\relax 
\bibstyle{plainnat}
\citation{BGV92}
\citation{Vap95}
\citation{FCC98}
\citation{Platt99}
\citation{BGV92}
\citation{OFG97}
\citation{Joachims99}
\@writefile{toc}{\contentsline {section}{\numberline {1}Introduction}{243}}
\citation{ChoiMonmaShanno}
\citation{Marxen}
\citation{GS1}
\citation{SS00}
\citation{WS01}
\@writefile{toc}{\contentsline {subsection}{\numberline {1.1}Notation}{245}}
\@writefile{toc}{\contentsline {section}{\numberline {2}An Interior Point Method}{246}}
\newlabel{sec:IPM}{{2}{246}}
\citation{Wright}
\citation{Mehrotra}
\@writefile{toc}{\contentsline {section}{\numberline {3}Low-rank updates}{248}}
\newlabel{sec:Low-rank_updates}{{3}{248}}
\citation{ChoiMonmaShanno}
\citation{Marxen}
\citation{Andersen}
\citation{SW}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.1}Sherman-Morrison-Woodbury update}{249}}
\newlabel{exmp:SMW}{{1}{249}}
\newlabel{solut}{{2}{250}}
\@writefile{toc}{\contentsline {subsection}{\numberline {3.2}Product-Form Cholesky Factorization}{250}}
\newlabel{pfcf}{{3.2}{250}}
\citation{Bennet}
\citation{FP}
\citation{GMS}
\newlabel{ltild}{{3}{251}}
\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces Rank-One update for Product-Form Cholesky Factorization}}{252}}
\newlabel{fig:RankOnePFCF}{{1}{252}}
\citation{GS1}
\citation{WSS98}
\citation{SS00}
\citation{OSS00}
\citation{SS00}
\citation{WS01}
\citation{GolubVanLoan}
\citation{GolubVanLoan}
\citation{Wright2}
\citation{GolubVanLoan}
\@writefile{toc}{\contentsline {section}{\numberline {4}Approximating the Kernel Matrix}{254}}
\newlabel{sec:Approximating_the_Kernel_Matrix}{{4}{254}}
\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces Column-wise Cholesky Factorization with Symmetric Pivoting}}{255}}
\newlabel{fig:CFSP}{{2}{255}}
\@writefile{toc}{\contentsline {subsection}{\numberline {4.1}Bound on the error in the optimal objective value}{256}}
\newlabel{subsec:bound}{{4.1}{256}}
\citation{FS2}
\citation{FNG01a}
\citation{JH99}
\citation{Platt99}
\newlabel{bound}{{1}{257}}
\@writefile{toc}{\contentsline {section}{\numberline {5}Experiments}{257}}
\newlabel{sec:Experiments}{{5}{257}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.1}Cholesky Product Form QP vs. SMO}{257}}
\citation{Joachims99}
\citation{BM98}
\@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces  Cholesky Product Form QP vs. SMO}}{258}}
\newlabel{CPFQPvisSMO}{{3}{258}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.2}Cholesky Product Form QP vs. SVM$^{light}$}{258}}
\newlabel{Sec_CPFQPvsSVMlight}{{5.2}{258}}
\citation{SS00}
\@writefile{lof}{\contentsline {figure}{\numberline {4}{\ignorespaces  Cholesky Product Form QP vs. SVM$^{light}$}}{259}}
\newlabel{CPFQPvsSVMlight}{{4}{259}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.3}Example of failure of the Sherman-Morrison-Woodbury update}{259}}
\@writefile{toc}{\contentsline {subsection}{\numberline {5.4}Incomplete Cholesky Factorization (ICF)}{259}}
\@writefile{lof}{\contentsline {figure}{\numberline {5}{\ignorespaces  Example for the failure of SMW update}}{260}}
\newlabel{smw_ex}{{5}{260}}
\@writefile{toc}{\contentsline {section}{\numberline {6}Concluding Remarks}{260}}
\newlabel{sec:Conclusions}{{6}{260}}
\@writefile{lof}{\contentsline {figure}{\numberline {6}{\ignorespaces  Incomplete Cholesky Factorization for Poly Kernel (input dim = 10, poly degree = 5) on the Abalone data set. The optimization problem perspective: optimal value of the objective function and norm of the separating hyperplane. X axis is the rank (note that the feature space dim $\approx 3000$).}}{261}}
\newlabel{BM_W_Obj}{{6}{261}}
\bibdata{cholqp_paper}
\bibcite{Andersen}{{1}{1996}{{Andersen}}{{}}}
\bibcite{Bennet}{{2}{1965}{{Bennet}}{{}}}
\@writefile{lof}{\contentsline {figure}{\numberline {7}{\ignorespaces  Incomplete Cholesky Factorization for Poly Kernel (input dim = 10, poly degree = 5) on the Abalone data set. The classification problem perspective: training and testing errors. X axis is the rank (note that the feature space dim $\approx 3000$).}}{262}}
\newlabel{BM_Err}{{7}{262}}
\bibcite{BM98}{{3}{1998}{{Blake and Merz}}{{}}}
\bibcite{BGV92}{{4}{1992}{{Boser et~al.}}{{Boser, Guyon, and Vapnik}}}
\bibcite{ChoiMonmaShanno}{{5}{1990}{{Choi et~al.}}{{Choi, Monma, and Shanno}}}
\bibcite{FNG01a}{{6}{2001}{{Fine et~al.}}{{Fine, Navr\'atil, and Gopinath}}}
\bibcite{FS2}{{7}{2001}{{Fine and Scheinberg}}{{}}}
\bibcite{FP}{{8}{1974}{{Fletcher and Powell}}{{}}}
\bibcite{FCC98}{{9}{1998}{{Friess et~al.}}{{Friess, Cristianini, and Campbell}}}
\bibcite{GMS}{{10}{1975}{{Gill et~al.}}{{Gill, Murray, and Saunders}}}
\bibcite{GS1}{{11}{1999}{{Goldfarb and Scheinberg}}{{}}}
\bibcite{GolubVanLoan}{{12}{1996}{{Golub and Van~Loan}}{{}}}
\bibcite{JH99}{{13}{1999}{{Jaakkola and Haussler}}{{}}}
\bibcite{Joachims99}{{14}{1999}{{Joachims}}{{}}}
\bibcite{Marxen}{{15}{1989}{{Marxen}}{{}}}
\bibcite{Mehrotra}{{16}{1992}{{Mehrotra}}{{}}}
\bibcite{OSS00}{{17}{2000}{{Oliver et~al.}}{{Oliver, Sch\"{o}lkopf, and Smola}}}
\bibcite{OFG97}{{18}{1997}{{Osuna et~al.}}{{Osuna, Freund, and Girosi}}}
\bibcite{Platt99}{{19}{1999}{{Platt}}{{}}}
\bibcite{SW}{{20}{2000}{{Scheinberg and Wright}}{{}}}
\bibcite{SS00}{{21}{2000}{{Smola and Sch\"{o}lkopf}}{{}}}
\bibcite{Vap95}{{22}{1995}{{Vapnik}}{{}}}
\bibcite{WS01}{{23}{2001}{{Williams and Seeger}}{{}}}
\bibcite{WSS98}{{24}{1998}{{Williamson et~al.}}{{Williamson, Smola, and Sch\"{o}lkopf}}}
\bibcite{Wright2}{{25}{1996}{{Wright}}{{}}}
\bibcite{Wright}{{26}{1997}{{Wright}}{{}}}