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This paper presents a general reproducing kernel Hilbert Spaces (RKHS) framework with its various applications in statistical learning area. This theory has been around for quite some time and has been widely used in nonlinear regression and classification problems. Kernel methods, which map data from low-dimensional space into higher-dimensional space (RKHS), can be transferred in many classical statistical learning algorithms. This paper can be roughly divided into two parts. In the first part, the writer attempts to take the reader from a very basic understanding of fields through Hilbert spaces, into reproducing kernel Hilbert spaces. In the second part, the writer want to show reader the abundant applications of kernel methods in statistical learning algorithms, with algorithms and read-world examples.

Keywords: RKHS, kernel methods, statistical learning, SVM

• Chapter 1: Introduction
• Chapter 2: Reproducing Kernel Hilbert Spaces
• Chapter 3: Applications in Different Statistical Learning Algorithms
• Chapter 4: Real-World Example: handwritten digits recognition
• Chapter 5: Conclusion

• Some of the data and code used in this paper comes from Cousera Online Course: Machine Learning, Andrew Ng.
• The algorithm of kernel PCA in Chapter 3 is created by Ambarish Jash.
• The algorithm of SVM in Chapter 4 relies on LIBSVM package.
• The algorithm of Neural Network in Chapter 4 relies on UFLDL Tutorial.

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