Kernel Fisher discriminant analysis

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

Intuitively, the idea of LDA is to find a projection where class separation is maximized. Given two sets of labeled data, \mathbf{C}_1 and \mathbf{C}_2, define the class means \mathbf{m}_1 and \mathbf{m}_2 to be

\mathbf{m}_i = \frac{1}{l_i}\sum_{n=1}^{l_i}\mathbf{x}_n^i,

where l_i is the number of examples of class \mathbf{C}_i. 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

J(\mathbf{w}) = \frac{\mathbf{w}^{\text{T}}\mathbf{S}_B\mathbf{w}}{\mathbf{w}^{\text{T}}\mathbf{S}_W\mathbf{w}},

where \mathbf{S}_B is the between-class covariance matrix and \mathbf{S}_W is the total within-class covariance matrix:

\begin{align} \mathbf{S}_B & = (\mathbf{m}_2-\mathbf{m}_1)(\mathbf{m}_2-\mathbf{m}_1)^{\text{T}} \\ \mathbf{S}_W & = \sum_{i=1,2}\sum_{n=1}^{l_i}(\mathbf{x}_n^i-\mathbf{m}_i)(\mathbf{x}_n^i-\mathbf{m}_i)^{\text{T}}. \end{align}

Differentiating J(\mathbf{w}) with respect to \mathbf{w}, setting equal to zero, and rearranging gives

(\mathbf{w}^{\text{T}}\mathbf{S}_B\mathbf{w})\mathbf{S}_W\mathbf{w} = (\mathbf{w}^{\text{T}}\mathbf{S}_W\mathbf{w})\mathbf{S}_B\mathbf{w}.

Since we only care about the direction of \mathbf{w} and \mathbf{S}_B\mathbf{w} has the same direction as (\mathbf{m}_2-\mathbf{m}_1) , \mathbf{S}_B\mathbf{w} can be replaced by (\mathbf{m}_2-\mathbf{m}_1) and we can drop the scalars (\mathbf{w}^{\text{T}}\mathbf{S}_B\mathbf{w}) and (\mathbf{w}^{\text{T}}\mathbf{S}_W\mathbf{w}) to give

\mathbf{w} \propto \mathbf{S}^{-1}_W(\mathbf{m}_2-\mathbf{m}_1).

Kernel trick with LDA

To extend LDA to non-linear mappings, the data, given as the \ell points \mathbf{x}_i, can be mapped to a new feature space, F, via some function \phi. In this new feature space, the function that needs to be maximized is[1]

J(\mathbf{w}) = \frac{\mathbf{w}^{\text{T}}\mathbf{S}_B^{\phi}\mathbf{w}}{\mathbf{w}^{\text{T}}\mathbf{S}_W^{\phi}\mathbf{w}},

where

\begin{align} \mathbf{S}_B^{\phi} & = (\mathbf{m}_2^{\phi}-\mathbf{m}_1^{\phi})(\mathbf{m}_2^{\phi}-\mathbf{m}_1^{\phi})^{\text{T}} \\ \mathbf{S}_W^{\phi} & = \sum_{i=1,2}\sum_{n=1}^{l_i}(\phi(\mathbf{x}_n^i)-\mathbf{m}_i^{\phi})(\phi(\mathbf{x}_n^i)-\mathbf{m}_i^{\phi})^{\text{T}}, \end{align}

and

\mathbf{m}_i^{\phi} = \frac{1}{l_i}\sum_{j=1}^{l_i}\phi(\mathbf{x}_j^i).

Further, note that \mathbf{w}\in F. Explicitly computing the mappings \phi(\mathbf{x}_i) and then performing LDA can be computationally expensive, and in many cases intractable. For example, F may be infinitely dimensional. Thus, rather than explicitly mapping the data to F, 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, k(\mathbf{x},\mathbf{y})=\phi(\mathbf{x})\cdot\phi(\mathbf{y}).

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

\mathbf{w} = \sum_{i=1}^l\alpha_i\phi(\mathbf{x}_i).

Then note that

\mathbf{w}^{\text{T}}\mathbf{m}_i^{\phi} = \frac{1}{l_i}\sum_{j=1}^{l}\sum_{k=1}^{l_i}\alpha_jk(\mathbf{x}_j,\mathbf{x}_k^i) = \mathbf{\alpha}^{\text{T}}\mathbf{M}_i,

where

(\mathbf{M}_i)_j = \frac{1}{l_i}\sum_{k=1}^{l_i}k(\mathbf{x}_j,\mathbf{x}_k^i).

