Sketching as a Tool for Numerical Linear Algebra

本章主要介绍精确回归算法(Exact Regression Algorithms)、草绘加速最小二乘回归(Sketching to speed up Least Squares Regression)、草绘以加速最小绝对偏差(l1)回归(Sketching to speed up Least Absolute Deviation (l1) Regression)、草绘加速低秩近似(Sketching to speed up Low Rank Approximation)几种处理方法。
展开查看详情

1.Sketching as a Tool for Numerical Linear Algebra David Woodruff IBM Almaden

2.Talk Outline  Exact Regression Algorithms  Sketching to speed up Least Squares Regression  Sketching to speed up Least Absolute Deviation (l1) Regression  Sketching to speed up Low Rank Approximation 2

3.Regression Linear Regression  Statistical method to study linear dependencies between variables in the presence of noise. Example  Ohm's law V = R ∙ I  Find linear function that best fits the data 3

4.Regression Standard Setting  One measured variable b  A set of predictor variables a1 ,…, a d  Assumption: b = x0 + a1 x1 + … + ad xd +   is assumed to be noise and the xi are model parameters we want to learn  Can assume x0 = 0  Now consider n observations of b 4

5.Regression analysis Matrix form Input: nd-matrix A and a vector b=(b1,…, bn) n is the number of observations; d is the number of predictor variables Output: x* so that Ax* and b are close  Consider the over-constrained case, when n À d  Can assume that A has full column rank 5

6.Regression analysis Least Squares Method  Find x* that minimizes |Ax-b|22 =  (bi – <Ai*, x>)²)²  Ai* is i-th row of A  Certain desirable statistical properties  Closed form solution: x = (ATA)-1 AT b Method of least absolute deviation (l1 -regression)  Find x* that minimizes |Ax-b|1 =  |bi – <Ai*, x>)²|  Cost is less sensitive to outliers than least squares  Can solve via linear programming Time complexities are at least n*d2, we want better! 6

7.Talk Outline  Exact Regression Algorithms  Sketching to speed up Least Squares Regression  Sketching to speed up Least Absolute Deviation (l1) Regression  Sketching to speed up Low Rank Approximation 7

8.Sketching to solve least squares regression  How to find an approximate solution x to minx |Ax-b|2 ?  Goal: output x‘ for which |Ax‘-b|2 · (1+ε) minx |Ax-b|2 with high probability  Draw S from a k x n random family of matrices, for a value k << n  Compute S*A and S*b  Output the solution x‘ to minx‘ |(SA)x-(Sb)|2 8

9.How to choose the right sketching matrix S?  Recall: output the solution x‘ to minx‘ |(SA)x-(Sb)|2  Lots of matrices work  S is d/ε2 x n matrix of i.i.d. Normal random variables  Computing S*A may be slow… 9

10.How to choose the right sketching matrix S? [S]  S is a Johnson Lindenstrauss Transform  S = P*H*D  D is a diagonal matrix with +1, -1 on diagonals  H is the Hadamard transform  P just chooses a random (small) subset of rows of H*D  S*A can be computed much faster 10

11.Even faster sketching matrices [CW,MM,NN]  CountSketch matrix  Define k x n matrix S, for k = d2/ε2  S is really sparse: single randomly chosen non-zero entry per column [ 0010 01 00 1000 00 00 0 0 0 -1 1 0 -1 0 [ Surprisingly, this works! 0-1 0 0 0 0 0 1 11

12.Talk Outline  Exact Regression Algorithms  Sketching to speed up Least Squares Regression  Sketching to speed up Least Absolute Deviation (l1) Regression  Sketching to speed up Low Rank Approximation 12

13.Sketching to solve l1-regression  How to find an approximate solution x to minx |Ax-b|1 ?  Goal: output x‘ for which |Ax‘-b|1 · (1+ε) minx |Ax-b|1 with high probability  Natural attempt: Draw S from a k x n random family of matrices, for a value k << n  Compute S*A and S*b  Output the solution x‘ to minx‘ |(SA)x-(Sb)|1  Turns out this does not work! 13

