  Spectral dimensionality reduction methods enable linear separations of
complex data with high-dimensional features in a reduced space. However, these
methods do not always give the desired results due to irregularities or
uncertainties of the data. Thus, we consider aggressively modifying the scales
of the features to obtain the desired classification. Using prior knowledge on
the labels of partial samples to specify the Fiedler vector, we formulate an
eigenvalue problem of a linear matrix pencil whose eigenvector has the feature
scaling factors. The resulting factors can modify the features of entire
samples to form clusters in the reduced space, according to the known labels.
In this study, we propose new dimensionality reduction methods supervised using
the feature scaling associated with the spectral clustering. Numerical
experiments show that the proposed methods outperform well-established
supervised methods for toy problems with more samples than features, and are
more robust regarding clustering than existing methods. Also, the proposed
methods outperform existing methods regarding classification for real-world
problems with more features than samples of gene expression profiles of cancer
diseases. Furthermore, the feature scaling tends to improve the clustering and
classification accuracies of existing unsupervised methods, as the proportion
of training data increases.