The numerator of J(\mathbf{w}) can then be written as:

\begin{align} \mathbf{w}^{\text{T}}\mathbf{S}_B^{\phi}\mathbf{w} & = \mathbf{w}^{\text{T}}(\mathbf{m}_2^{\phi}-\mathbf{m}_1^{\phi})(\mathbf{m}_2^{\phi}-\mathbf{m}_1^{\phi})^{\text{T}}\mathbf{w} \\ & = \mathbf{\alpha}^{\text{T}}\mathbf{M}\mathbf{\alpha}, \end{align}

where \mathbf{M} = (\mathbf{M}_2-\mathbf{M}_1)(\mathbf{M}_2-\mathbf{M}_1)^{\text{T}}. Similarly, the denominator can be written as

\mathbf{w}^{\text{T}}\mathbf{S}_W^{\phi}\mathbf{w}=\mathbf{\alpha}^{\text{T}}\mathbf{N}\mathbf{\alpha},

where

\mathbf{N} = \sum_{j=1,2}\mathbf{K}_j(\mathbf{I}-\mathbf{1}_{l_j})\mathbf{K}_j^{\text{T}},

with the n^{\text{th}}, m^{\text{th}} component of \mathbf{K}_j defined as k(\mathbf{x}_n,\mathbf{x}_m^j), \mathbf{I} is the identity matrix, and \mathbf{1}_{l_j} the matrix with all entries equal to 1/l_j. This identity can be derived by starting out with the expression for \mathbf{w}^{\text{T}}\mathbf{S}_W^{\phi}\mathbf{w} and using the expansion of \mathbf{w} and the definitions of \mathbf{S}_W^{\phi} and \mathbf{m}_i^{\phi}

\begin{align} \mathbf{w}^{\text{T}}\mathbf{S}_W^{\phi}\mathbf{w} & = \left(\sum_{i=1}^l\alpha_i\phi^{\text{T}}(\mathbf{x}_i)\right)\left(\sum_{j=1,2}\sum_{n =1}^{l_j}(\phi(\mathbf{x}_n^j)-\mathbf{m}_j^{\phi})(\phi(\mathbf{x}_n^j)-\mathbf{m}_j^{\phi})^{\text{T}}\right) \left(\sum_{k=1}^l\alpha_k\phi(\mathbf{x}_k)\right)\\ & = \sum_{j=1,2}\sum_{i=1}^l\sum_{n =1}^{l_j}\sum_{k=1}^l\alpha_i\phi^{\text{T}}(\mathbf{x}_i)(\phi(\mathbf{x}_n^j)-\mathbf{m}_j^{\phi})(\phi(\mathbf{x}_n^j)-\mathbf{m}_j^{\phi})^{\text{T}} \alpha_k\phi(\mathbf{x}_k) \\ & = \sum_{j=1,2}\sum_{i=1}^l\sum_{n =1}^{l_j}\sum_{k=1}^l \left(\alpha_ik(\mathbf{x}_i,\mathbf{x}_n^j)-\frac{1}{l_j}\sum_{p=1}^{l_j}\alpha_ik(\mathbf{x}_i,\mathbf{x}_p^j)\right) \left(\alpha_kk(\mathbf{x}_k,\mathbf{x}_n^j)-\frac{1}{l_j}\sum_{q=1}^{l_j}\alpha_kk(\mathbf{x}_k,\mathbf{x}_q^j)\right) \\ & = \sum_{j=1,2}\left( \sum_{i=1}^l\sum_{n =1}^{l_j}\sum_{k=1}^l\Bigg( \alpha_i\alpha_kk(\mathbf{x}_i,\mathbf{x}_n^j)k(\mathbf{x}_k,\mathbf{x}_n^j)\right.\\ & \left.{} - \frac{2\alpha_i\alpha_k}{l_j}\sum_{p=1}^{l_j}k(\mathbf{x}_i,\mathbf{x}_n^j)k(\mathbf{x}_k,\mathbf{x}_p^j) \left. + \frac{\alpha_i\alpha_k}{l_j^2}\sum_{p=1}^{l_j}\sum_{q=1}^{l_j}k(\mathbf{x}_i,\mathbf{x}_p^j)k(\mathbf{x}_k,\mathbf{x}_q^j) \right)\right) \\ & = \sum_{j=1,2}\left( \sum_{i=1}^l\sum_{n =1}^{l_j}\sum_{k=1}^l\left( \alpha_i\alpha_kk(\mathbf{x}_i,\mathbf{x}_n^j)k(\mathbf{x}_k,\mathbf{x}_n^j) - \frac{\alpha_i\alpha_k}{l_j}\sum_{p=1}^{l_j}k(\mathbf{x}_i,\mathbf{x}_n^j)k(\mathbf{x}_k,\mathbf{x}_p^j) \right)\right) \\ & = \sum_{j=1,2} \mathbf{\alpha}^{\text{T}} \mathbf{K}_j\mathbf{K}_j^{\text{T}}\mathbf{\alpha} - \mathbf{\alpha}^{\text{T}} \mathbf{K}_j\mathbf{1}_{l_j}\mathbf{K}_j^{\text{T}}\mathbf{\alpha} \\ & = \mathbf{\alpha}^{\text{T}}\mathbf{N}\mathbf{\alpha}. \end{align}

With these equations for the numerator and denominator of J(\mathbf{w}), the equation for J can be rewritten as

J(\mathbf{\alpha}) = \frac{\mathbf{\alpha}^{\text{T}}\mathbf{M}\mathbf{\alpha}}{\mathbf{\alpha}^{\text{T}}\mathbf{N}\mathbf{\alpha}}.

