Wavelet support vector machine

An admissible support vector (SV) kernel (the wavelet kernel), by which we can construct a wavelet support vector machine (SVM), is presented. The wavelet kernel is a kind of multidimensional wavelet function that can approximate arbitrary nonlinear functions. The existence of wavelet kernels is proven by results of theoretic analysis. Computer simulations show the feasibility and validity of wavelet support vector machines (WSVMs) in regression and pattern recognition.


I. INTRODUCTION
T HE SUPPORT vector machine (SVM) is a new universal learning machine proposed by Vapnik et al. [6], [8], which is applied to both regression [1], [2] and pattern recognition [2], [5].An SVM uses a device called kernel mapping to map the data in input space to a high-dimensional feature space in which the problem becomes linearly separable [10].The decision function of an SVM is related not only to the number of SVs and their weights but also to the a priori chosen kernel that is called the support vector kernel [1], [9], [10].There are many kinds of kernels can be used, such as the Gaussian and polynomial kernels.
Since the wavelet technique shows promise for both nonstationary signal approximation and classification [3], [4], it is valuable for us to study the problem of whether a better performance could be obtained if we combine the wavelet technique with SVMs.An admissible SV kernel, which is a wavelet kernel constructed in this paper, implements the combination of the wavelet technique with SVMs.In theory, wavelet decomposition emerges as a powerful tool for approximation [11]- [16]; that is to say the wavelet function is a set of bases that can approximate arbitrary functions.Here, the wavelet kernel has the same expression as a multidimensional wavelet function; therefore, the goal of the WSVMs is to find the optimal approximation or classification in the space spanned by multidimensional wavelets or wavelet kernels.Experiments show the feasibility and validity of WSVMs in approximation and classification.

II. SUPPORT VECTOR MACHINES (SVMS)
SVMs use SV kernel to map the data in input space to a highdimensional feature space in which we can process a problem in linear form.[1], [2] Let and , where represents input space.By some nonlinear mapping , is mapped into a feature space We seek to estimate (1) based on independent uniformly distributed data by finding a function with a small risk.Vapnik et al. suggested using the following regularized risk functional to obtain a small risk [6], [8]:

A. SVM for Regression
where is a constant, and is a small positive number.The second term can be defined as if otherwise.
(3) By using Lagrange multiplier techniques, the minimization of (2) leads to the following dual optimization problem.Maximize The resulting regression estimates are linear.Then, the regression takes the form (6) A kernel is called an SV kernel if it satisfies a certain conditions, which will discussed in detail in Section II-C.[2], [5] It is similar to SVM for regression.The training procedure of SVM for pattern recognition is to solve a constrained quadratic optimization problem as well.The only difference between them is the expression of the optimization problem.Given an i.i.

C. Conditions for Support Vector Kernel
The formation of an SV kernel is a kernel of dot-product type in some feature space .The Mercer theorem (see [7]) gives the conditions that a dot product kernel must satisfy.
Theorem 1: Suppose ( denotes the input space) such that the integral operator (9) is positive.Let be the eigenfunction of associated with the eigenvalue and normalized such that .Let denote its complex conjugate.Then, we have the following.

3)
holds for almost all , where the series converges absolutely and uniformly for almost all .In (9), denotes a measure defined on some measurable set.This theorem means that if (Mercer's condition, [6], [9]) (10) holds we can write as a dot product in some feature space.Translation invariant kernels, i.e., derived in [9] are admissive SV kernels if they satisfy Mercer's condition.However, it is difficult to decompose the translation invariant kernels into the product of two functions and then to prove them as SV kernels.Now, we state a necessary and sufficient condition for translation invariant kernels [1], [9].
Theorem 2: A translation invariant kernel is an admissible SV kernels if and only if the Fourier transform (11) is non-negative.
The theorems stated above can be useful for both checking whether a kernel is an admissible SV kernel and actually constructing new kernels.

III. WAVELET SUPPORT VECTOR MACHINES
In this section, we will propose WSVMs and construct wavelet kernels, which are admissible SV kernels.It is the wavelet kernel that combines the wavelet technique with SVMs.

A. Wavelet Analysis
The idea behind the wavelet analysis is to express or approximate a signal or function by a family of functions generated by dilations and translations of a function called the mother wavelet: (12) where , is a dilation factor, and is a translation factor (In wavelet analysis, the translation factor is denoted by , but here, is used for expressing the threshold in SVMs.)Therefore, the wavelet transform of a function can be written as (13) In the right-hand side of of ( 13), denotes the dot product in .Equation ( 13) means the decomposition of a function on a wavelet basis .For a mother wavelet , it is necessary to satisfy the condition [3], [12] (14 where is the Fourier transform of .We can reconstruct as follows: (15) If we take the finite terms to approximate (15) [3], then Here, is approximated by .For a common multidimensional wavelet function, we can write it as the product of one-dimensional (1-D) wavelet functions [3]: (17) where . Here, every 1-D mother wavelet must satisfy (14).For wavelet analysis and theory, see [17]- [19].

