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Browsing Department of Mathematics by Author "Adams, Henry, advisor"
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Item Open Access Generic support vector machines and Radon's theorem(Colorado State University. Libraries, 2019) Carr, Brittany M., author; Adams, Henry, advisor; Shipman, Patrick, committee member; Fremstad, Anders, committee memberA support vector machine, (SVM), is an algorithm which finds a hyperplane that optimally separates labeled data points in Rn into positive and negative classes. The data points on the margin of this separating hyperplane are called \emph{support vectors}. We study the possible configurations of support vectors for points in general position. In particular, we connect the possible configurations to Radon's theorem, which provides guarantees for when a set of points can be divided into two classes (positive and negative) whose convex hulls intersect. If the positive and negative support vectors in a generic SVM configuration are projected to the separating hyperplane, then these projected points will form a Radon configuration.Item Open Access Iterative matrix completion and topic modeling using matrix and tensor factorizations(Colorado State University. Libraries, 2021) Kassab, Lara, author; Adams, Henry, advisor; Fosdick, Bailey, committee member; Kirby, Michael, committee member; Peterson, Chris, committee memberWith the ever-increasing access to data, one of the greatest challenges that remains is how to make sense out of this abundance of information. In this dissertation, we propose three techniques that take into account underlying structure in large-scale data to produce better or more interpretable results for machine learning tasks. One of the challenges that arise when it comes to analyzing large-scale datasets is missing values in data, which could be challenging to handle without efficient methods. We propose adjusting an iteratively reweighted least squares algorithm for low-rank matrix completion to take into account sparsity-based structure in the missing entries. We also propose an iterative gradient-projection-based implementation of the algorithm, and present numerical experiments showcasing the performance of the algorithm compared to standard algorithms. Another challenge arises while performing a (semi-)supervised learning task on high-dimensional data. We propose variants of semi-supervised nonnegative matrix factorization models and provide motivation for these models as maximum likelihood estimators. The proposed models simultaneously provide a topic model and a model for classification. We derive training methods using multiplicative updates for each new model, and demonstrate the application of these models to document classification (e.g., 20 Newsgroups dataset). Lastly, although many datasets can be represented as matrices, datasets also often arise as high-dimensional arrays, known as higher-order tensors. We show that nonnegative CANDECOMP/PARAFAC tensor decomposition successfully detects short-lasting topics in temporal text datasets, including news headlines and COVID-19 related tweets, that other popular methods such as Latent Dirichlet Allocation and Nonnegative Matrix Factorization fail to fully detect.Item Open Access Metric thickenings and group actions(Colorado State University. Libraries, 2020) Heim, Mark T., author; Adams, Henry, advisor; Peterson, Chris, advisor; Neilson, James, committee memberLet G be a group acting properly and by isometries on a metric space X; it follows that the quotient or orbit space X/G is also a metric space. We study the Vietoris–Rips and Čech complexes of X/G. Whereas (co)homology theories for metric spaces let the scale parameter of a Vietoris–Rips or Čech complex go to zero, and whereas geometric group theory requires the scale parameter to be sufficiently large, we instead consider intermediate scale parameters (neither tending to zero nor to infinity). As a particular case, we study the Vietoris–Rips and Čech thickenings of projective spaces at the first scale parameter where the homotopy type changes.Item Open Access Molecular configurations and persistence: branched alkanes and additive energies(Colorado State University. Libraries, 2022) Story, Brittany M., author; Adams, Henry, advisor; Shipman, Patrick, committee member; Achter, Jeff, committee member; Fremstad, Anders, committee memberEnergy landscapes are high-dimensional functions that encapsulate how certain molecular properties affect the energy of a molecule. Chemists use disconnectivity graphs to find transition paths, the lowest amount of energy needed to transfer from one energy minimum to another. But disconnectivity graphs fail to show not only some lower-dimensional features, such as transition paths with an energy value only slightly higher than the minimum transition path, but also all higher-dimensional features. Sublevelset persistent homology is a tool that can be used to capture other relevant features, including all transition paths. In this paper, we will use sublevelset persistent homology to find the structure of the energy landscapes of branched alkanes: tree-like molecules consisting of only carbons and hydrogens. We derive complete characterizations of the sublevelset persistent homology of the OPLS-UA energy function on two different