# Reproducing kernel hilbert space

## Spectral decomposition

Gaussian kernel matrix can be factorized into $$(\Phi \textbf{X})^\textbf{H} \Phi \textbf{X} =\textbf{X}^\textbf{H} \Phi^\textbf{H} \Phi \textbf{X} = \textbf{X}^\textbf{H}\textbf{X}$$, where $$\Phi$$ is Gaussian kernel basis matrix and $$\textbf{X}$$ is coefficients matrix of reproducing kernel Hilbert space $$K(\cdot,x) \in \mathcal{H}_K$$ https://www.jkangpathology.com/post/reproducing-kernel-hilbert-space/. A matrix is a system. A system takes input and gives output. A matrix is a linear system. Differentiation and Integration are linear systems. Fourier transformation matches input basis and operator (differentiation) basis.

## Reproducing Kernel Hilbert Space

Finally arrive at reproducing kernel Hilbert space. https://nzer0.github.io/reproducing-kernel-hilbert-space.html The above post introduces RKHS in Korean. It was helpful. I had struggled to understand some concepts in RKHS. What does mean Hilbert space in terms of feature expansion? ($$f:\mathcal{X} \to \mathbb{R}$$, $$f \in \mathcal{H}_K$$) It was confusing the difference between $$f$$ and $$f(x)$$. $$f$$ means the function in Hilbert space and $$f(x)$$ is evaluation. I thought that the function can be represented by the inner product of the basis of feature space $$K(\cdot,x)$$ and coefficients $$f$$, and the coefficients are vectors in feature space.

## Analysis

The reproducing kernel hilbert space (RKHS) was my motivation to study analysis. The hilbert space is a orthogonal normed vector space. I still do not know about the meaning of “reproducing kernal”. The RKHS appeared in the book titled An Introduction to Statisitical Learning written by Hastie. I began to google the meaning of the spaces such as the Hilbert, Banarch. I decided to read the Understaing Analysis written by Abbott.