B. Wavelet Kernels and WSVMS
Theorem 3: Let be a mother wavelet, and let and denote the dilation and translation, respectively.
. If , then dot-product wavelet kernels are (18)  and translation-invariant wavelet kernels that satisfy the translation invariant kernel theorem are (19) The proof of Theorem 3 is given in Appendix A. Without loss of generality, in the following, we construct a translation-invariant wavelet kernel by a wavelet function adopted in [4].(20) Theorem 4: Given the mother wavelet (20) and the dilation , , . If , the wavelet kernel of this mother wavelet is (21) which is an admissible SV kernel.
The proof of Theorem 4 is shown in Appendix B. From the expression of wavelet kernels, we can take them as a kind of multidimensional wavelet function.The goal of our WSVM is to find the optimal wavelet coefficients in the space spanned by the multidimensional wavelet basis.Thereby, we can obtain the optimal estimate function or decision function.Now, we give the estimate function of WSVMs for the approximation (22)    For comparison, we showed the results obtained by wavelet kernel and Gaussian kernel, respectively.The Gaussian kernel is one of the first SV kernels investigated for most of learning problems.Its expression is , where is a parameter chosen by user.Since SVMs cannot optimize the parameters of kernels, it is difficult to determine

parameters
. For the sake of simplicity, let such that the number of parameters becomes 1.The parameters for wavelet kernel and for the Gaussian kernel are selected by using cross validation that is in wide use [20], [21].

A. Approximation of a Single-Variable Function
In this experiment, we approximate the following single-variable function [3] . (24) We have a uniformly sampled examples of 148 points, 74 of which are taken as training examples and others testing examples.We adopted the approximation error defined in [3] as where (25) where denotes the desired output for and the approximation output.Table I lists the approximation errors using the two kernels.The approximation results are plotted in Figs. 1 and 2, respectively.The solid lines represent the function and the dashed lines show the approximations.

B. Approximation of Two-Variable Function
This experiment is to approximate a two-variable function [3] (26) over the domain .We take 81 points as the training examples, and 1600 points as the testing examples.Fig. 3 shows the original function , and Figs. 4 and 5 show the approximation results obtained by Gaussian and wavelet kernel, respectively.Table II gives the approximation errors.

C. Recognition of Radar Target
This task is to recognize the 1-D images of three-class planes B-52, J-6, and J-7.Our data is acquired in a microwave anechonic chamber with imaging angle from 0 to 160 .Here, the dimension of the input space of the 1-D image recognition problem is 64.The 1-D images of B-52, J-6, and J-7 under 0      We have compared the approximation and recognition results obtained by Gaussian and wavelet kernel, respectively.In the three experiments, our wavelet kernel has better results than the Gaussian kernel.

V. CONCLUSION AND DISCUSSION
In this paper, wavelet kernels by which we can combine the wavelet technique with SVMs to construct WSVMs are presented.The existence of wavelet kernels is proven by results of theoretic analysis.Our wavelet kernel is a kind of multidimensional wavelet function that can approximate arbitrary functions.It is not surprising that wavelet kernel gives better approximation than Gaussian kernel, which is shown by Computer simulations.From ( 22) and ( 23), the decision function and regression estimation function can be expressed as the linear combination of wavelet kernel as well as the Gaussian kernel.Notice that the wavelet kernel is orthonormal (or orthonormal approximately), whereas the Gaussian kernel is not.In other words, the Gaussian kernel is correlative or even redundancy, which is the possible reason why the training speed of the wavelet kernel SVM is slightly faster than the Gaussian kernel SVM.

APPENDIX A PROOF OF THEOREM 3
Proof: We prove first that dot-product wavelet kernels (18) are admissible SV kernels.For , we have Hence, dot-product kernels (18) satisfy Mercer's condition.Therefore, this part of Theorem 3 is proved.Now, we prove that translation-invariant wavelet kernels (19) are admissible kernels.Kernels (19) satisfy Theorem 2 [or condition (11)], which is a necessary and sufficient condition for translation invariant kernels; therefore, they are admissible ones.
This completes the proof of Theorem 3.
Manuscript received February 3, 2001; revised November 5, 2001.This work was supported by the National Natural Science Foundation of China under Grants 60073053, 60133010, and 69831040.This paper was recommended by Associate Editor P. Bhattacharya.The authors are with the National Key Laboratory for Radar Signal Processing, Xidian University, Xi'an, 710071, China.Digital Object Identifier 10.1109/TSMCB.2003.811113 in which a linear estimate function is defined (1) d. training example set , where , .Kernel mapping can map the training examples in input space into a feature space in which the mapped training examples are linearly separable.For pattern recognition problem, SVM becomes the following dual optimization problem

TABLE I RESULTS
OF APPROXIMATIONS

TABLE II APPROXIMATION
RESULTS OF TWO-VARIABLE FUNCTION

TABLE III NUMBER
OF TRAINING AND TESTING EXAMPLES

TABLE IV RESULTS
OF RADAR TARGET RECOGNITION