# nLab kernel method

Contents

### Context

#### Measure and probability theory

measure theory

probability theory

# Contents

## Idea

In the context of machine learning, a kernel method is effectively the analysis of a data set $D$ through a choice of a distance/metric-function $d \;\colon\; D \times D \to \mathbb{R}$ on it. The corresponding integral kernels $\exp(- \lambda \cdot d(-,-))$ turn out to contain useful information, if chosen correctly.

As a special case, if the data set $D$ is a “choice of rankings”, hence the set of permutations, of $n \in \mathbb{N}$ elements, distance functions come from the graph distance of Cayley graphs of the symmetric group, such as the Kendall distance (or Cayley distance) with associated kernels such as the Mallows kernel. With these choices, the kernel method overlaps with geometric group theory.

## Details

A kernel on a set $X$ is a function $K: X \times X \to \mathbb{R}$ (although complex-valued kernels are also used) which is symmetric and positive definite, in the sense that for any $N \geq 1$ and any $x_1,..., x_N \in X$, the matrix $K_{i j} = K(x_i, x_j)$ is positive definite, i.e., $\sum_{i, j} c_i c_j K_{i j} \geq 0$ for all $c_1, ..., c_N \in \mathbb{R}$.

This may be reformulated as a mapping $\phi : X \to H$, where $H$ is a reproducing kernel Hilbert space, a function space in which pointwise evaluation is a continuous linear functional.

The ‘kernel’ and ‘feature map to Hilbert space’ stories relate to each other as follows:

$K(x, .) = \phi (x).$

The ‘reproducing’ aspect of $H$ means that

$\langle f(.), K(x, .) \rangle = f(x),$

and this is continuous. So we have

$K(x, y) = \langle \phi (x), \phi(y) \rangle.$

$H$ is the span of the set of functions $\{K(x, .)| x \in X\}$, and, under certain conditions, when we find the $f \in H$ for which a functional is minimised, it takes the form

$f(x) = \sum_{i = 1}^m c_i K(x_i, x).$

Many linear algebra techniques in $H$ just involve the inner product, and so can be conducted as a form of nonlinear algebra back in $X$, using the ‘kernel trick’.

Kernels characterise the similarity of the input set. But how do they get chosen? Often a Gaussian radial-basis function (RBF) kernel is selected:

$K(x, y) = e^{-\|x - y\|^2 / \sigma^2}.$

Note that there’s a Bayesian version of kernel methods which uses Gaussian processes.

## Example

Consider binary classification where we look to form a classifier which accurately finds the labels in $\{\pm 1\}$ of a sample from a distribution over $X \times \{\pm 1\}$ on the basis of their $X$ values, given a set of labelled training data. The support vector machine approach to this task looks to find the hyperplane in the associated Hilbert space, $H$, which best separates the images of data points $\phi (x_i)$, so that those with the same label $(y_i)$ fall in the same half space. The control for overfitting comes from finding such a hyperplane that classifies the training data accurately with the largest perpendicular distance to the nearest of them (the support vectors).

### General

Survey and introduction:

On positive definiteness via Fourier transform/Bochner's theorem:

Proof that the Mallows kernel and Kendall kernel are positive definite:

Discussion of weighted variants:

### In topological machine learning

On kernel methods applicable to persistence diagrams/barcodes for making topological data analysis amenable to “topologicalmachine learning:

• Jan Reininghaus, Stefan Huber, Ulrich Bauer, Roland Kwitt, A Stable Multi-Scale Kernel for Topological Machine Learning, NIPS’15: Proceedings of the 28th International Conference on Neural Information Processing System, 2 (2015 3070–3078 $[$arXiv:1412.6821$]$

• Roland Kwitt, Stefan Huber, Marc Niethammer, Weili Lin, Ulrich Bauer, Statistical Topological Data Analysis – A Kernel Perspective, in: Advances in Neural Information Processing Systems (NIPS 2015) $[$ISBN:9781510825024, doi:10.5555/2969442.2969582$]$

• Bastian Rieck, Filip Sadlo, Heike Leitte, Topological Machine Learning with Persistence Indicator Functions, In: Topological Methods in Data Analysis and Visualization V TopoInVis (2017) 87-101 Mathematics and Visualization. Springer, $[$arXiv:1907.13496, doi:10.1007/978-3-030-43036-8_6$]$

• Raphael Reinauer, Matteo Caorsi, Nicolas Berkouk, Persformer: A Transformer Architecture for Topological Machine Learning $[$arXiv:2112.15210$]$