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.
翻译:我们用非对称方式调查最小移动距离的使用情况,以提取数据中未知的基本模式和结构的特征。 我们开发了一个通用和计算高效的框架, 以使用使用数字数据使用的许多机器学习方法的最小移动距离。 我们既计算所有天体的对称最小移动距离, 也计算所有天体与固定(测试)天体之间的对称最小移动距离。 我们首先有效地计算天体之间的对称最小移动距离, 使用图形和最小树上构建的最低天体距离的等值。 然后, 我们将对称小型移动距离嵌入一个新的矢量空间, 使这些天体在新空间的正方形 Eucloidean距离与原始天体的对称最小移动距离相等。 我们还研究使用多对称小型天体矩阵而不是单一天体。 因此, 我们建议通过最小型移动矩阵的等值距离进行嵌入, 然后进行最小值降天体距离的最小值降低天体距离定位, 来测量每个直径的直径位置, 我们用最短的直径进行最短的直径的直径直径直, 。