Learning a proper representation is usually the first step in every machine learning and data analytic tasks. Some recent representation learning methods have been developed in the context of deep learning [1], which are highly parameterized and require a huge amount of labeled data for training. On the other hand, there are methods that learn a proper representation in an unsupervised way and usually do not require learning free parameters. A category of unsupervised representations and distance measures, called link-based distance [2, 3], take into account all the paths between the objects represented in a graph. These distance measures are often obtained by inverting the Laplacian of the base distance matrix in the context of Markov diffusion kernel [2].

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 framework to employ Minimax distances with many machine learning methods that perform on numerical data. For this purpose, first, we 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. In the following, we 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. Finally, we perform several experimental studies to illustrate the effectiveness of our framework.

Minimax distance measures provide an effective way to capture the unknown underlying patterns and classes of the data in a non-parametric way. We develop a general-purpose framework to employ Minimax distances with any classification method that performs on numerical data. For this purpose, we establish a two-step strategy. First, we 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 are equal to their Minimax distances in the original space. We also consider the cases where multiple pairwise Minimax matrices are given, instead of a single one. Thereby, we propose an embedding via first summing up the centered matrices and then performing an eigenvalue decomposition. We experimentally validate our framework on different synthetic and real-world datasets.