A fast randomized algorithm for computing an approximate null space
Taejun Park, Yuji Nakatsukasa
Abstract
Abstract Randomized algorithms in numerical linear algebra can be fast, scalable and robust. This paper examines the effect of sketching on the right singular vectors corresponding to the smallest singular values of a tall–skinny matrix. We analyze a fast algorithm by Gilbert, Park and Wakin for finding the trailing right singular vectors using randomization by examining the quality of the solution using multiplicative perturbation theory. For an $$m\times n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>×</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> </mml:math> ( $$m\ge n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>≥</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> </mml:math> ) matrix, the algorithm runs with complexity $$O(mn\log n +n^3)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:mi>m</mml:mi> <mml:mi>n</mml:mi> <mml:mo>log</mml:mo> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:msup> <mml:mi>n</mml:mi> <mml:mn>3</mml:mn> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> which is faster than the standard $$O(mn^2)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:mi>m</mml:mi> <mml:msup> <mml:mi>n</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> methods. In applications, numerical experiments show great speedups including a $$30\times $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>30</mml:mn> <mml:mo>×</mml:mo> </mml:mrow> </mml:math> speedup for the AAA algorithm and $$10\times $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>10</mml:mn> <mml:mo>×</mml:mo> </mml:mrow> </mml:math> speedup for the total least squares problem.