14.Sketching to solve l1-regression [SW]  Why doesn’t outputting the solution x‘ to minx‘ |(SA)x- (Sb)|1 work?  Don‘t know of k x n matrices S with small k for which if x‘ is solution to minx |(SA)x-(Sb)|1 then |Ax‘-b|1 · (1+ε) minx |Ax-b|1 with high probability  Instead: can find an S so that |Ax‘-b|1 · (d log d) minx |Ax-b|1  S is a matrix of i.i.d. Cauchy random variables 14

15.Cauchy random variables  Cauchy random variables not as nice as Normal (Gaussian) random variables  They don’t have a mean and have infinite variance  Ratio of two independent Normal random variables is Cauchy 15

16.Sketching to solve l1-regression  How to find an approximate solution x to minx |Ax-b|1 ?  Want x‘ for which if x‘ is solution to minx |(SA)x-(Sb)|1 , then |Ax‘-b|1 · (1+ε) minx |Ax-b|1 with high probability  For d log d x n matrix S of Cauchy random variables: |Ax‘-b|1 · (d log d) minx |Ax-b|1  For this “poor” solution x’, let b’ = Ax’-b  Might as well solve regression problem with A and b’ 16

17.Sketching to solve l1-regression  Main Idea: Compute a QR-factorization of S*A  Q has orthonormal columns and Q*R = S*A  A*R-1 turns out to be a “well-conditioning” of original matrix A  Compute A*R-1 and sample d3.5/ε2 rows of [A*R-1 , b’] where the i-th row is sampled proportional to its 1-norm  Solve regression problem on the (reweighted) samples 17

18.Sketching to solve l1-regression [MM]  Most expensive operation is computing S*A where S is the matrix of i.i.d. Cauchy random variables  All other operations are in the “smaller space”  Can speed this up by choosing S as follows: [ [ 0010 01 00 1000 00 00 0 0 0 -1 1 0 -1 0 [ ¢ C1 C2 C3 [ 0-1 0 0 0 0 0 1 … Cn 18

19.Further sketching improvements [WZ]  Can show you need a fewer number of sampled rows in later steps if instead choose S as follows  Instead of diagonal of Cauchy random variables, choose diagonal of reciprocals of exponential random variables [ [ 0010 01 00 1000 00 00 0 0 0 -1 1 0 -1 0 [ ¢ 1/E1 1/E2 1/E3 [ 0-1 0 0 0 0 0 1 … 1/En 19

20.Talk Outline  Exact regression algorithms  Sketching to speed up Least Squares Regression  Sketching to speed up Least Absolute Deviation (l1) Regression  Sketching to speed up Low Rank Approximation 20

21.Low rank approximation  A is an n x n matrix  Typically well-approximated by low rank matrix  E.g., only high rank because of noise  Want to output a rank k matrix A’, so that |A-A’|F · (1+ε) |A-Ak|F, w.h.p., where Ak = argminrank k matrices B |A-B|F  For matrix C, |C|F = (Σi,j Ci,j2)1/2 21

22. Solution to low-rank approximation [S]  Given n x n input matrix A Most time- consuming  Compute S*A using a sketching matrix S with k << n rows. S*A takes random linear combinations of rows of A step is computing S*A A SScan canbe bematrix matrixof ofi.i.d. i.i.d. Normals Normals SScan canbe beaaFast FastJohnson Johnson SA Lindenstrauss LindenstraussMatrix Matrix  Project rows of A onto SA, then find best rank-k SScan can be be aaCountSketch CountSketch approximation to points inside of SA. matrix matrix 22

23.Conclusion  Gave fast sketching-based algorithms for  Least Squares Regression  Least Absolute Deviation (l1) Regression  Low Rank Approximation  Sketching also provides “dimensionality reduction”  Communication-efficient solutions for these problems 23