IAM775: Reproducing Kernel Hilbert Spaces

Semester: Spring 2024
Level: Graduate
Department: Institute of Applied Mathematics, METU
Role: Teaching Assistant

Overview

Mathematical foundations of reproducing kernel Hilbert spaces with applications to machine learning, approximation theory, and functional analysis.

Sample notation

A symmetric positive definite kernel $k: X \times X \to \mathbb{R}$ induces a unique RKHS $\mathcal{H}_k$ in which evaluation is continuous and the reproducing property holds:

f(x) = \langle f,\, k(\cdot, x) \rangle_{\mathcal{H}_k} \quad\text{for all } f \in \mathcal{H}_k,\ x \in X.

The representer theorem says that regularized empirical risk minimizers in $\mathcal{H}_k$ admit a finite expansion $f^\star = \sum_{i=1}^n c_i\, k(\cdot, x_i)$ .

Status

This offering is archived. Materials may still be useful as a reference for kernel methods.