We investigate the use of Minimax distances to extract in a nonparametric way
the features that capture the unknown underlying patterns and structures in the
data. We develop a general-purpose and computationally efficient framework to
employ Minimax distances with many machine learning methods that perform on
numerical data. We study both computing the pairwise Minimax distances for all
pairs of objects and as well as computing the Minimax distances of all the
objects to/from a fixed (test) object.
We first efficiently compute the pairwise Minimax distances between the
objects, using the equivalence of Minimax distances over a graph and over a
minimum spanning tree constructed on that. Then, we perform an embedding of the
pairwise Minimax distances into a new vector space, such that their squared
Euclidean distances in the new space equal to the pairwise Minimax distances in
the original space. We also study the case of having multiple pairwise Minimax
matrices, instead of a single one. Thereby, we propose an embedding via first
summing up the centered matrices and then performing an eigenvalue
decomposition to obtain the relevant features.
In the following, we study computing Minimax distances from a fixed (test)
object which can be used for instance in K-nearest neighbor search. Similar to
the case of all-pair pairwise Minimax distances, we develop an efficient and
general-purpose algorithm that is applicable with any arbitrary base distance
measure. Moreover, we investigate in detail the edges selected by the Minimax
distances and thereby explore the ability of Minimax distances in detecting
outlier objects.
Finally, for each setting, we perform several experiments to demonstrate the
effectiveness of our framework.Comment: 32 page
