Randomized Approximation of Linear Least Squares Regression at Sub-linear Cost

We prove that with a high probability nearly optimal solution of the highly important problem of Linear Least Squares Regression can be computed at sub-linear cost for a random input. Our extensive tests are in good accordance with this result.

[1]  Paulo Cortez,et al.  Modeling wine preferences by data mining from physicochemical properties , 2009, Decis. Support Syst..

[2]  Victor Y. Pan,et al.  Low Rank Approximation of a Matrix at Sub-linear Cost , 2019, ArXiv.

[3]  Victor Y. Pan,et al.  Refinement of Low Rank Approximation of a Matrix at Sub-linear Cost. , 2019 .

[4]  A. Levine,et al.  New estimates of the storage permanence and ocean co-benefits of enhanced rock weathering , 2023, PNAS nexus.

[5]  Tamás Sarlós,et al.  Improved Approximation Algorithms for Large Matrices via Random Projections , 2006, 2006 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS'06).

[6]  Victor Y. Pan,et al.  Low Rank Approximation Directed by Leverage Scores and Computed at Sub-linear Cost , 2019, ArXiv.

[7]  David P. Woodruff,et al.  Low rank approximation and regression in input sparsity time , 2013, STOC '13.

[8]  R. Pace,et al.  Sparse spatial autoregressions , 1997 .

[9]  David P. Woodruff,et al.  Numerical linear algebra in the streaming model , 2009, STOC '09.

[10]  Victor Y. Pan,et al.  N A ] 5 J un 2 01 6 Fast Low-rank Approximation of a Matrix : Novel Insights , Novel Multipliers , and Extensions ∗ , 2016 .

[11]  Sivan Toledo,et al.  Blendenpik: Supercharging LAPACK's Least-Squares Solver , 2010, SIAM J. Sci. Comput..

[12]  Victor Y. Pan,et al.  Random multipliers numerically stabilize Gaussian and block Gaussian elimination: Proofs and an extension to low-rank approximation ☆ , 2014, 1406.5802.

[13]  Michael W. Mahoney Boyd,et al.  Randomized Algorithms for Matrices and Data , 2010 .

[14]  David P. Woodruff.  Sketching as a Tool for Numerical Linear Algebra , 2014, Found. Trends Theor. Comput. Sci..

[15]  Victor Y. Pan,et al.  Low Rank Approximation at Sublinear Cost by Means of Subspace Sampling , 2019, ArXiv.

[16]  Michael W. Mahoney.  Randomized Algorithms for Matrices and Data , 2011, Found. Trends Mach. Learn..

[17]  Victor Y. Pan,et al.  Primitive and Cynical Low Rank Approximation, Preprocessing and Extensions , 2016 .

[18]  V. Rokhlin,et al.  A fast randomized algorithm for overdetermined linear least-squares regression , 2008, Proceedings of the National Academy of Sciences.