Linguistic Geometries for Unsupervised Dimensionality Reduction
Table 2: Three evaluation measures (i), (ii), and (iii) (see the beginning of the section for description) for convex combinations (8) using different values of α. The first four rows represent methods A, B, C, and D. The bottom row represents a convex combination whose coefficients were obtained by searching for the minimizer of measure (iii). Interestingly the minimizer also performs well on measure (i) and more impressively on the labeled measure (iii).
['( [ITALIC] α1, [ITALIC] α2, [ITALIC] α3, [ITALIC] α4)', '(i)', '(ii)', '(iii) (k=5)']
We also examined convex combinations α1HA+α2HB+α3HC+α4HD (8) with ∑αi=1 and αi≥0. The beginning of the section provides more information on these measures. The first four rows correspond to the “pure” methods A,B,C,D. The bottom row correspond to a convex combination found by minimizing the unsupervised evaluation measure (ii). Note that the convex combination found also outperforms A, B, C, and D on measure (i) and more impressively on measure (iii) which is a supervised measure that uses labeled data (the search for the optimal combination was done based on (ii) which does not require labeled data). We conclude that combining heterogeneous domain knowledge may improve the quality of dimensionality reduction for visualization, and that the search for an improved convex combination may be accomplished without the use of labeled data.