Fast Truncated Singular Value Decomposition and Principal Components Analysis for Large Dense and ….

Fast and memory efficient methods for truncated singular value decomposition and principal components analysis of large sparse and dense matrices.


Implicitly-restarted Lanczos methods for fast truncated singular value decomposition of sparse and dense matrices (also referred to as partial SVD). IRLBA stands for Augmented, /bmplicitly Restarted /banczos /bidiagonalization Algorithm. The package provides the following functions (see help on each for details and examples).

  • irlba() partial SVD function
  • ssvd() l1-penalized matrix decompoisition for sparse PCA (based on Shen and Huang's algorithm)
  • prcomp_irlba() principal components function similar to the prcomp function in stats package for computing the first few principal components of large matrices
  • svdr() alternate partial SVD function based on randomized SVD (see also the rsvd package by N. Benjamin Erichson for an alternative implementation)
  • partial_eigen() a very limited partial eigenvalue decomposition for symmetric matrices (see the RSpectra package for more comprehensive truncated eigenvalue decomposition)

Help documentation for each function includes extensive documentation and examples. Also see the package vignette, vignette("irlba", package="irlba").

An overview web page is here:

New in 2.3.2

  • Fixed a regression in prcomp_irlba() discovered by Xiaojie Qiu, see, and other related problems reported in
  • Added rchk testing to pre-CRAN submission tests.
  • Fixed a sign bug in ssvd() found by Alex Poliakov.

What's new in Version 2.3.1?

  • Fixed an irlba() bug associated with centering (PCA), see
  • Fixed irlba() scaling to conform to scale, see
  • Improved prcomp_irlba() from a suggestion by N. Benjamin Erichson, see
  • Significanty changed/improved svdr() convergence criterion.
  • Added a version of Shen and Huang's Sparse PCA/SVD L1-penalized matrix decomposition (ssvd()).
  • Fixed valgrind errors.

Deprecated features

I will remove partial_eigen() in a future version. As its documentation states, users are better off using the RSpectra package for eigenvalue computations (although not generally for singular value computations).

The mult argument is deprecated and will be removed in a future version. We now recommend simply defining a custom class with a custom multiplcation operator. The example below illustrates the old and new approaches.

A <- matrix(rnorm(100), 10)

# ------------------ old way ----------------------------------------------
# A custom matrix multiplication function that scales the columns of A
# (cf the scale option). This function scales the columns of A to unit norm.
col_scale <- sqrt(apply(A, 2, crossprod))
mult <- function(x, y)
          # check if x is a  vector
          if (is.vector(x))
            return((x %*% y) / col_scale)
          # else x is the matrix
          x %*% (y / col_scale)
irlba(A, 3, mult=mult)$d
## [1] 1.820227 1.622988 1.067185

# Compare with:
irlba(A, 3, scale=col_scale)$d
## [1] 1.820227 1.622988 1.067185

# Compare with:
svd(sweep(A, 2, col_scale, FUN=`/`))$d[1:3]
## [1] 1.820227 1.622988 1.067185

# ------------------ new way ----------------------------------------------
setClass("scaled_matrix", contains="matrix", slots=c(scale="numeric"))
setMethod("%*%", signature(x="scaled_matrix", y="numeric"), function(x ,y) [email protected] %*% (y / [email protected]))
setMethod("%*%", signature(x="numeric", y="scaled_matrix"), function(x ,y) (x %*% [email protected]) / [email protected])
a <- new("scaled_matrix", A, scale=col_scale)

irlba(a, 3)$d
## [1] 1.820227 1.622988 1.067185

We have learned that using R's existing S4 system is simpler, easier, and more flexible than using custom arguments with idiosyncratic syntax and behavior. We've even used the new approach to implement distributed parallel matrix products for very large problems with amazingly little code.

Wishlist / help wanted...

  • More Matrix classes supported in the fast code path
  • Help improving the solver for singular values in tricky cases (basically, for ill-conditioned problems and especially for the smallest singular values); in general this may require a combination of more careful convergence criteria and use of harmonic Ritz values; Dmitriy Selivanov has proposed alternative convergence criteria in for example.


  • Baglama, James, and Lothar Reichel. "Augmented implicitly restarted Lanczos bidiagonalization methods." SIAM Journal on Scientific Computing 27.1 (2005): 19-42.
  • Halko, Nathan, Per-Gunnar Martinsson, and Joel A. Tropp. "Finding structure with randomness: Stochastic algorithms for constructing approximate matrix decompositions." (2009).
  • 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, Daniela M., Robert Tibshirani, and Trevor Hastie. "A penalized matrix decomposition, with applications to sparse principal components and canonical correlation analysis." Biostatistics 10.3 (2009): 515-534.




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