ssvd {irlba}

Sparse regularized low-rank matrix approximation.

Description

Estimate an {\ell}1-penalized singular value or principal components decomposition (SVD or PCA) that introduces sparsity in the right singular vectors based on the fast and memory-efficient sPCA-rSVD algorithm of Haipeng Shen and Jianhua Huang.

Usage

ssvd(x, k = 1, n = 2, maxit = 500, tol = 0.001, center = FALSE,
  scale. = FALSE, alpha = 0, tsvd = NULL, ...)

Arguments

x

A numeric real- or complex-valued matrix or real-valued sparse matrix.

k

Matrix rank of the computed decomposition (see the Details section below).

n

Number of nonzero components in the right singular vectors. If k > 1, then a single value of n specifies the number of nonzero components in each regularized right singular vector. Or, specify a vector of length k indicating the number of desired nonzero components in each returned vector. See the examples.

maxit

Maximum number of soft-thresholding iterations.

tol

Convergence is determined when \|U_j - U_{j-1}\|_F < tol, where U_j is the matrix of estimated left regularized singular vectors at iteration j.

center

a logical value indicating whether the variables should be shifted to be zero centered. Alternately, a centering vector of length equal the number of columns of x can be supplied. Use center=TRUE to perform a regularized sparse PCA.

scale.

a logical value indicating whether the variables should be scaled to have unit variance before the analysis takes place. Alternatively, a vector of length equal the number of columns of x can be supplied.

The value of scale determines how column scaling is performed (after centering). If scale is a numeric vector with length equal to the number of columns of x, then each column of x is divided by the corresponding value from scale. If scale is TRUE then scaling is done by dividing the (centered) columns of x by their standard deviations if center=TRUE, and the root mean square otherwise. If scale is FALSE, no scaling is done. See scale for more details.

alpha

Optional scalar regularization parameter between zero and one (see Details below).

tsvd

Optional initial rank-k truncated SVD or PCA (skips computation if supplied).

...

Additional arguments passed to irlba.

Details

The ssvd function implements a version of an algorithm by Shen and Huang that computes a penalized SVD or PCA that introduces sparsity in the right singular vectors by solving a penalized least squares problem. The algorithm in the rank 1 case finds vectors u, w that minimize

\|x - u w^T\|_F^2 + \lambda \|w\|_1

such that \|u\| = 1, and then sets v = w / \|w\| and d = u^T x v; see the referenced paper for details. The penalty \lambda is implicitly determined from the specified desired number of nonzero values n. Higher rank output is determined similarly but using a sequence of \lambda values determined to maintain the desired number of nonzero elements in each column of v specified by n. Unlike standard SVD or PCA, the columns of the returned v when k > 1 may not be orthogonal.

Value

A list containing the following components:

• u regularized left singular vectors with orthonormal columns

• d regularized upper-triangluar projection matrix so that x %*% v == u %*% d

• v regularized, sparse right singular vectors with columns of unit norm

• center, scale the centering and scaling used, if any

• lambda the per-column regularization parameter found to obtain the desired sparsity

• iter number of soft thresholding iterations

• n value of input parameter n

• alpha value of input parameter alpha

Note

Our ssvd implementation of the Shen-Huang method makes the following choices:

1. The l1 penalty is the only available penalty function. Other penalties may appear in the future.

2. Given a desired number of nonzero elements in v, value(s) for the \lambda penalty are determined to achieve the sparsity goal subject to the parameter alpha.

3. An experimental block implementation is used for results with rank greater than 1 (when k > 1) instead of the deflation method described in the reference.

4. The choice of a penalty lambda associated with a given number of desired nonzero components is not unique. The alpha parameter, a scalar between zero and one, selects any possible value of lambda that produces the desired number of nonzero entries. The default alpha = 0 selects a penalized solution with largest corresponding value of d in the 1-d case. Think of alpha as fine-tuning of the penalty.

