Kernel Fisher discriminant analysis

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In statistics, kernel Fisher discriminant analysis (KFD),[1] also known as generalized discriminant analysis[2] and kernel discriminant analysis,[3] is a kernelized version of linear discriminant analysis (LDA). It is named after Ronald Fisher. Using the kernel trick, LDA is implicitly performed in a new feature space, which allows non-linear mappings to be learned.

Linear discriminant analysis[edit]

Intuitively, the idea of LDA is to find a projection where class separation is maximized. Given two sets of labeled data, and , define the class means and to be

where is the number of examples of class . The goal of linear discriminant analysis is to give a large separation of the class means while also keeping the in-class variance small.[4] This is formulated as maximizing, with respect to , the following ratio:

where is the between-class covariance matrix and is the total within-class covariance matrix:

The maximum of the above ratio is attained at

as can be shown by the Lagrange multiplier method (sketch of proof):

Maximizing is equivalent to maximizing

subject to

This, in turn, is equivalent to maximizing , where is the Lagrange multiplier.

At the maximum, the derivatives of with respect to and must be zero. Taking yields

which is trivially satisfied by and


Kernel trick with LDA[edit]

To extend LDA to non-linear mappings, the data, given as the points can be mapped to a new feature space, via some function In this new feature space, the function that needs to be maximized is[1]

where

and

Further, note that . Explicitly computing the mappings and then performing LDA can be computationally expensive, and in many cases intractable. For example, may be infinitely dimensional. Thus, rather than explicitly mapping the data to , the data can be implicitly embedded by rewriting the algorithm in terms of dot products and using the kernel trick in which the dot product in the new feature space is replaced by a kernel function,.

LDA can be reformulated in terms of dot products by first noting that will have an expansion of the form[5]

Then note that

where

The numerator of can then be written as:

Similarly, the denominator can be written as

with the component of defined as is the identity matrix, and the matrix with all entries equal to . This identity can be derived by starting out with the expression for and using the expansion of and the definitions of and

With these equations for the numerator and denominator of , the equation for can be rewritten as

Then, differentiating and setting equal to zero gives

Since only the direction of , and hence the direction of matters, the above can be solved for as

Note that in practice, is usually singular and so a multiple of the identity is added to it [1]

Given the solution for , the projection of a new data point is given by[1]

Multi-class KFD[edit]

The extension to cases where there are more than two classes is relatively straightforward.[2][6][7] Let be the number of classes. Then multi-class KFD involves projecting the data into a -dimensional space using discriminant functions

This can be written in matrix notation

where the are the columns of .[6] Further, the between-class covariance matrix is now

where is the mean of all the data in the new feature space. The within-class covariance matrix is

The solution is now obtained by maximizing

The kernel trick can again be used and the goal of multi-class KFD becomes[7]

where and

The are defined as in the above section and is defined as

can then be computed by finding the leading eigenvectors of .[7] Furthermore, the projection of a new input, , is given by[7]

where the component of is given by .

Classification using KFD[edit]

In both two-class and multi-class KFD, the class label of a new input can be assigned as[7]

where is the projected mean for class and is a distance function.

Applications[edit]

Kernel discriminant analysis has been used in a variety of applications. These include:

See also[edit]

References[edit]

  1. ^ a b c d e Mika, S; Rätsch, G.; Weston, J.; Schölkopf, B.; Müller, KR (1999). "Fisher discriminant analysis with kernels". Neural Networks for Signal Processing. IX: 41–48. doi:10.1109/NNSP.1999.788121. 
  2. ^ a b c Baudat, G.; Anouar, F. (2000). "Generalized discriminant analysis using a kernel approach". Neural Computation. 12 (10): 2385–2404. doi:10.1162/089976600300014980. 
  3. ^ a b Li, Y.; Gong, S.; Liddell, H. (2003). "Recognising trajectories of facial identities using kernel discriminant analysis". Image and Vision Computing. 21 (13-14): 1077–1086. doi:10.1016/j.imavis.2003.08.010. 
  4. ^ Bishop, CM (2006). Pattern Recognition and Machine Learning. New York, NY: Springer. 
  5. ^ Scholkopf, B; Herbrich, R.; Smola, A. (2001). "A generalized representer theorem". Computational learning theory. 
  6. ^ a b Duda, R.; Hart, P.; Stork, D. (2001). Pattern Classification. New York, NY: Wiley. 
  7. ^ a b c d e Zhang, J.; Ma, K.K. (2004). "Kernel fisher discriminant for texture classification". 
  8. ^ Liu, Q.; Lu, H.; Ma, S. (2004). "Improving kernel Fisher discriminant analysis for face recognition". IEEE Transactions on Circuits and Systems for Video Technology. 14 (1): 42–49. doi:10.1109/tcsvt.2003.818352. 
  9. ^ Liu, Q.; Huang, R.; Lu, H.; Ma, S. (2002). "Face recognition using kernel-based Fisher discriminant analysis". IEEE International Conference on Automatic Face and Gesture Recognition. 
  10. ^ Kurita, T.; Taguchi, T. (2002). "A modification of kernel-based Fisher discriminant analysis for face detection". IEEE International Conference on Automatic Face and Gesture Recognition. 
  11. ^ Feng, Y.; Shi, P. (2004). "Face detection based on kernel fisher discriminant analysis". IEEE International Conference on Automatic Face and Gesture Recognition. 
  12. ^ Yang, J.; Frangi, AF; Yang, JY; Zang, D., Jin, Z. (2005). "KPCA plus LDA: a complete kernel Fisher discriminant framework for feature extraction and recognition". IEEE Transactions on Pattern Analysis and Machine Intelligence. 27 (2). doi:10.1109/tpami.2005.33. 
  13. ^ Wang, Y.; Ruan, Q. (2006). "Kernel fisher discriminant analysis for palmprint recognition". International Conference on Pattern Recognition. 
  14. ^ Wei, L.; Yang, Y.; Nishikawa, R.M.; Jiang, Y. (2005). "A study on several machine-learning methods for classification of malignant and benign clustered microcalcifications". IEEE Transactions on Medical Imaging. 24 (3): 371–380. doi:10.1109/tmi.2004.842457. 

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