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Showing 1 to 11 of 11 for “"matrix factorizations"”.
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Learning with matrix factorizations
… or high-dimensional data. Models based on matrix factorization (Factor Analysis, PCA) have been extensively used in statistical analysis and machine learning for over a century, with many new formulations and models suggested in recent years (Latent Semantic Indexing, Aspect Models, …
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Matrix Factorizations With More Than Two Factors
… of $S$. In this thesis, we study the notion of a matrix factorization of $f$ with $d\ge 2$ factors, that is, we consider tuples of square matrices $(\phi_1,\phi_2,\dots,\phi_d)$, with entries in $S$, such that their product is $f$ times an identity matrix of the appropriate size. These objects …
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Matrix Factorizations, Triadic Matrices, and Modified Cholesky Factorizations for Optimization
This thesis focuses on the Cholesky-related factorizations of symmetric matrices and their application to Newton-type optimization. A matrix is called triadic if it has at most two nonzero off-diagonal elements in each column. Tridiagonal matrices are a special case of these. We prove that the …
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Betti numbers of Koszul algebras and codimension two matrix factorizations
… In Chapter 4, we study the codimension two matrix factorizations of Eisenbud and Peeva. Each matrix factorization compactly encodes the data of a free resolution of its corresponding matrix factorization module. By showing that each matrix factorization also encodes a canonical system of …
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Thom-Sebastiani and duality for matrix factorizations, and results on the higher structures of the Hochschild invariants
… structure on its category of singularities (as matrix factorizations). We prove a Thom-Sebastiani type Theorem, identifying the k[[beta]]-linear tensor products of these dg categories with coherent complexes on the zero locus of the sum potential on the product (with a support condition), and …
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Linear Algebra, Random Matrices and Lie Theory
… group. In the second part, we derive a series of matrix factorizations from the generalized Cartan decomposition introduced by Flensted-Jensen and Hoogenboom. The generalized Cartan decomposition applied to structured matrices proves the existence of several known matrix factorizations at once and …
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Matrix Factorization: Nonnegativity, Sparsity and Independence
Matrix factorization arises in a wide range of application domains and is useful for extracting the latent features in the dataset. Examples include recommender systems, brain data analysis, and document clustering. In this dissertation, we are interested in matrix factorizations which impose the …
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Pattern extraction and clustering for high-dimensional discrete data
We explore connections of low-rank matrix factorizations with interesting problems in data mining and machine learning. We propose a framework for solving several low-rank matrix factorization problems, including binary matrix factorization, constrained binary matrix factorization, weighted …
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Homology over a Complete Intersection Ring via the Generic Hypersurface
… resolutions are eventually 2-periodic, given by matrix factorizations, and are thus relatively easy to understand. We approach this relationship in two ways. First, we give a correspondence between the two rings in the graded setting, where existing results are insufficient for preserving graded …
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Continuous low-rank tensor decompositions, with applications to stochastic optimal control and data assimilation
… with the function-train requires continuous matrix factorizations and continuous numerical linear algebra. Continuous analogues are presented for performing cross approximation; rounding; multilinear algebra operations such as addition, multiplication, integration, and differentiation; and …
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Geometric optimization algorithms for linear regression on fixed-rank matrices
… regression in nonlinear and high-dimensional matrix search spaces. Our purpose is to efficiently exploit the geometric structure of the search space in the design of scalable linear regression algorithms. Our search space of main interest will be the set of low-rank matrices. Learning a …