Improvised Admissible Kernel Function for Support Vector Machines in Banach Space for Multiclass Data

Classification based on supervised learning theory is one of the most significant tasks frequently accomplished by so-called Intelligent Systems. Contrary to the traditional classification techniques that are used to validate or contradict a predefined hypothesis, kernel based classifiers offer the possibility to frame new hypotheses using statistical learning theory (Sangeetha and Kalpana, 2010). Support Vector Machine (SVM) is a standard kernel based learning algorithm where it improves the learning ability through experience. It is highly accurate, robust and optimal kernel based classification technique that is well-suited to many real time applications. In this paper, kernel functions related to Hilbert space and Banach Space are explained. Here, the experimental results are carried out using benchmark multiclass datasets which are taken from UCI Machine Learning Repository and their performance are compared using various metrics like support vector, support vector percentage, training time and accuracy.


INTRODUCTION
Despite of about 50 years of research investigation, developing a unified theory for a pattern recognizer has been an elusive task (Dutt et al. 2011). Three processes in pattern recognition are data acquisition, data analysis and pattern classification. Data acquisition is a first step in which the acquired raw data is converted into information, where it can be given as an input to the computing machine for next step in processing. Data analysis is a process where the collected information and pattern classes are analyzed. Based on the knowledge derived from data analysis step, pattern classification categorizes the data into different classes.
Data that are given as an input to the pattern classification is splitted into two types of datasets i.e. training dataset and testing dataset. Pattern classification (Sharma and Kishor, 2007) technique first takes the training dataset as an input to train the system and then the effectiveness of the system is tested by presenting the test data to it. The paper is organized as follows. Section 2 explains the multiclass classification in SVM and Section 3 describes the hybrid kernel function and proposed kernel function. Section 4 discusses the experimental results and Section 5 concludes the work.

MULTICLASS CLASSIFICATION (Vojtech Franc and Václav Hlavác, 2009)
Multiclass SVM can be solved by combining the binary classification decision functions. Multiclass SVM is of two types namely One against One (OAO) and One against All (OAA) decomposition. The OAA decomposition transforms the multiclass problem into a series of c binary subtasks that can be trained by the binary SVM. Let the training set The training set is constructed for all g=c(c−1)/2 combinations of classes The binary SVM rules qj , j = 1, . . . , g are trained on the data . j XY T

KERNEL FUNCTION
From the empirical analysis, the two kernel functions that have given better results for the datasets when compared to other kernels are selected to construct a hybrid kernel. The chosen kernel functions from the experimental analysis are radial basis kernel and polynomial kernel. Therefore, a kernel function is designed by hybridizing the translation invariant and rotation invariant kernels i.e. RBF and Polynomial. The hybrid kernel is given by the following equation where K 1 is RBF kernel , K 2 is polynomial kernel and λ is a box constraint that control and balance these two kernels.
From an extensive analysis (Song et al. 2012), admissible kernel functions in Banach space are designed either based on exponential distribution or Brownian bridge distribution. But, these distributions have been devoted to the class of processes with stochastic integrals, geometry of Banach space, stationary processes and independent increments called as Lévy processes (Applebaum, 2007;FAJ Ueltzhöfer, 2012). Lévy process is an origin of the mainly used distributions in Banach space since lévy is strongly interrelated with Banach space geometry e.g. Brownian process and poisson process. Thus, a novel kernel is constructed in RKBS using lévy distribution (Sangeetha and Kalpana, 2013).
Novel kernel function using lévy distribution is defined by the following equation O c t 5 , 2 0 1 3 where C is regularization parameter and s,t Є R.

EXPERIMENTAL RESULTS
The experimentation for multiclass datasets using kernel functions are carried out using MATLAB and C. The multiclass datasets are taken from the UCI machine learning repository which is detailed in Table 1. Before applying the classification process, the actual datasets are preprocessed and its dimensionality is reduced. Here, the proposed and hybrid kernel functions are evaluated using the performance metrics like number of Support Vectors (SV), support vector percentage, training error, classification error and time. They are given by the following equations

Number of Support Vectors
Vectors x i for which y i (w • x i + b) = 1, where x is an input data, y is the class label, w is weight vector, b is bias value, ) ( x f is predicted class label, N is number of samples, L is loss function to estimate an error.
Classification time and computational complexity for the SVM multiclass classifier depend on the number of support vectors required for the classification process. As the number of support vector increases, it leads to increase in computational requirements such as addition, multiplication and floating point. In SVM, for multiclass classification the required memory to store the support vectors is directly proportional to the number of support vectors. Hence, the number of support vectors must be reduced to speed up the classification and to minimize the computational and hardware resources required for the classification process. O c t 5 , 2 0 1 3 Here, the proposed kernel function and hybrid kernel function are executed with multiclass benchmark datasets. To overcome the demerits of predefined kernel functions in Hilbert space, a hybrid kernel function in Hilbert space is designed. But from the observed results, it is evident that there are some limitations in Hilbert space. So, the motivation for constructing a novel and admissible kernel functions in Banach space has emerged.