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Another challenging issue in applying KDA is the selection of an appropriate kernel function. Recall that kernel methods work by embedding the input data into some high-dimensional feature space. The key fact underlying the success of kernel methods is that the embedding into feature space can be determined uniquely by specifying a kernel function that computes the dot product between data points in the feature space. In other words, the kernel function implicitly de nes the nonlinear mapping to the feature space and expensive computations in the high-dimensional feature space can be avoided by evaluating the kernel function. Thus one of the central issues in kernel methods is the selection of kernels. To automate kernel-based learning algorithms, it is desirable to integrate the tuning of kernels into the learning process. This problem has been addressed from different perspectives recently. Lanckriet et al. [58] pioneered the work of multiple kernel learning (MKL) in which the optimal kernel matrix is obtained as a linear combination of prespeci ed kernel matrices. It was shown [58] that the coef cients in MKL can be determined by solving convex programs in the case of Support Vector Machines (SVM). While most existing work focuses on learning kernels for SVM, Fung et al. [59] proposed to learn kernels for discriminant analysis. Based on ideas from MKL, this problem was reformulated as a semide nite program (SDP) [60] in reference 61 for binary-class problems. By optimizing an alternative criterion, an SDP formulation for the KDA kernel learning problem in the multiclass case was proposed in reference 62. To reduce the computational cost of the SDP formulation, an approximate scheme was also developed. Furthermore, it was shown that the regularization parameter for KDA can also be learned automatically in this framework [62]. Although the approximate SDP formulation in reference 62 is scalable in terms of the number of classes, interior point algorithms [63] for solving SDP have an inherently large time complexity and thus it can not be applied to large-scale problems. To improve the ef ciency of this formulation, a quadratically constrained quadratic program (QCQP) [63] formulation was proposed in reference 64 and it is more scalable than the SDP formulations.
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Sparsity has recently received much attention for extending existing algorithms to induce sparse solutions [65 67]. L1 -norm penalty has been used in regression [68], known as LASSO, and SVM [69, 70] to achieve model sparsity. Sparsity often leads to easy interpretation and good generalization ability of the resulting model. Sparse Fisher LDA has been proposed in reference 35, for binary-class problems. Based on the equivalence relationship between LDA and MLR, a multiclass sparse LDA formulation was proposed in reference 71 and an entire solution path for LDA was also obtained through the LARS algorithm [72]. The discussions in this chapter focus on supervised approaches. In the unsupervised setting, LDA can be applied to nd the discriminant subspace for clustering, such as K-means clustering. In this case, an iterative algorithm can be derived alternating between clustering and discriminant subspace learning via LDA [73 75].
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Interestingly, it can be shown that this iterative procedure can be simpli ed and is essentially equivalent to kernel K-means with a speci c kernel Gram matrix [76]. When the data in question are given as high-order representations such as 2D and 3D images, it is natural to encode them using high-order tensors. Discriminant tensor factorization, which is a two-dimensional extension of LDA, for a collection of twodimensional images has been studied [77]. It was further extended to higher-order tensors in reference 78. However, the computational convergency of these iterative algorithms [77, 78] is not guaranteed. Recently, a novel discriminant tensor factorization procedure with the convergency property was proposed [79]. Other recent extensions on discriminant tensor factorization as well as their applications to image analysis can be found in reference 80.
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