3,079 research outputs found
Cholesky-factorized sparse Kernel in support vector machines
Support Vector Machine (SVM) is one of the most powerful machine learning algorithms due to its convex optimization formulation and handling non-linear classification. However, one of its main drawbacks is the long time it takes to train large data sets. This limitation is often aroused when applying non-linear kernels (e.g. RBF Kernel) which are usually required to obtain better separation for linearly inseparable data sets. In this thesis, we study an approach that aims to speed-up the training time by combining both the better performance of RBF kernels and fast training by a linear solver, LIBLINEAR. The approach uses an RBF kernel with a sparse matrix which is factorized using Cholesky decomposition. The method is tested on large artificial and real data sets and compared to the standard RBF and linear kernels where both the accuracy and training time are reported. For most data sets, the result shows a huge training time reduction, over 90\%, whilst maintaining the accuracy
On kernel engineering via PaleyâWiener
A radial basis function approximation takes the form
where the coefficients a 1,âŚ,a n are real numbers, the centres b 1,âŚ,b n are distinct points in â d , and the function Ď:â d ââ is radially symmetric. Such functions are highly useful in practice and enjoy many beautiful theoretical properties. In particular, much work has been devoted to the polyharmonic radial basis functions, for which Ď is the fundamental solution of some iterate of the Laplacian. In this note, we consider the construction of a rotation-invariant signed (Borel) measure Îź for which the convolution Ď=Îź Ď is a function of compact support, and when Ď is polyharmonic. The novelty of this construction is its use of the PaleyâWiener theorem to identify compact support via analysis of the Fourier transform of the new kernel Ď, so providing a new form of kernel engineering
Stochastic expansions using continuous dictionaries: L\'{e}vy adaptive regression kernels
This article describes a new class of prior distributions for nonparametric
function estimation. The unknown function is modeled as a limit of weighted
sums of kernels or generator functions indexed by continuous parameters that
control local and global features such as their translation, dilation,
modulation and shape. L\'{e}vy random fields and their stochastic integrals are
employed to induce prior distributions for the unknown functions or,
equivalently, for the number of kernels and for the parameters governing their
features. Scaling, shape, and other features of the generating functions are
location-specific to allow quite different function properties in different
parts of the space, as with wavelet bases and other methods employing
overcomplete dictionaries. We provide conditions under which the stochastic
expansions converge in specified Besov or Sobolev norms. Under a Gaussian error
model, this may be viewed as a sparse regression problem, with regularization
induced via the L\'{e}vy random field prior distribution. Posterior inference
for the unknown functions is based on a reversible jump Markov chain Monte
Carlo algorithm. We compare the L\'{e}vy Adaptive Regression Kernel (LARK)
method to wavelet-based methods using some of the standard test functions, and
illustrate its flexibility and adaptability in nonstationary applications.Comment: Published in at http://dx.doi.org/10.1214/11-AOS889 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Gagliardo-Nirenberg Inequalities for Differential Forms in Heisenberg Groups
The L 1-Sobolev inequality states that the L n/(n--1)-norm of a compactly
supported function on Euclidean n-space is controlled by the L 1-norm of its
gradient. The generalization to differential forms (due to Lanzani & Stein and
Bourgain & Brezis) is recent, and states that a the L n/(n--1)-norm of a
compactly supported differential h-form is controlled by the L 1-norm of its
exterior differential du and its exterior codifferential u (in special
cases the L 1-norm must be replaced by the H 1-Hardy norm). We shall extend
this result to Heisenberg groups in the framework of an appropriate complex of
differential forms
On the spectral distribution of kernel matrices related to\ud radial basis functions
This paper focuses on the spectral distribution of kernel matrices related to radial basis functions. The asymptotic behaviour of eigenvalues of kernel matrices related to radial basis functions with different smoothness are studied. These results are obtained by estimated the coefficients of an orthogonal expansion of the underlying kernel function. Beside many other results, we prove that there are exactly (k+dâ1/d-1) eigenvalues in the same order for analytic separable kernel functions like the Gaussian in Rd. This gives theoretical support for how to choose the diagonal scaling matrix in the RBF-QR method (Fornberg et al, SIAM J. Sci. Comput. (33), 2011) which can stably compute Gaussian radial basis function interpolants
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