Why are kernel methods with RBFs effective for handwritten digits (letters) classification? The question emerged while reading Ch. 3 of Rasmussen & Williams . In the end of this chapter, the authors gave results for the problem of handwritten digits classification (16x16 greyscale pictures); features are 256 pixel intensities + bias. I was surprised that in such a high-dimensional problem, 'metric' methods, like Gaussian processes with squared exponential kernel, or SVM with the same kernel, behave quite nice without any dimension reduction preceeded.
Also, I heard sometimes that SVM is good for [essentially bag-of-word] text classification. Why aren't they suffering from the curse of dimensionality?
 A: From a machine learning perspective, 257 dimensions is far from high dimensional. A wide range of problems, including text classification, are solved in thousands of dimensions. Currently, problems that are considered high dimensional are in millions of dimensions.
Both Gaussian processes and SVMs are part of a larger class of algorithms called kernel methods. The combination of the kernel trick, the representer theorem and proper use of regularization make these methods robust against the curse of dimensionality. 
Kernel methods always work on distances between points, regardless of the dimensionality in which this distance is defined. The dimensionality can be infinite, for example with the Gaussian kernel (see e.g. slide 11 of this presentation by Chih-Jen Lin).
From the representer theorem we know that the solution of any kernel method can be written in terms of instances, and is therefore limited in dimensionality and complexity.
A: To put this interesting discussion on the ground, one can modify the toy script from http://scikit-learn.org/stable/auto_examples/plot_digits_classification.html to get some simple statistics.
For example, let us look at the last test vector, and how far it is from 519 support vectors (the space has 8*6=42 dimensions).
Distances are big enough, laying in the interval [26.7394839142 64.88451279]. However, gamma is small (0.001), i.e. variance is big so "effective distance" is small (and that is the key, I think!).
As a consequence, scalar product (i.e. exp(-gamma*|test-support|^2) with all 519 support vectors lays in the interval [0.937175688934 0.973614850818]
