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6.3. Kernel Approximation

This submodule contains functions that approximate the feature mappings that correspond to certain kernels, as they are used for example in support vector machines (see Support Vector Machines). The following feature functions perform non-linear transformations of the input, which can serve as a basis for linear classification or other algorithms.

The advantage of using approximate explicit feature maps compared to the kernel trick, which makes use of feature maps implicitly, is that explicit mappings can be better suited for online learning and can significantly reduce the cost of learning with very large datasets. Standard kernelized SVMs do not scale well to large datasets, but using an approximate kernel map it is possible to use much more efficient linear SVMs. In particularly the combination of kernel map approximations with SGDClassifier can make nonlinear learning on large datasets possible.

Since there has not been much empirical work using approximate embeddings, it is advisable to compare results against exact kernel methods when possible.

6.3.1. Radial Basis Function Kernel

The RBFSampler constructs an approximate mapping for the radial basis function kernel.

The mapping relies on a Monte Carlo approximation to the kernel values. The fit function performs the Monte Carlo sampling, whereas the transform method performs the mapping of the data. Because of the inherent randomness of the process, results may vary between different calls to the fit function.

The fit function takes two arguments: n_components, which is the target dimensionality of the feature transform, and gamma, the parameter of the RBF-kernel. A higher n_components will result in a better approximation of the kernel and will yield results more similar to those produced by a kernel SVM. Note that “fitting” the feature function does not actually depend on the data given to the fit function. Only the dimensionality of the data is used. Details on the method can be found in [RR2007].

../_images/plot_kernel_approximation_21.png

Comparing an exact RBF kernel (left) with the approximation (right)

6.3.2. Additive Chi Squared Kernel

The chi squared kernel is a kernel on histograms, often used in computer vision.

The chi squared kernel is given by

k(x, y) = \sum_i \frac{2x_iy_i}{x_i+y_i}

Since the kernel is additive, it is possible to treat all components x_i separately for embedding. This makes it possible to sample the Fourier transform in regular intervals, instead of approximating using Monte Carlo sampling.

The class AdditiveChi2Sampler implements this component wise deterministic sampling. Each component is sampled n times, yielding 2n+1 dimensions per input dimension (the multiple of two stems from the real and complex part of the Fourier transform). In the literature, n is usually choosen to be 1 or 2, transforming the dataset to size n_samples x 5 * n_features (in the case of n=2).

The approximate feature map provided by AdditiveChi2Sampler can be combined with the approximate feature map provided by RBFSampler to yield an approximate feature map for the exponentiated chi squared kernel. See the [VZ2010] for details and [VVZ2010] for combination with the RBFSampler.

6.3.3. Skewed Chi Squared Kernel

The skewed chi squared kernel is given by:

k(x,y) = \prod_i \frac{2\sqrt{x_i+c}\sqrt{y_i+c}}{x_i + y_i + 2c}

It has properties that are similar to the exponentiated chi squared kernel often used in computer vision, but allows for a simple Monte Carlo approximation of the feature map.

The usage of the SkewedChi2Sampler is the same as the usage described above for the RBFSampler. The only difference is in the free parameter, that is called c. For a motivation for this mapping and the mathematical details see [LS2010].

6.3.4. Mathematical Details

Kernel methods like support vector machines or kernelized PCA rely on a property of reproducing kernel Hilbert spaces. For any positive definite kernel function k (a so called Mercer kernel), it is guaranteed that there exists a mapping \phi into a Hilbert space \mathcal{H}, such that

k(x,y) = < \phi(x), \phi(y)>

Where < \cdot, \cdot > denotes the inner product in the Hilbert space.

If an algorithm, such as a linear support vector machine or PCA, relies only on the scalar product of data points x_i, one may use the value of k(x_i, x_j), which corresponds to applying the algorithm to the mapped data points \phi(x_i). The advantage of using k is that the mapping \phi never has to be calculated explicitly, allowing for arbitrary large features (even infinite).

One drawback of kernel methods is, that it might be necessary to store many kernel values k(x_i, x_j) during optimization. If a kernelized classifier is applied to new data y_j, k(x_i, y_j) needs to be computed to make predictions, possibly for many different x_i in the training set.

The classes in this submodule allow to approximate the embedding \phi, thereby working explicitly with the representations \phi(x_i), which obviates the need to apply the kernel or store training examples.

References:

[RR2007]“Random features for large-scale kernel machines” Rahimi, A. and Recht, B. - Advances in neural information processing 2007,
[LS2010]“Random Fourier approximations for skewed multiplicative histogram kernels” Random Fourier approximations for skewed multiplicative histogram kernels - Lecture Notes for Computer Sciencd (DAGM)
[VZ2010]“Efficient additive kernels via explicit feature maps” Vedaldi, A. and Zisserman, A. - Computer Vision and Pattern Recognition 2010
[VVZ2010]“Generalized RBF feature maps for Efficient Detection” Vempati, S. and Vedaldi, A. and Zisserman, A. and Jawahar, CV - 2010
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