL++ontologies by distributed representation learning. Specifically, concepts within EL++theories have been represented as n-balls within an n-dimensional embedding space. However, the intersectional closure is not satisfied when using n-balls to represent concepts because the intersection of two n-balls is not an n-ball. This leads to challenges when measuring the distance between concepts and inferring equivalence between concepts. To this end, we developed EL Box Embedding (ELBE) to learn Description Logic EL++embeddings using axis-parallel boxes. We generate specially designed box-based geometric constraints from EL++axioms for model training. Since the intersection of boxes remains as a box, the intersectional closure is satisfied. We report extensive experimental results on three datasets and present a case study to demonstrate the effectiveness of the proposed method.

Indiana University-Purdue University Indianapolis (IUPUI) / A significant body of work has demonstrated the importance of diversity and representation in racial and ethnic minority jobseekers’ organizational judgments. While representation is often conceptualized as the general percentage or count of underrepresented minorities (URM) within an organization, a broader definition has been proposed that distinguishes this general or numerical representation from hierarchical representation which considers the placement of those URM employees within an organization. Although the separate effects of these two forms of representation have been evaluated, the present study extends on earlier work by considering the interactive effect. Additionally, the current research considered a potential mechanism to explain the influence of these forms of representation on URM’s organizational judgements. As expected, results showed that an organization depicting more URM employees (high numerical representation) and including Black leadership personnel (hierarchical representation) increased URM’s identity-safety relative to those which had low numerical representation and only White leadership. Moreover, and importantly, both representation effects could be explained indirectly via feelings of anticipated tokenism.

Recent advancements in inverse rendering have exhibited promising results for 3D representation, novel view synthesis, scene parameter reconstruction, and direct graphical asset generation and editing. Inverse rendering attempts to recover the scene parameters of interest from a set of camera observations by optimizing the photometric error between rendering model output and the true observation with appropriate regularization. The objective of this dissertation is to study inverse problems from several perspectives: (1) Software Framework: the general differentiable pipeline for solving physically-based or neural-based rendering problems, (2) Closed-Form: efficient and closed-form solutions in specific condition in inverse problems, (3) Representation Structure: hybrid 3D scene representation for efficient training and adaptive resource allocation, and (4) Robustness: enhanced robustness and accuracy from controlled lighting aspect. We aim to solve the following tasks: 1. How to address the challenge of rendering and optimizing scene parameters such as geometry, texture, and lighting, while considering multiple viewpoints from physically-based or neural 3D representations. To this end, we present a comprehensive software toolkit that provides support for diverse ray-based sampling and tracing schemes that enable the optimization of a wide range of targeting scene parameters. Our approach emphasizes the importance of maintaining differentiability throughout the entire pipeline to ensure efficient and effective optimization of the desired parameters. 2. Is there a 3D representation that has a fixed computational complexity or closed-form solution for forward rendering when the target has specific geometry or simplified lighting cases for better relaxing computational problems or reducing complexity. We consider multi-bounce reflection inside the plane transparent medium, and design differentiable polarization simulation engine that jointly optimize medium's parameters as well as the polarization state of reflection and transmission light. 3. How can we use our hybrid, learned 3D scene representation to solve inverse rendering problems for scene reconstruction and novel view synthesis, with a particular interest in several scientific fields, including density, radiance field, signed distance function, etc. 4. Unknown lighting condition significantly influence object appearance, to enhance the robustness of inverse rendering, we adopt invisible co-located lighting to further control lighting and suppress unknown lighting by jointly optimize separated channels of RGB and near infrared light, and enable accurate all scene parameters reconstruction from wider application environment. We also demonstrate the visually and quantitatively improved results for the aforementioned tasks and make comparisons with other state-of-the-art methods to demonstrate superior performance on representation and reconstruction tasks.

Recent work has demonstrated the impressive efficacy of computing representations in hyperbolic space rather than in Euclidean space. This is especially true for multi-relational data and for data containing latent hierarchical structures. In this work, we seek to understand why this is the case. We reflect on the intrinsic properties of hyperbolic geometry and then zero in on one of these as a possible explanation for the performance improvements --- projection. To validate this hypothesis, we propose our projected cone model, $\mathcal{PC}$. This model is designed to capture the effects of projection while not exhibiting other distinguishing properties of hyperbolic geometry. We define the $\mathcal{PC}$ model and determine all of the properties we need in order to conduct machine learning experiments with it. The model is defined as the stereographic projection of a cone into a unit disk. This is analogous to the construction of the Beltrami-Poincaré model of hyperbolic geometry by stereographic projection of one sheet of a two-sheet hyperboloid into the unit disk. We determine the mapping formulae between the cone and the unit disk, its Riemannian metric, and the distance formula between two points in the $\mathcal{PC}$ model. We investigate the learning capacity of our model. Finally, we generalize our model to higher dimensions so that we can perform representation learning in higher dimensions with our $\mathcal{PC}$ model. Because generalizing models into higher dimensions can be difficult, we also introduce a baseline model for comparison. This is a product space model, $\mathcal{PCP}$. It is built up from our rigourously developed, two-dimensional version of the $\mathcal{PC}$ model. We run experiments and compare our results with those obtained by others using the Beltrami-Poincaré model. We find that our model performs almost as well as their Beltrami-Poincaré model, far outperforming representation learning in Euclidean space. We thus conclude that projection indeed is key in explaining the success which hyperbolic geometry brings to representation learning.