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.

中文翻译：

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使用 Minimax 距离度量的无监督表示学习

我们研究了使用 Minimax 距离以非参数方式提取捕获数据中未知潜在模式和结构的特征。我们开发了一个通用且计算效率高的框架，以将 Minimax 距离与许多对数值数据执行的机器学习方法结合使用。我们研究计算所有对象对的成对 Minimax 距离，以及计算所有对象到/来自固定（测试）对象的 Minimax 距离。我们首先使用图上的 Minimax 距离和在其上构建的最小生成树的等效性，有效地计算对象之间的成对 Minimax 距离。然后，我们将成对的 Minimax 距离嵌入到一个新的向量空间中，使得它们在新空间中的平方欧几里得距离等于原始空间中的成对极小极大距离。我们还研究了具有多个成对极小极大矩阵而不是单个矩阵的情况。因此，我们通过首先对居中矩阵求和然后执行特征值分解以获得相关特征来提出嵌入。在下文中，我们研究从固定（测试）对象计算 Minimax 距离，该距离可用于例如 K-最近邻搜索。与所有对成对极小极大距离的情况类似，我们开发了一种适用于任何任意基本距离度量的高效通用算法。此外，我们详细研究了由 Minimax 距离选择的边缘，从而探索了 Minimax 距离在检测异常对象方面的能力。