2-families of lines in P 3 and pairs of congruent lines in P by Kruglyakov I.Z. PDF

By Kruglyakov I.Z.

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But, in three-dimensional cases and higher, you can position the nth axis perpendicular to the plane constructed by any two axes. The model is constrained by the position of the first axis, which is positioned in the direction with the highest variance. This results in a new feature space that compresses the swarm of points into the axes of high variance. You may select the axes with higher variances and eliminate the axes with lower variances. Figure 2-4 illustrates the new feature space, reduced from a dataset with 160 featuresto 59 components (axes).

Using N1 examples, OAA will learn in cN 3, whereas OAO will require 4(c – 1)N 3/ c2. Although the SVM parametric model allows for adjustments when constructing the discriminant function, for multiclass problems these parameters do not always fit across the entire dataset. For this reason, it is sometimes preferable to partition the data into subgroups with similar features and derive the classifier parameters separately. This process results in a multistage SVM (MSVM), or hierarchical SVM, which can produce greater generalization accuracy and reduce the likelihood of overfitting, as shown by Stockman (2010).

SVM has been extensively used for classification, regression, novelty detection tasks, and feature reduction. This chapter focuses on SVM for supervised classification tasks only, providing SVM formulations for when the input space is linearly separable or linearly nonseparable and when the data are unbalanced, along with examples. The chapter also presents recent improvements to and extensions of the original SVM formulation. A case study concludes the chapter. SVM from a Geometric Perspective In classification tasks a discriminant machine learning technique aims at finding, based on an independent and identically distributed (iid) training dataset, a discriminant function that can correctly predict labels for newly acquired instances.

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2-families of lines in P 3 and pairs of congruent lines in P 3 by Kruglyakov I.Z.

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