A short summary of this paper. Kwok jamesk cse. Tsang ivor cse. The major concern of this letter is to extend the main idea of SVDD to pattern denoising. Combining the geodesic projection to the spherical decision boundary resulting from the SVDD, together with solving the preimage problem, we propose a new method for pattern denoising.
We first solve SVDD for the training data and then for each noisy test pattern, obtain its denoised feature by moving its feature vector along the geodesic on the manifold to the nearest decision boundary of the SVDD ball. Finally we find the location of the denoised pattern by obtaining the pre-image of the denoised feature. The applicability of the proposed method is illustrated by a number of toy and real-world data sets.
Park et al. In one-class classification problems, we are given only the training data for the normal class, and after the training phase is finished, we are required to decide whether each test vector belongs to the normal or the abnormal class.
One-class classification problems are often called outlier detection problems or novelty detection prob- lems. In the SVDD, balls are used for expressing the region for the normal class.
Since balls on the input domain can express only a limited class of regions, the SVDD in general enhances its expressing power by utilizing balls on the feature space instead of the balls on the input domain.
Combining the movement to the spherical decision boundary resulting from the SVDD together with a solver for the preimage problem, we propose a new method for pattern denoising that consists of the following steps. First, we solve the SVDD for the training data consisting of the prototype patterns. Second, for each noisy test pattern, we obtain its denoised feature by moving its feature vector along the geodesic to the spherical decision boundary of the SVDD ball on the feature space.
Finally in the third step, we recover the location of the denoised pattern by obtaining the preimage of the denoised feature following the strategy of Kwok and Tsang The remaining parts of this letter are organized as follows.
In section 2, preliminaries are provided regarding the SVDD. Our main results on pat- tern denoising based on the SVDD are presented in section 3. In section 4, the applicability of the proposed method is illustrated by a number of toy and real-world data sets. Finally, in section 5, concluding remarks are given. The dual problem of equa- tion 2.
The main idea of this letter is to utilize the ball-shaped support on the feature space for correcting test inputs distorted by noise. More precisely, with the trade-off constant C set appropriately,1 we can find a region where the normal objects without noise generally reside.
When an object which was originally normal is given as a test input x in a distorted form, the network resulting from the SVDD is supposed to judge that the distorted object x does not belong to the normal class. The role of the SVDD has been conventional up to this point, and the problem of curing the distortion might be thought of as beyond the scope of the SVDD. In the following, we present the proposed method more precisely with mathematical details. The proposed method consists of three steps.
First, we solve the SVDD, equation 2. Second, we consider each test pattern x. When the decision function f F of equation 2. Otherwise, the test input x is considered to be abnormal and distorted by noise. In principle, any kernel can be used here. However, as we will show, closed-form solutions can be obtained when stationary kernels such as the gaussian kernel are used.
In the following, the proposed method will be presented only for the gaussian kernel, where all the points are mapped to the unit ball in the fea- ture space. Extension to other stationary kernels is straightforward. Thus, it should satisfy equation 3. However, the exact preimage typically does not exist Mika et al. Thus, we need to seek an approximate solution instead. Using the kernel trick and the simple relation, equation 3. Generally, the distances with neighbors are the most important in determining the location of any point.
For simplicity, we denote the proposed method by SVDD. We first use a toy example to illustrate the proposed method and compare its reconstruction performance with PCA.
The setup is similar to that in Mika et al. For each source, 30 points are generated to form the training data and 5 points to form the clean test data. Note that ratios larger than one indicate that the proposed SVDD-based method performs better compared to the other one.
Simulations were also performed for a two-dimensional version of the toy example see Figure 4a , and the denoised results were shown in Figures 4b and 4c. For PCA, we used only one eigenvector if two eigen- vectors were used, the result is just a change of basis and thus not useful. From Table 1 and Figures 4b and 4c, one can see that in the considered examples, the proposed method yielded better performance than the PCA- based method.
We first normalized each feature value to the range [0, 1]. For each digit, we randomly chose 60 exam- ples to form the training set and examples as the test set see Figure 5. Two types of additive noise were added to the test set. Denoising was applied to each digit separately. KPCA Mika et. Bottom Salt-and-pepper noise with noise level p.
The proposed approach is compared with the following standard meth- ods: r Kernel PCA denoising, using the preimage finding method in Mika et al.
For wavelet denoising, the image is first decomposed into wavelet coef- ficients using the discrete wavelet transform Mallat, These wavelet coefficients are then compared with a given threshold value, and those that are close to zero are shrunk so as to remove the effect of noise in the data. The denoised image is then reconstructed from the shrunken wavelet coefficients by using the inverse discrete wavelet transform.
The choice of the threshold value can be important to denoising performance. Moreover, the Symlet6 wavelet basis, with two levels of decomposition, is used. The methods of Mika et al. In the experiments, the number of principal com- ponents is varied from 5 to 60 the maximum number of PCA components that can be obtained on this data set.
Figure 6 shows the average SNR values obtained for the various methods. SVDD always achieves the best performance. When more PCs are used, the performance of denoising using kernel PCA in- creases, while the performance of PCA first increases and then decreases as some noisy PCs are included, which corrupts the resultant images. Note that one advantage of the wavelet denoising methods is that they do not require training.
But a subsequent disadvantage is that they cannot utilize the training set, and so both do not perform well here. Samples of the de- noised images that correspond to the best setting of each method are shown in Figure 7.
As the performance of denoising using kernel PCA appears improving with the number of PCs, we also experimented with a larger data set so that even more PCs can be used. Figure 8 shows the SNR values for the various methods. On this larger data set, denoising using kernel PCA does perform better than the others when a suitable number of PCs are chosen.
This demonstrates that the proposed denoising procedure is comparatively more effective on small training sets. Results are reported in Figure 9. The proposed method shows robust performance around the range of parameters used. In the previous experiments, denoising was applied to each digit sep- arately, which means one must know what the digit is before applying denoising. To investigate how well the proposed method denoises when the true digit is unknown, we follow the same setup but combine all the digits with a total of digits for training.
Results are shown in Figures 10 and Consumption: These are taxes levied on goods and services. Classification Based on Incidence of Tax An incidence of tax is the impact of tax on the person who pays tax to the Government. Under this classification of tax, two forms of taxes are evident. Direct Taxes: These are taxes collected directly from the income of individuals and companies whose incidence and burden is on the individuals or the companies that pay the tax to the Government.
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For example Mr. N, and Mr. In this situation, income tax is progressive, as the tax rate has direct relationship with the tax base i.
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