Topics in matrix approximation
dc.rights.license  CC BYNCND. Users may download and share copies with attribution in accordance with a Creative Commons AttributionNoncommercialNo Derivative Works 3.0 License. No commercial use or derivatives are permitted without the explicit approval of the author.  
dc.contributor  Kramer, Peter Roland, 1971  
dc.contributor  MagdonIsmail, Malik  
dc.contributor  McLaughlin, Joyce  
dc.contributor  Mitchell, John E.  
dc.contributor.author  Nambirajan, Srinivas  
dc.date.accessioned  20211103T08:34:52Z  
dc.date.available  20211103T08:34:52Z  
dc.date.created  20160407T12:14:09Z  
dc.date.issued  201512  
dc.identifier.uri  https://hdl.handle.net/20.500.13015/1651  
dc.description  December 2015  
dc.description  School of Science  
dc.description.abstract  A fundamental need in computational linear algebra is computing with matrices quickly but approximately. This is commonly achieved by approximating matrices, either deterministically or randomly such that the structure in these matrices essential to computation is preserved well. We study two useful and natural problems in this area, one involving deterministic, lowrank approximation of a matrix, and the other involving randomized approximation.  
dc.description.abstract  Next, we study a randomized approximation of a matrix to obtain good preconditioners to it. A ubiquitous operation in computational linear algebra is the solution of a linear system $\A \x = \b$. The technique used to quickly obtain relativeerror solutions to such systems with high probability is finding good randomized preconditioners to $\A$ for use in an appropriate iterative algorithm  Chebyshev or Conjugate Gradient, for instance. An established result for such preconditioning of symmetric, diagonally dominant (SDD) matrices has recently been extended to finite element matrices arising from finite element meshes for elliptic PDEs. The computation of such preconditioners is expensive, requiring $O(rn^2 + n^3)$ operations for a matrix $\A \in \reals^{n, n}$ for an $r > n$, of the order of the number of elements in the finite element mesh. We provide a method that computes these preconditioners in $\tilde{O}(n^3 \log (rn))$ (where $\tilde{O}$ hides polylogarithmic factors), which is a significant improvement for $r = \omega(n)$.  
dc.description.abstract  First, we study the lowrank approximation of a matrix, $\C \in \reals^{m, n}$, using a matrix of rank at most $k< \min (m, n)$ under spectral (operator) norm with the additional constraint that the approximation contains columns belonging to a specified, $r$dimensional subspace $\sB$. We derive a closed form expression for the solution to this problem and present an algorithm to compute it. A similarly constrained approximation under the \emph{Frobenius} norm allows a quick solution obtained in $O(T_{svd}(\B))$, where $T_{svd}(\B)$ is the number of operations taken to compute the full singular value decomposition of a matrix $\B \in \reals^{m, n}$ whose range is $\sB$. However, there was no known algorithm for the problem in \emph{spectral} norm. We provide the first closed form solution to the problem and an algorithm to compute it that runs in $O(T_{svd}(\C))$. We use this algorithm to then improve an existing result in lowrank approximation drastically: The best known result in computing a general lowrank approximation of a matrix guarantees only a \emph{relative error} approximation; we guarantee the existence of \emph{optimal} lowrank approximations.  
dc.language.iso  ENG  
dc.publisher  Rensselaer Polytechnic Institute, Troy, NY  
dc.relation.ispartof  Rensselaer Theses and Dissertations Online Collection  
dc.rights  AttributionNonCommercialNoDerivs 3.0 United States  * 
dc.rights.uri  http://creativecommons.org/licenses/byncnd/3.0/us/  * 
dc.subject  Applied mathematics  
dc.title  Topics in matrix approximation  
dc.type  Electronic thesis  
dc.type  Thesis  
dc.digitool.pid  177190  
dc.digitool.pid  177191  
dc.digitool.pid  177192  
dc.rights.holder  This electronic version is a licensed copy owned by Rensselaer Polytechnic Institute, Troy, NY. Copyright of original work retained by author.  
dc.description.degree  PhD  
dc.relation.department  Dept. of Mathematical Sciences 
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Except where otherwise noted, this item's license is described as CC BYNCND. Users may download and share copies with attribution in accordance with a Creative Commons AttributionNoncommercialNo Derivative Works 3.0 License. No commercial use or derivatives are permitted without the explicit approval of the author.