3,655 research outputs found
Randomized QR with Column Pivoting
The dominant contribution to communication complexity in factorizing a matrix
using QR with column pivoting is due to column-norm updates that are required
to process pivot decisions. We use randomized sampling to approximate this
process which dramatically reduces communication in column selection. We also
introduce a sample update formula to reduce the cost of sampling trailing
matrices. Using our column selection mechanism we observe results that are
comparable in quality to those obtained from the QRCP algorithm, but with
performance near unpivoted QR. We also demonstrate strong parallel scalability
on shared memory multiple core systems using an implementation in Fortran with
OpenMP.
This work immediately extends to produce low-rank truncated approximations of
large matrices. We propose a truncated QR factorization with column pivoting
that avoids trailing matrix updates which are used in current implementations
of level-3 BLAS QR and QRCP. Provided the truncation rank is small, avoiding
trailing matrix updates reduces approximation time by nearly half. By using
these techniques and employing a variation on Stewart's QLP algorithm, we
develop an approximate truncated SVD that runs nearly as fast as truncated QR
Fast and Accurate Proper Orthogonal Decomposition using Efficient Sampling and Iterative Techniques for Singular Value Decomposition
In this paper, we propose a computationally efficient iterative algorithm for
proper orthogonal decomposition (POD) using random sampling based techniques.
In this algorithm, additional rows and columns are sampled and a merging
technique is used to update the dominant POD modes in each iteration. We derive
bounds for the spectral norm of the error introduced by a series of merging
operations. We use an existing theorem to get an approximate measure of the
quality of subspaces obtained on convergence of the iteration. Results on
various datasets indicate that the POD modes and/or the subspaces are
approximated with excellent accuracy with a significant runtime improvement
over computing the truncated SVD. We also propose a method to compute the POD
modes of large matrices that do not fit in the RAM using this iterative
sampling and merging algorithms
Blocked rank-revealing QR factorizations: How randomized sampling can be used to avoid single-vector pivoting
Given a matrix of size , the manuscript describes a algorithm
for computing a QR factorization where is a permutation matrix,
is orthonormal, and is upper triangular. The algorithm is blocked, to allow
it to be implemented efficiently. The need for single vector pivoting in
classical algorithms for computing QR factorizations is avoided by the use of
randomized sampling to find blocks of pivot vectors at once. The advantage of
blocking becomes particularly pronounced when is very large, and possibly
stored out-of-core, or on a distributed memory machine. The manuscript also
describes a generalization of the QR factorization that allows to be a
general orthonormal matrix. In this setting, one can at moderate cost compute a
\textit{rank-revealing} factorization where the mass of is concentrated to
the diagonal entries. Moreover, the diagonal entries of closely approximate
the singular values of . The algorithms described have asymptotic flop count
, just like classical deterministic methods. The scaling
constant is slightly higher than those of classical techniques, but this is
more than made up for by reduced communication and the ability to block the
computation
Sampling and multilevel coarsening algorithms for fast matrix approximations
This paper addresses matrix approximation problems for matrices that are
large, sparse and/or that are representations of large graphs. To tackle these
problems, we consider algorithms that are based primarily on coarsening
techniques, possibly combined with random sampling. A multilevel coarsening
technique is proposed which utilizes a hypergraph associated with the data
matrix and a graph coarsening strategy based on column matching. Theoretical
results are established that characterize the quality of the dimension
reduction achieved by a coarsening step, when a proper column matching strategy
is employed. We consider a number of standard applications of this technique as
well as a few new ones. Among the standard applications we first consider the
problem of computing the partial SVD for which a combination of sampling and
coarsening yields significantly improved SVD results relative to sampling
alone. We also consider the Column subset selection problem, a popular low rank
approximation method used in data related applications, and show how multilevel
coarsening can be adapted for this problem. Similarly, we consider the problem
of graph sparsification and show how coarsening techniques can be employed to
solve it. Numerical experiments illustrate the performances of the methods in
various applications
High-Performance Out-of-core Block Randomized Singular Value Decomposition on GPU
Fast computation of singular value decomposition (SVD) is of great interest
in various machine learning tasks. Recently, SVD methods based on randomized
linear algebra have shown significant speedup in this regime. This paper
attempts to further accelerate the computation by harnessing a modern computing
architecture, namely graphics processing unit (GPU), with the goal of
processing large-scale data that may not fit in the GPU memory. It leads to a
new block randomized algorithm that fully utilizes the power of GPUs and
efficiently processes large-scale data in an out-of- core fashion. Our
experiment shows that the proposed block randomized SVD (BRSVD) method
outperforms existing randomized SVD methods in terms of speed with retaining
the same accuracy. We also show its application to convex robust principal
component analysis, which shows significant speedup in computer vision
applications
Subspace Iteration Randomization and Singular Value Problems
A classical problem in matrix computations is the efficient and reliable
approximation of a given matrix by a matrix of lower rank. The truncated
singular value decomposition (SVD) is known to provide the best such
approximation for any given fixed rank. However, the SVD is also known to be
very costly to compute. Among the different approaches in the literature for
computing low-rank approximations, randomized algorithms have attracted
researchers' recent attention due to their surprising reliability and
computational efficiency in different application areas. Typically, such
algorithms are shown to compute with very high probability low-rank
approximations that are within a constant factor from optimal, and are known to
perform even better in many practical situations. In this paper, we present a
novel error analysis that considers randomized algorithms within the subspace
iteration framework and show with very high probability that highly accurate
low-rank approximations as well as singular values can indeed be computed
quickly for matrices with rapidly decaying singular values. Such matrices
appear frequently in diverse application areas such as data analysis, fast
structured matrix computations and fast direct methods for large sparse linear
systems of equations and are the driving motivation for randomized methods.