Then, differentiating and setting equal to zero gives

(\mathbf{\alpha}^{\text{T}}\mathbf{M}\mathbf{\alpha})\mathbf{N}\mathbf{\alpha} = (\mathbf{\alpha}^{\text{T}}\mathbf{N}\mathbf{\alpha})\mathbf{M}\mathbf{\alpha}.

Since only the direction of \mathbf{w}, and hence the direction of \mathbf{\alpha}, matters, the above can be solved for \mathbf{\alpha} as

\mathbf{\alpha} = \mathbf{N}^{-1}(\mathbf{M}_2- \mathbf{M}_1).

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

\mathbf{N}_{\epsilon} = \mathbf{N}+\epsilon\mathbf{I}.

Given the solution for \mathbf{\alpha}, the projection of a new data point is given by[1]

y(\mathbf{x}) = (\mathbf{w}\cdot\phi(\mathbf{x})) = \sum_{i=1}^l\alpha_ik(\mathbf{x}_i,\mathbf{x}).

Multi-class KFD

The extension to cases where there are more than two classes is relatively straightforward.[2][6][7] Let c be the number of classes. Then multi-class KFD involves projecting the data into a (c-1)-dimensional space using (c-1) discriminant functions

y_i = \mathbf{w}_i^{\text{T}}\phi(\mathbf{x}) \qquad i= 1,\ldots,c-1.

This can be written in matrix notation

\mathbf{y} = \mathbf{W}^{\text{T}}\phi(\mathbf{x}),

where the \mathbf{w}_i are the columns of \mathbf{W}.[6] Further, the between-class covariance matrix is now

\mathbf{S}_B^{\phi} = \sum_{i=1}^c l_i(\mathbf{m}_i^{\phi}-\mathbf{m}^{\phi})(\mathbf{m}_i^{\phi}-\mathbf{m}^{\phi})^{\text{T}},

where \mathbf{m}^\phi is the mean of all the data in the new feature space. The within-class covariance matrix is

\mathbf{S}_W^{\phi} = \sum_{i=1}^c \sum_{n=1}^{l_i}(\phi(\mathbf{x}_n^i)-\mathbf{m}_i^{\phi})(\phi(\mathbf{x}_n^i)-\mathbf{m}_i^{\phi})^{\text{T}},

The solution is now obtained by maximizing

J(\mathbf{W}) = \frac{\left|\mathbf{W}^{\text{T}}\mathbf{S}_B^{\phi}\mathbf{W}\right|}{\left|\mathbf{W}^{\text{T}}\mathbf{S}_W^{\phi}\mathbf{W}\right|}.

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

\mathbf{A}^* = \underset{\mathbf{A}}{\operatorname{argmax}} = \frac{\left|\mathbf{A}^{\text{T}}\mathbf{M}\mathbf{A}\right|}{\left|\mathbf{A}^{\text{T}}\mathbf{N}\mathbf{A}\right|},

where A = [\mathbf{\alpha}_1,\ldots,\mathbf{\alpha}_{c-1}] and

\begin{align} M & = \sum_{j=1}^cl_j(\mathbf{M}_j-\mathbf{M}_{*})(\mathbf{M}_j-\mathbf{M}_{*})^{\text{T}} \\ N & = \sum_{j=1}^c\mathbf{K}_j(\mathbf{I}-\mathbf{1}_{l_j})\mathbf{K}_j^{\text{T}}. \end{align}

The \mathbf{M}_i are defined as in the above section and \mathbf{M}_{*} is defined as

(\mathbf{M}_{*})_j = \frac{1}{l}\sum_{k=1}^{l}k(\mathbf{x}_j,\mathbf{x}_k).

\mathbf{A}^{*} can then be computed by finding the (c-1) leading eigenvectors of \mathbf{N}^{-1}\mathbf{M}.[7] Furthermore, the projection of a new input, \mathbf{x}_t, is given by[7]

\mathbf{y}(\mathbf{x}_t) = \left(\mathbf{A}^{*}\right)^{\text{T}}\mathbf{K}_t,

where the i^{th} component of \mathbf{K}_t is given by k(\mathbf{x}_i,\mathbf{x}_t).

Classification using KFD

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

f(\mathbf{x}) = arg\min_j D(\mathbf{y}(\mathbf{x}),\bar{\mathbf{y}}_j),

where \bar{\mathbf{y}}_j is the projected mean for class j and D(\cdot,\cdot) is a distance function.

Applications

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

See also

References

  1. 1 2 3 4 5 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. 1 2 3 Baudat, G.; Anouar, F. (2000). "Generalized discriminant analysis using a kernel approach". Neural Computation 12 (10): 2385–2404. doi:10.1162/089976600300014980.
  3. 1 2 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. 1 2 Duda, R.; Hart, P.; Stork, D. (2001). Pattern Classification. New York, NY: Wiley.
  7. 1 2 3 4 5 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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