5. Our method returns an upper-triangular matrix d when k > 1 so that x %*% v == u %*% d. Non-zero elements above the diagonal result from non-orthogonality of the v matrix, providing a simple interpretation of cumulative information, or explained variance in the PCA case, via the singular value decomposition of d %*% t(v).

What if you have no idea for values of the argument n (the desired sparsity)? The reference describes a cross-validation and an ad-hoc approach; neither of which are in the package yet. Both are prohibitively computationally expensive for matrices with a huge number of columns. A future version of this package will include a revised approach to automatically selecting a reasonable sparsity constraint.

Compare with the similar but more general functions SPC and PMD in the PMA package by Daniela M. Witten, Robert Tibshirani, Sam Gross, and Balasubramanian Narasimhan. The PMD function can compute low-rank regularized matrix decompositions with sparsity penalties on both the u and v vectors. The ssvd function is similar to the PMD(*, L1) method invocation of PMD or alternatively the SPC function. Although less general than PMD(*), the ssvd function can be faster and more memory efficient for the basic sparse PCA problem. See https://bwlewis.github.io/irlba/ssvd.html for more information.

(* Note that the s4vd package by Martin Sill and Sebastian Kaiser, https://cran.r-project.org/package=s4vd, includes a fast optimized version of a closely related algorithm by Shen, Huang, and Marron, that penalizes both u and v.)

References

• Shen, Haipeng, and Jianhua Z. Huang. "Sparse principal component analysis via regularized low rank matrix approximation." Journal of multivariate analysis 99.6 (2008): 1015-1034.

• Witten, Tibshirani and Hastie (2009) A penalized matrix decomposition, with applications to sparse principal components and canonical correlation analysis. _Biostatistics_ 10(3): 515-534.

Examples

set.seed(1)
u <- matrix(rnorm(200), ncol=1)
v <- matrix(c(runif(50, min=0.1), rep(0,250)), ncol=1)
u <- u / drop(sqrt(crossprod(u)))
v <- v / drop(sqrt(crossprod(v)))
x <- u %*% t(v) + 0.001 * matrix(rnorm(200*300), ncol=300)
s <- ssvd(x, n=50)
table(actual=v[, 1] != 0, estimated=s$v[, 1] != 0)
oldpar <- par(mfrow=c(2, 1))
plot(u, cex=2, main="u (black circles), Estimated u (blue discs)")
points(s$u, pch=19, col=4)
plot(v, cex=2, main="v (black circles), Estimated v (blue discs)")
points(s$v, pch=19, col=4)

# Let's consider a trivial rank-2 example (k=2) with noise. Like the
# last example, we know the exact number of nonzero elements in each
# solution vector of the noise-free matrix. Note the application of
# different sparsity constraints on each column of the estimated v.
# Also, the decomposition is unique only up to sign, which we adjust
# for below.
set.seed(1)
u <- qr.Q(qr(matrix(rnorm(400), ncol=2)))
v <- matrix(0, ncol=2, nrow=300)
v[sample(300, 15), 1] <- runif(15, min=0.1)
v[sample(300, 50), 2] <- runif(50, min=0.1)
v <- qr.Q(qr(v))
x <- u %*% (c(2, 1) * t(v)) + .001 * matrix(rnorm(200 * 300), 200)
s <- ssvd(x, k=2, n=colSums(v != 0))

# Compare actual and estimated vectors (adjusting for sign):
s$u <- sign(u) * abs(s$u)
s$v <- sign(v) * abs(s$v)
table(actual=v[, 1] != 0, estimated=s$v[, 1] != 0)
table(actual=v[, 2] != 0, estimated=s$v[, 2] != 0)
plot(v[, 1], cex=2, main="True v1 (black circles), Estimated v1 (blue discs)")
points(s$v[, 1], pch=19, col=4)
plot(v[, 2], cex=2, main="True v2 (black circles), Estimated v2 (blue discs)")
points(s$v[, 2], pch=19, col=4)
par(oldpar)