families of branched alkanes. More generally, we explain how the sublevelset persistent homology of any additive energy landscape can be computed from the individual terms comprising that landscape.Item Open Access Multidimensional scaling: infinite metric measure spaces(Colorado State University. Libraries, 2019) Kassab, Lara, author; Adams, Henry, advisor; Kirby, Michael, committee member; Fosdick, Bailey, committee memberMultidimensional scaling (MDS) is a popular technique for mapping a finite metric space into a low-dimensional Euclidean space in a way that best preserves pairwise distances. We study a notion of MDS on infinite metric measure spaces, along with its optimality properties and goodness of fit. This allows us to study the MDS embeddings of the geodesic circle S1 into Rm for all m, and to ask questions about the MDS embeddings of the geodesic n-spheres Sn into Rm. Furthermore, we address questions on convergence of MDS. For instance, if a sequence of metric measure spaces converges to a fixed metric measure space X, then in what sense do the MDS embeddings of these spaces converge to the MDS embedding of X? Convergence is understood when each metric space in the sequence has the same finite number of points, or when each metric space has a finite number of points tending to infinity. We are also interested in notions of convergence when each metric space in the sequence has an arbitrary (possibly infinite) number of points.Item Open Access Persistence and simplicial metric thickenings(Colorado State University. Libraries, 2024) Moy, Michael, author; Adams, Henry, advisor; Patel, Amit, committee member; Peterson, Christopher, committee member; Ben-Hur, Asa, committee memberThis dissertation examines the theory of one-dimensional persistence with an emphasis on simplicial metric thickenings and studies two particular filtrations of simplicial metric thickenings in detail. It gives self-contained proofs of foundational results on one-parameter persistence modules of vector spaces, including interval decomposability, existence of persistence diagrams and barcodes, and the isometry theorem. These results are applied to prove the stability of persistent homology for sublevel set filtrations, simplicial complexes, and simplicial metric thickenings. The filtrations of simplicial metric thickenings studied in detail are the Vietoris–Rips and anti-Vietoris–Rips metric thickenings of the circle. The study of the Vietoris–Rips metric thickenings is motivated by persistent homology and its use in applied topology, and it builds on previous work on their simplicial complex counterparts. On the other hand, the study of the anti-Vietoris–Rips metric thickenings is motivated by their connections to graph colorings. In both cases, the homotopy types of these spaces are shown to be odd-dimensional spheres, with dimensions depending on the scale parameters.Item Open Access Persistence stability for metric thickenings(Colorado State University. Libraries, 2021) Moy, Michael, author; Adams, Henry, advisor; King, Emily, committee member; Ben-Hur, Asa, committee memberPersistent homology often begins with a filtered simplicial complex, such as the Vietoris–Ripscomplex or the Čech complex, the vertex set of which is a metric space. An important result, the stability of persistent homology, shows that for certain types of filtered simplicial complexes, two metric spaces that are close in the Gromov–Hausdorff distance result in persistence diagrams that are close in the bottleneck distance. The recent interest in persistent homology has motivated work to better understand the homotopy types and persistent homology of these commonly used simplicial complexes. This has led to the definition of metric thickenings, which agree with simplicial complexes for finite vertex sets but may have different topologies for infinite vertex sets. We prove Vietoris–Rips metric thickenings and Čech metric thickenings have the same persistence diagrams as their corresponding simplicial complexes for all totally bounded metric spaces. This immediately implies the stability of persistent homology for these metric thickenings.Item Open Access Persistent homology of products and Gromov-Hausdorff distances between hypercubes and spheres(Colorado State University. Libraries, 2023) Vargas-Rosario, Daniel, author; Adams, Henry, advisor; Hulpke, Alexander, committee member; Duflot-Miranda, Jeanne, committee member; Bacon, Joel, committee memberAn exploration in the first half of this dissertation of the relationships among spectral sequences, persistent homology, and products of simplices, including the development of a new concept in categorical product filtration, is followed in the second half by new determinations of a) lower bounds for the Gromov-Hausdorff distance between n-spheres and (n + 1)-hypercubes equipped with the geodesic metric and of b) new lower bounds for the coindexes of the Vietoris-Rips complexes of hypercubes equipped with the Hamming metric. In their paper, "Spectral Sequences, Exact Couples, and Persistent Homology of Filtrations", Basu and Parida worked on building an n-derived exact couple from an increasing filtration X of simplicial complexes, C(n)(X) = {D(n)(X), E(n)(X), i(n), j(n), ∂(n)}. The terms E(n)∗,∗ (X) are the bigraded vector spaces of a spectral sequence that has differentials d(r)(X), and the