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Department of Mathematics

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https://hdl.handle.net/10217/100468
These digital collections include faculty/student publications, theses, and dissertations from the Department of Mathematics.

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Browsing Department of Mathematics by Subject "algorithm"

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    Classifying simplicial dissections of convex polyhedra with symmetry
    (Colorado State University. Libraries, 2021) Mukthineni, Tarun, author; Betten, Anton, advisor; Peterson, Chris, committee member; Rajopadhye, Sanjay, committee member
    A convex polyhedron is the convex hull of a finite set ofpoints in R3. A triangulation of a convex polyhedron is a decomposition into a finite number of 3-simplices such that any two intersect in a common face or are disjoint. A simplicial dissection is a decomposition into a finite number of 3-simplices such that no two share an interior point. We present an algorithm to classify the simplicial dissections of a regular polyhedron under the symmetry group of the polyhedron.
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    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 member
    With 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.