Unsupervised representation learning with Minimax distance measures
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Unsupervised representation learning with Minimax distance measures Morteza Haghir Chehreghani1 Received: 27 June 2019 / Revised: 2 March 2020 / Accepted: 4 June 2020 © The Author(s) 2020
Abstract 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. Keywords Representation learning · Distance measure · Computational efficiency · Minimax distances
Editors: Larisa Soldatova, Joaquin Vanschoren. * Morteza Haghir Chehreghani [email protected] 1
Department of Computer Science and Engineering (CSE), Chalmers University of Technology, Gothenburg, Sweden
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Vol.:(0123456789)
Machine Learning
1 Introduction Data is usually described by a set of objects and a corresponding representation. The representation can be for example the vectors in a vector space or the pairwise dissimilarities between the objects. In real-world applications, the data is often very complicated and a priori unknown. Thus, the basic representation, e.g., Euclidean distance, Mahalanobis distance, cosine similarity and Pearson correlation, might fail to correctly capture the underlying patterns or classes. Thereby, the raw data needs to be processed further in order to obtain a more sophisticated representation. Kernel methods are a common approach to enrich the basic representation of the data and model the underlying patterns (ShaweTaylor and Cristianini 2004; H
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