terms D(n)∗,∗ (X) are the persistent homology groups H∗,∗∗ (X). They proved that there exists a long exact sequence whose groups are H∗,∗ ∗ (X) and whose bigraded vector spaces are (E∗∗, ∗(X), d∗(X)). We establish in Section 3 of this dissertation a new, similar theorem in the case of the categorical product filtration X × Y that states that there exists a long exact sequence consisting of ⊕(l+j=n) H∗,∗ l (X) ⊗ H∗,∗j (Y) and of the bigraded vector spaces E∗ ∗,∗(X × Y) of (E∗ ∗,∗(X × Y ),d∗(X × Y)), and prove it in part using Künneth formulas on homology. The emphasis on product spaces continues in Section 5, where we establish new lower bounds for the Gromov-Hausdorff distance between n-spheres and (n+1)-hypercubes, I(n+1), when both are equipped with the geodesic distance. From these lower bounds, we conjecture new lower bounds for the coindices of the Vietoris-Rips complexes of hypercubes when equipped with the Hamming metric. We then determine new lower bounds for the coindices of the Vietoris-Rips complexes of hypercubes, a) by producing a map between spheres and the geometric realizations of Vietoris-Rips complexes of hypercubes using abstract convex combination and balanced sets, and b) by decomposing hollow n-cubes (homotopically equivalent to the above-mentioned spheres) into simplices of smaller dimension and smaller diameter.Item Open Access Topological, geometric, and combinatorial aspects of metric thickenings(Colorado State University. Libraries, 2021) Bush, Johnathan E., author; Adams, Henry, advisor; Patel, Amit, committee member; Peterson, Chris, committee member; Luong, Gloria, committee memberThe geometric realization of a simplicial complex equipped with the 1-Wasserstein metric of optimal transport is called a simplicial metric thickening. We describe relationships between these metric thickenings and topics in applied topology, convex geometry, and combinatorial topology. We give a geometric proof of the homotopy types of certain metric thickenings of the circle by constructing deformation retractions to the boundaries of orbitopes. We use combinatorial arguments to establish a sharp lower bound on the diameter of Carathéodory subsets of the centrally-symmetric version of the trigonometric moment curve. Topological information about metric thickenings allows us to give new generalizations of the Borsuk–Ulam theorem and a selection of its corollaries. Finally, we prove a centrally-symmetric analog of a result of Gilbert and Smyth about gaps between zeros of homogeneous trigonometric polynomials.Item Open Access Using mathematical techniques to leverage domain knowledge in image analysis for earth science(Colorado State University. Libraries, 2023) Ver Hoef, Lander, author; Adams, Henry, advisor; King, Emily J., advisor; Hagman, Jess Ellis, committee member; Ebert-Uphoff, Imme, committee memberWhen presented with the power of modern machine learning techniques, there is a belief that we can simply let these algorithms loose on the data and see what they can find, unconstrained by human choice or bias. While such approaches can be useful, they are (of course) not fully free of bias or choice. Moreover, by utilizing the deep store of knowledge built up by scientific domains over decades or centuries, we can make architectural choices in our machine learning algorithms that focus the learning on features that we already know are important and informative, leading to more efficient, explainable, and interpretable methods. In this work, we present three examples of this approach. In the first project, to make use of the knowledge that texture is an important attribute of clouds, we use tools from topological data analysis focusing on the texture of satellite imagery, which leads to an effective and highly interpretable classifier of mesoscale cloud organization. This project resulted in a paper that has been published as a journal article. In the second project, we compare a rotationally invariant convolutional neural network against a conventional CNN both with and without data augmentation in their performance and behaviors on the task of predicting the major and minor axes lengths of storms in forecast data. Finally, in the third project, we explore three different techniques from harmonic analysis to enhance the signature of gravity waves in satellite imagery.Item Open Access Vietoris–Rips metric thickenings and Wasserstein spaces(Colorado State University. Libraries, 2020) Mirth, Joshua, author; Adams, Henry, advisor; Peterson, Christopher, committee member; Patel, Amit, committee member; Eykholt, Richard, committee memberIf the vertex set, X, of a simplicial complex, K, is a metric space, then K can be interpreted as a subset of the Wasserstein space of probability measures on X. Such spaces are called simplicial metric thickenings, and a prominent example is the Vietoris–Rips metric thickening. In this work we study these spaces from three perspectives: metric geometry, optimal transport, and category theory. Using the geodesic structure of Wasserstein space we give a novel proof of Hausmann's theorem for Vietoris–Rips metric thickenings. We also prove the first Morse lemma in Wasserstein space and relate it to the geodesic perspective. Finally we study the category of simplicial metric thickenings and determine effects of certain limits and colimits on homotopy type.