Название: Learning with Fractional Orthogonal Kernel Classifiers in Support Vector Machines: Theory, Algorithms and Applications
Автор: Jamal Amani Rad, Kourosh Parand, Snehashish Chakraverty
Издательство: Springer
Серия: Industrial and Applied Mathematics
Год: 2023
Страниц: 312
Язык: английский
Формат: pdf (true), epub
Размер: 42.2 MB

This book contains select chapters on support vector algorithms from different perspectives, including mathematical background, properties of various kernel functions, and several applications. The main focus of this book is on orthogonal kernel functions, and the properties of the classical kernel functions―Chebyshev, Legendre, Gegenbauer, and Jacobi―are reviewed in some chapters. Moreover, the fractional form of these kernel functions is introduced in the same chapters, and for ease of use for these kernel functions, a tutorial on a Python package named ORSVM is presented. The book also exhibits a variety of applications for support vector algorithms, and in addition to the classification, these algorithms along with the introduced kernel functions are utilized for solving ordinary, partial, integro, and fractional differential equations.

On the other hand, nowadays, the real-time and big data applications of support vector algorithms are growing. Consequently, the Compute Unified Device Architecture (CUDA) parallelizing the procedure of support vector algorithms based on orthogonal kernel functions is presented. The book sheds light on how to use support vector algorithms based on orthogonal kernel functions in different situations and gives a significant perspective to all Machine Learning and scientific Machine Learning researchers all around the world to utilize fractional orthogonal kernel functions in their pattern recognition or scientific computing problems.

In recent years, Machine Learning has been applied in different areas of science and engineering including Computer Science, medical science, cognitive science, psychology, and so on. In these fields, Machine Learning non-specialists utilize it to address their problems. One of the most popular families of algorithms in Machine Learning is support vector machine (SVM) algorithms. Traditionally, these algorithms are used for binary classification problems. But recently, the SVM algorithms have been utilized in various areas including numerical analysis, computer vision, and so on. Therefore, the popularity of SVM algorithms has risen in recent years.

The main part of the SVM algorithms is the kernel function and the performance of a given SVM is related to the power of the kernel function. Different kernels provide different capabilities to the SVM algorithms; therefore, understanding the properties of the kernel functions is a crucial part of utilizing the SVM algorithms. Up until now, various kernel functions have been developed by researchers. One of the most significant families of the kernel functions is the orthogonal kernel function which has been attracting much attention. The computational power of this family of kernel functions has been illustrated by the researchers in the last few years. But despite the computational power of orthogonal kernel functions they have not been used in real application problems. This issue has various reasons, some of which are summarized in the following:

• 1. The mathematical complexity of orthogonal kernel functions formulation.
• 2. Lack of a simple and comprehensive resource for expressing the orthogonal kernels and their properties.
• 3. Implementation difficulties of these kernels and lack of a convenient package that implements these kernels.

For the purpose of solving the aforementioned issues, in this book, we are going to present a simple and comprehensive tutorial on orthogonal kernel functions and a Python package that is named ORSVM that contains the orthogonal kernel functions. The reason we chose Python as the language of the ORSVM package is Python is open source, very popular, easy to learn, and there are lots of tutorial for it:

• 1. Python has a lot of packages for manipulating data and they can be used besides the ORSVM for solving a machine learning problem.
• 2. Python is a multi-platform language and can be launched on different operating systems.

In addition to the developed orthogonal kernels, we aim to introduce some new kernel functions which are called fractional orthogonal kernels. The name fractional comes from the order x being a positive real number instead of being an integer. In fact, the fractional orthogonal kernels are extensions of integer order orthogonal functions. All introduced fractional orthogonal kernels in this book are implemented in the ORSVM package and their performance is illustrated by testing on some real datasets.

This book contains 12 chapters, including an appendix at the end of the book to cover any programming preliminaries. Chapter 1 includes the fundamental concepts of machine learning. In this chapter, we explain the definitions of pattern and similarity and then a geometrical intuition of the SVM algorithm is presented. At the end of this chapter, a historical review of the SVM and the current applications of SVM are discussed.

In Chap. 2, we present the basics of SVM and least-square SVM (LS-SVM). The mathematical background of SVM is presented in this chapter in detail. Moreover, Mercer’s theorem and kernel trick are discussed too. In the last part of this chapter, function approximation using the SVM is illustrated.

In Chap. 3, the discussion is about Chebyshev polynomials. At first, the properties of Chebyshev polynomials and fractional Chebyshev functions are explained. After that, a review of Chebyshev kernel functions is presented and the fractional Chebyshev kernel functions are introduced. In the final section of this chapter, the performance of fractional Chebyshev kernels on real datasets is illustrated and compared with other state-of-the-art kernels.

In Chap. 4, the Legendre polynomials are considered. In the beginning, the properties of the Legendre polynomials and fractional Legendre functions are explained. In the next step after reviewing the Legendre kernel functions, the fractional Legendre kernel functions are introduced. Finally, the performance of fractional Legendre kernels is illustrated by applying them to real datasets.

Another orthogonal polynomial series is discussed in Chap. 5 (the Gegenbauer polynomials). Similar to the previous chapters, this chapter includes properties of the Gegenbauer polynomials, properties of the fractional Gegenbauer functions, a review on Gegenbauer kernels, introducing fractional Gegenbauer kernels, and showing the performance of fractional Gegenbauer kernels on real datasets.

In Chap. 6, we focus on Jacobi polynomials. This family of polynomials is the general form of the previous polynomials which are presented in Chaps. 3–5. Therefore, the relations between the polynomials and the kernels are discussed in this chapter. In addition to the relations between the polynomials, other parts of this chapter are similar to the three previous chapters.

In Chaps. 7 and 8, some applications of the SVM in scientific computing are presented and the procedure of using LS-SVM for solving ordinary/partial differential equations is explained. Mathematical basics of ordinary/partial differential equations and traditional numerical algorithms for approximating the solution of ordinary/partial differential equations are discussed in these chapters too.

Chapter 9 consists of the procedure of using the LS-SVM algorithm for solving integral equations, basics of integral equations, traditional analytical and numerical algorithms for solving integral equations, and a numerical algorithm based on LS-SVR for solving various kinds of integral equations.