Nyström Approximation¶
While linear techniques operating in the primal (input) space are able to achieve good generalization capabilities in some specific application areas, one cannot in general approximate with the linear model more complex or highly nonlinear functions. We apply a FixedSize approach [FS2010] and Nyström approximation [WS2001] to approximate a kernelinduced feature map with some higher dimensional explicit and approximate feature vector.
We select prototype vectors (a small working sample of size ) and construct, for instance an RBF kernel matrix with
By following the approach in [WS2001] an expression for the entries of the approximated feature map , with is given by
where and denote the ith eigenvalue and the ith eigenvector of .
Available API¶

AFEm
(Xs, kernel, X)¶ Performs Automatic Feature Extraction (AFE) by Nyström method [WS2001] using a subsample . We restrict
kernel <: Kernel
to be a subclass ofKernel
, for instanceRBFKernel
.Parameters:  Xs – subset which is used to construct kernel matrix
 kernel – kernel function, e.g.
RBFKernel()
, used to construct kernel matrix  X – full dataset
Returns: new dataset derived from stacking together feature maps for every

entropy_subset
(X, kernel, subset_size)¶ Performs maximization of the quadratic Rényi Entropy by the representative points selection from
X
which can be supplied toAFEm
asXs
subset.Parameters:  X – full dataset
 kernel – kernel function, e.g.
RBFKernel()
, used to construct kernel matrix over which we compute Rényi Entropy  subset_size – number of representative data points
Available Kernel Functions¶

LinearKernel
()¶ Defines an implementation of the Linear Kernel, i.e. .

PolynomialKernel
()¶ Defines an implementation of the Polynomial Kernel, i.e. .

RBFKernel
()¶ Defines an implementation of the Radial Basis Function (RBF) Kernel, i.e. .
[WS2001]  (1, 2, 3) Williams C. and Seeger M., “Using the Nyström method to speed up kernel machines”, in Proceedings of the 14th Annual Conference on Neural Information Processing (NIPS), pp. 682688, 2001. 