Furthermore, we show that the low-rank approximations computed by these
randomized algorithms are actually rank-revealing approximations, and the
special case of a rank-1 approximation can also be used to correctly estimate
matrix 2-norms with very high probability. Our numerical experiments are in
full support of our conclusions.Comment: 45 pages, 5 figure
Pass-Efficient Randomized Algorithms for Low-Rank Matrix Approximation Using Any Number of Views
This paper describes practical randomized algorithms for low-rank matrix
approximation that accommodate any budget for the number of views of the
matrix. The presented algorithms, which are aimed at being as pass efficient as
needed, expand and improve on popular randomized algorithms targeting efficient
low-rank reconstructions. First, a more flexible subspace iteration algorithm
is presented that works for any views , instead of only allowing an
even . Secondly, we propose more general and more accurate single-pass
algorithms. In particular, we propose a more accurate memory efficient
single-pass method and a more general single-pass algorithm which, unlike
previous methods, does not require prior information to assure near peak
performance. Thirdly, combining ideas from subspace and single-pass algorithms,
we present a more pass-efficient randomized block Krylov algorithm, which can
achieve a desired accuracy using considerably fewer views than that needed by a
subspace or previously studied block Krylov methods. However, the proposed
accuracy enhanced block Krylov method is restricted to large matrices that are
either accessed a few columns or rows at a time. Recommendations are also given
on how to apply the subspace and block Krylov algorithms when estimating either
the dominant left or right singular subspace of a matrix, or when estimating a
normal matrix, such as those appearing in inverse problems. Computational
experiments are carried out that demonstrate the applicability and
effectiveness of the presented algorithms
On Truncated-SVD-like Sparse Solutions to Least-Squares Problems of Arbitrary Dimensions
We describe two algorithms for computing a sparse solution to a least-squares
problem where the coefficient matrix can have arbitrary dimensions. We show
that the solution vector obtained by our algorithms is close to the solution
vector obtained via the truncated SVD approach.Comment: This paper has been withdrawn by the author. This article has been
replaced by another submission: arXiv:1312.749
Literature survey on low rank approximation of matrices
Low rank approximation of matrices has been well studied in literature.
Singular value decomposition, QR decomposition with column pivoting, rank
revealing QR factorization (RRQR), Interpolative decomposition etc are
classical deterministic algorithms for low rank approximation. But these
techniques are very expensive operations are required for matrices). There are several randomized algorithms available in the
literature which are not so expensive as the classical techniques (but the
complexity is not linear in n). So, it is very expensive to construct the low
rank approximation of a matrix if the dimension of the matrix is very large.
There are alternative techniques like Cross/Skeleton approximation which gives
the low-rank approximation with linear complexity in n . In this article we
review low rank approximation techniques briefly and give extensive references
of many techniques
Frequent Directions : Simple and Deterministic Matrix Sketching
We describe a new algorithm called Frequent Directions for deterministic
matrix sketching in the row-updates model. The algorithm is presented an
arbitrary input matrix one row at a time. It performed
operations per row and maintains a sketch matrix such that for any
and .
Here, stands for the minimizer of over all rank
matrices (similarly ) and is the rank matrix resulting
from projecting on the row span of . We show both of these bounds are
the best possible for the space allowed. The summary is mergeable, and hence
trivially parallelizable. Moreover, Frequent Directions outperforms exemplar
implementations of existing streaming algorithms in the space-error tradeoff.Comment: 28 pages , This paper contains Frequent Directions algorithm (see
arXiv:1206.0594) and relative error bound on it (see arXiv:1307.7454
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