Incremental SVD Package
Overview Documentation

## The Problem

Many applications can be characterized by the requirement for a subset of the singular values/vectors of a given matrix. Three distinct approaches exist for this problem:
• Compute the full SVD and truncate the parts that are not needed. This is wasteful, and often times, prohibitively expensive.
• Transform the singular value problem into a symmetric eigenvalue problem, apply an iterative eigensolver to compute the relevant part of the spectrum, back-transform the eigenvalue solutions into the desired singular value solutions. This is the most popular approach for large SVD problems, especially where the data matrix is sparse.
• Attack directly the singular value problem via some specialized solver:
• JD-SVD method of Hochstenbach. This is analogous to the Jacobi-Davidson eigensolver. It operates by using a Newton method to compute an update to the current approximation that attempts to set the SVD problem residuals to zero. This is wrapped by a Davidson-type two-sided subspace acceleration strategy, which serves to improve the rate of convergence and globalize the Newton method. JD-SVD is best applied to finding either the largest or smallest singular values.
• The computation of the dominant singular vectors can be characterized as the maximization of a particular function over a Riemannian manifold. This allows the application of a number of Riemannian optimization techniques. See the GenRTR page for more information. There is currently no analogous method for computing the smallest singular values.
• There exist a number of efforts which compute the dominant singular vectors via a neural network. Try this or this.
• A family of so-called low-rank incremental SVD methods allow the approximation of the dominant or dominated singular subspaces in a pass-efficient manner. These methods are the focus of this webpage.

## Incremental SVD Publications

The most recent talk of Baker provides a good introduction the the familiy of incremental/streaming SVD methods:

The family of low-rank incremental SVD methods were originally described in the following papers:

Baker's thesis described a generalization of these methods, with an emphasis on efficient implementations:

The following papers present further analysis of these methods, including descriptions of the multipass methods:

## IncPACK Software

IncPACK contains implementations of the low-rank incremental SVD methods for the approximation of either the largest or smallest singular values of a matrix, along with the associated singular vectors. Currently, the package provides a single solver, an implementation of the of the Sequential Karhunen-Loeve (Levy and Lindenbaum, 2000). This solver, as described by the authors, computes approximations for the left singular subspace. It has been modified to compute also the right singular subspace, and to improve on these approximations via multiple passes through the data matrix.

IncPACK currently provides implementations in MATLAB, released under an open-source modified BSD license. An MPI-based implementation in C++ is available in Trilinos in the RBGen package (development branch only). A beta-implementation for NVIDIA GPUs using CUDA is available below.

Version Date .zip .tgz
0.1.2 18 Dec 2012 IncPACK-0.1.2.zip IncPACK-0.1.2.tgz
Updated to include Modified BSD software license.
0.1.1 11 Apr 2008 IncPACK-0.1.1.zip IncPACK-0.1.1.tgz
Minor changes. Threshholding logic for 'S' case was not correct. Improved documentation.
0.1.0 8 Apr 2008 IncPACK-0.1.zip IncPACK-0.1.tgz
Initial public release of MATLAB IncSVD package. Provides implementation of sequential Karhunen-Loeve, with the following multipass strategies:
• simple restarting
• steepest descent (variant A)
• steepest descent (variant B)
Version Date .zip .tgz
1.0 31 Dec 2012 G-IncSVD-1.0.zip G-IncSVD-1.0.tgz
Initial release of GPU IncSVD (G-IncSVD), under modified BSD software license.

## Authors

The authors of the codes are:

## Funding

Funding for this work came in part from:

• National Science Foundation Award 032944: "Collaborative Research: Model Reduction of Dynamical Systems for Real Time Control"
• National Science Foundation Award 9912415: "Efficient Algorithms for Large Scale Dynamical Systems"

## Related software

• The JD Gateway contains information on the JD-SVD method, which can be used to compute the dominant or dominated SVD.
• The GenRTR package provides a solver for computing the dominant SVD.
• The RTRESGEV package provides eigensolvers, which can be used to compute dominant or dominated SVD.