23,243 research outputs found
Revisiting the Nystrom Method for Improved Large-Scale Machine Learning
We reconsider randomized algorithms for the low-rank approximation of
symmetric positive semi-definite (SPSD) matrices such as Laplacian and kernel
matrices that arise in data analysis and machine learning applications. Our
main results consist of an empirical evaluation of the performance quality and
running time of sampling and projection methods on a diverse suite of SPSD
matrices. Our results highlight complementary aspects of sampling versus
projection methods; they characterize the effects of common data preprocessing
steps on the performance of these algorithms; and they point to important
differences between uniform sampling and nonuniform sampling methods based on
leverage scores. In addition, our empirical results illustrate that existing
theory is so weak that it does not provide even a qualitative guide to
practice. Thus, we complement our empirical results with a suite of worst-case
theoretical bounds for both random sampling and random projection methods.
These bounds are qualitatively superior to existing bounds---e.g. improved
additive-error bounds for spectral and Frobenius norm error and relative-error
bounds for trace norm error---and they point to future directions to make these
algorithms useful in even larger-scale machine learning applications.Comment: 60 pages, 15 color figures; updated proof of Frobenius norm bounds,
added comparison to projection-based low-rank approximations, and an analysis
of the power method applied to SPSD sketche
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
Matrix completion and extrapolation via kernel regression
Matrix completion and extrapolation (MCEX) are dealt with here over
reproducing kernel Hilbert spaces (RKHSs) in order to account for prior
information present in the available data. Aiming at a faster and
low-complexity solver, the task is formulated as a kernel ridge regression. The
resultant MCEX algorithm can also afford online implementation, while the class
of kernel functions also encompasses several existing approaches to MC with
prior information. Numerical tests on synthetic and real datasets show that the
novel approach performs faster than widespread methods such as alternating
least squares (ALS) or stochastic gradient descent (SGD), and that the recovery
error is reduced, especially when dealing with noisy data
Sharp analysis of low-rank kernel matrix approximations
We consider supervised learning problems within the positive-definite kernel
framework, such as kernel ridge regression, kernel logistic regression or the
support vector machine. With kernels leading to infinite-dimensional feature
spaces, a common practical limiting difficulty is the necessity of computing
the kernel matrix, which most frequently leads to algorithms with running time
at least quadratic in the number of observations n, i.e., O(n^2). Low-rank
approximations of the kernel matrix are often considered as they allow the
reduction of running time complexities to O(p^2 n), where p is the rank of the
approximation. The practicality of such methods thus depends on the required
rank p. In this paper, we show that in the context of kernel ridge regression,
for approximations based on a random subset of columns of the original kernel
matrix, the rank p may be chosen to be linear in the degrees of freedom
associated with the problem, a quantity which is classically used in the
statistical analysis of such methods, and is often seen as the implicit number
of parameters of non-parametric estimators. This result enables simple
algorithms that have sub-quadratic running time complexity, but provably
exhibit the same predictive performance than existing algorithms, for any given
problem instance, and not only for worst-case situations
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