# Sharp analysis of low-rank kernel matrix approximations

@article{Bach2013SharpAO, title={Sharp analysis of low-rank kernel matrix approximations}, author={Francis R. Bach}, journal={ArXiv}, year={2013}, volume={abs/1208.2015} }

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…

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## References

SHOWING 1-10 OF 62 REFERENCES

### Efficient SVM Training Using Low-Rank Kernel Representations

- Computer ScienceJ. Mach. Learn. Res.
- 2001

This work shows that for a low rank kernel matrix it is possible to design a better interior point method (IPM) in terms of storage requirements as well as computational complexity and derives an upper bound on the change in the objective function value based on the approximation error and the number of active constraints (support vectors).

### Predictive low-rank decomposition for kernel methods

- Computer ScienceICML
- 2005

This paper presents an algorithm that can exploit side information (e.g., classification labels, regression responses) in the computation of low-rank decompositions for kernel matrices and presents simulation results that show that the algorithm yields decomposition of significantly smaller rank than those found by incomplete Cholesky decomposition.

### On the Impact of Kernel Approximation on Learning Accuracy

- Computer ScienceAISTATS
- 2010

Stability bounds based on the norm of the kernel approximation for these algorithms, including SVMs, KRR, and graph Laplacian-based regularization algorithms, are given to determine the degree of approximation that can be tolerated in the estimation of thekernel matrix.

### Improved Bounds for the Nyström Method With Application to Kernel Classification

- Computer ScienceIEEE Transactions on Information Theory
- 2013

A kernel classification approach based on the Nyström method is presented and it is shown that when the eigenvalues of the kernel matrix follow a p-power law, the number of support vectors can be reduced to N2p/(p2 - 1), which is sublinear in N when p > 1+√2, without seriously sacrificing its generalization performance.

### Compressed Least-Squares Regression

- Computer Science, MathematicsNIPS
- 2009

It is shown that solving the problem in the compressed domain instead of the initial domain reduces the estimation error at the price of an increased (but controlled) approximation error.

### Optimal Rates for the Regularized Least-Squares Algorithm

- Mathematics, Computer ScienceFound. Comput. Math.
- 2007

A complete minimax analysis of the problem is described, showing that the convergence rates obtained by regularized least-squares estimators are indeed optimal over a suitable class of priors defined by the considered kernel.

### Breaking the curse of kernelization: budgeted stochastic gradient descent for large-scale SVM training

- Computer ScienceJ. Mach. Learn. Res.
- 2012

Comprehensive empirical results show that BSGD achieves higher accuracy than the state-of-the-art budgeted online algorithms and comparable to non-budget algorithms, while achieving impressive computational efficiency both in time and space during training and prediction.

### Random Features for Large-Scale Kernel Machines

- Computer ScienceNIPS
- 2007

Two sets of random features are explored, provided convergence bounds on their ability to approximate various radial basis kernels, and it is shown that in large-scale classification and regression tasks linear machine learning algorithms applied to these features outperform state-of-the-art large- scale kernel machines.

### A high-dimensional Wilks phenomenon

- Mathematics
- 2011

A theorem by Wilks asserts that in smooth parametric density estimation the difference between the maximum likelihood and the likelihood of the sampling distribution converges toward a Chi-square…

### Fast Sparse Gaussian Process Methods: The Informative Vector Machine

- Computer ScienceNIPS
- 2002

A framework for sparse Gaussian process (GP) methods which uses forward selection with criteria based on information-theoretic principles, which allows for Bayesian model selection and is less complex in implementation is presented.