119,704 research outputs found
A Scalable CUR Matrix Decomposition Algorithm: Lower Time Complexity and Tighter Bound
The CUR matrix decomposition is an important extension of Nystr\"{o}m
approximation to a general matrix. It approximates any data matrix in terms of
a small number of its columns and rows. In this paper we propose a novel
randomized CUR algorithm with an expected relative-error bound. The proposed
algorithm has the advantages over the existing relative-error CUR algorithms
that it possesses tighter theoretical bound and lower time complexity, and that
it can avoid maintaining the whole data matrix in main memory. Finally,
experiments on several real-world datasets demonstrate significant improvement
over the existing relative-error algorithms.Comment: accepted by NIPS 201
Efficient Algorithms and Error Analysis for the Modified Nystrom Method
Many kernel methods suffer from high time and space complexities and are thus
prohibitive in big-data applications. To tackle the computational challenge,
the Nystr\"om method has been extensively used to reduce time and space
complexities by sacrificing some accuracy. The Nystr\"om method speedups
computation by constructing an approximation of the kernel matrix using only a
few columns of the matrix. Recently, a variant of the Nystr\"om method called
the modified Nystr\"om method has demonstrated significant improvement over the
standard Nystr\"om method in approximation accuracy, both theoretically and
empirically.
In this paper, we propose two algorithms that make the modified Nystr\"om
method practical. First, we devise a simple column selection algorithm with a
provable error bound. Our algorithm is more efficient and easier to implement
than and nearly as accurate as the state-of-the-art algorithm. Second, with the
selected columns at hand, we propose an algorithm that computes the
approximation in lower time complexity than the approach in the previous work.
Furthermore, we prove that the modified Nystr\"om method is exact under certain
conditions, and we establish a lower error bound for the modified Nystr\"om
method.Comment: 9-page paper plus appendix. In Proceedings of the 17th International
Conference on Artificial Intelligence and Statistics (AISTATS) 2014,
Reykjavik, Iceland. JMLR: W&CP volume 3
Optimal CUR Matrix Decompositions
The CUR decomposition of an matrix finds an
matrix with a subset of columns of together with an matrix with a subset of rows of as well as a
low-rank matrix such that the matrix approximates the matrix
that is, , where
denotes the Frobenius norm and is the best matrix
of rank constructed via the SVD. We present input-sparsity-time and
deterministic algorithms for constructing such a CUR decomposition where
and and rank. Up to constant
factors, our algorithms are simultaneously optimal in and rank.Comment: small revision in lemma 4.
On sparse representations of linear operators and the approximation of matrix products
Thus far, sparse representations have been exploited largely in the context
of robustly estimating functions in a noisy environment from a few
measurements. In this context, the existence of a basis in which the signal
class under consideration is sparse is used to decrease the number of necessary
measurements while controlling the approximation error. In this paper, we
instead focus on applications in numerical analysis, by way of sparse
representations of linear operators with the objective of minimizing the number
of operations needed to perform basic operations (here, multiplication) on
these operators. We represent a linear operator by a sum of rank-one operators,
and show how a sparse representation that guarantees a low approximation error
for the product can be obtained from analyzing an induced quadratic form. This
construction in turn yields new algorithms for computing approximate matrix
products.Comment: 6 pages, 3 figures; presented at the 42nd Annual Conference on
Information Sciences and Systems (CISS 2008
Spatial Random Sampling: A Structure-Preserving Data Sketching Tool
Random column sampling is not guaranteed to yield data sketches that preserve
the underlying structures of the data and may not sample sufficiently from
less-populated data clusters. Also, adaptive sampling can often provide
accurate low rank approximations, yet may fall short of producing descriptive
data sketches, especially when the cluster centers are linearly dependent.
Motivated by that, this paper introduces a novel randomized column sampling
tool dubbed Spatial Random Sampling (SRS), in which data points are sampled
based on their proximity to randomly sampled points on the unit sphere. The
most compelling feature of SRS is that the corresponding probability of
sampling from a given data cluster is proportional to the surface area the
cluster occupies on the unit sphere, independently from the size of the cluster
population. Although it is fully randomized, SRS is shown to provide
descriptive and balanced data representations. The proposed idea addresses a
pressing need in data science and holds potential to inspire many novel
approaches for analysis of big data
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