compute a low rank soft-thresholded svd by alternating orthogonal ridge regression

Share:

Description

fit a low-rank svd to a complete matrix by alternating orthogonal ridge regression. Special sparse-matrix classes available for very large matrices, including "SparseplusLowRank" versions for row and column centered sparse matrices.

Usage

1
2
svd.als(x, rank.max = 2, lambda = 0, thresh = 1e-05, maxit = 100,
        trace.it = FALSE, warm.start = NULL, final.svd = TRUE)

Arguments

x

An m by n matrix. Large matrices can be in "sparseMatrix" format, as well as "SparseplusLowRank". The latter arise after centering sparse matrices, for example with biScale, as well as in applications such as softImpute.

rank.max

The maximum rank for the solution. This is also the dimension of the left and right matrices of orthogonal singular vectors. 'rank.max' should be no bigger than 'min(dim(x)'.

lambda

The regularization parameter. lambda=0 corresponds to an accelerated version of the orthogonal QR-algorithm. With lambda>0 the algorithm amounts to alternating orthogonal ridge regression.

thresh

convergence threshold, measured as the relative changed in the Frobenius norm between two successive estimates.

maxit

maximum number of iterations.

trace.it

with trace.it=TRUE, convergence progress is reported.

warm.start

an svd object can be supplied as a warm start. If the solution requested has higher rank than the warm start, the additional subspace is initialized with random Gaussians (and then orthogonalized wrt the rest).

final.svd

Although in theory, this algorithm converges to the solution to a nuclear-norm regularized low-rank matrix approximation problem, with potentially some singular values equal to zero, in practice only near-zeros are achieved. This final step does one more iteration with lambda=0, followed by soft-thresholding.

Details

This algorithm solves the problem

\min ||X-M||_F^2 +λ ||M||_*

subject to rank(M)≤q r, where ||M||_* is the nuclear norm of M (sum of singular values). It achieves this by solving the related problem

\min ||X-AB'||_F^2 +λ/2 (||A||_F^2+||B||_F^2)

subject to rank(A)=rank(B)≤q r. The solution is a rank-restricted, soft-thresholded SVD of X.

Value

An svd object is returned, with components "u", "d", and "v".

u

an m by rank.max matrix with the left orthogonal singular vectors

d

a vector of length rank.max of soft-thresholded singular values

v

an n by rank.max matrix with the right orthogonal singular vectors

Author(s)

Trevor Hastie, Rahul Mazumder
Maintainer: Trevor Hastie hastie@stanford.edu

References

Rahul Mazumder, Trevor Hastie and Rob Tibshirani (2010) Spectral Regularization Algorithms for Learning Large Incomplete Matrices, http://www.stanford.edu/~hastie/Papers/mazumder10a.pdf
Journal of Machine Learning Research 11 (2010) 2287-2322

See Also

biScale, softImpute, Incomplete, lambda0, impute, complete

Examples

 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
14
15
16
17
18
19
20
#create a matrix and run the algorithm
set.seed(101)
n=100
p=50
J=25
np=n*p
x=matrix(rnorm(n*J),n,J)%*%matrix(rnorm(J*p),J,p)+matrix(rnorm(np),n,p)/5
fit=svd.als(x,rank=25,lambda=50)
fit$d
pmax(svd(x)$d-50,0)
# now create a sparse matrix and do the same
nnz=trunc(np*.3)
inz=sample(seq(np),nnz,replace=FALSE)
i=row(x)[inz]
j=col(x)[inz]
x=rnorm(nnz)
xS=sparseMatrix(x=x,i=i,j=j)
fit2=svd.als(xS,rank=20,lambda=7)
fit2$d
pmax(svd(as.matrix(xS))$d-7,0)