Mathematics (Aug 2022)
The Algorithm That Maximizes the Accuracy of <i>k</i>-Classification on the Set of Representatives of the <i>k</i> Equivalence Classes
Abstract
The article formulates the Dictionary Recognition problem, which is relevant for a wide range of applied problems: word recognition in a noisy audio signal for natural language processing tasks or in a noisy electromagnetic signal, recognition of visual patterns in limited visibility, and much more. A Dictionary Recognition problem is finding a set of words from a given set to maximize the classification accuracy of the words in the dictionary without losing semantic representation. The idea of solving the problem is to represent a set of objects (encoded as a sequence of symbols or visual sequences) in the form of a k-partite graph, where each partite of the graph corresponds to a group of objects with a certain common feature (equivalence class). The task is to find such a set of representatives of the k equivalence classes on which the k-classification accuracy by the classifier H meets certain criteria: (1) maximum classification accuracy; (2) maximin accuracy—the binary classification accuracy of every two objects is not lower than a certain value. The proposed Maximin Algorithm provides k-partite cliques with a maximin worst-case classification accuracy and belongs to the P-class. The Maximal Algorithm provides k-partite cliques with the maximum total weight (the problem belongs to the NP-hard class). The presented algorithms select a set of representatives optimally in terms of classification accuracy for the certain classifier and runtime. The algorithms increase classification accuracy when using classical classification methods without additional optimization of the classifiers themselves. We tested the algorithms on simulated data and provide an open-source project on GitHub. The results of the Maximin and Maximal Algorithms give 4-, 8- and 16-classification accuracy close to the best accuracy (obtained by brute-force enumeration) and better than the median accuracy by more than 20% for the support vector machine classifiers. Furthermore, the algorithms increase the selection speed of representatives by five orders of magnitude compared to the brute-force algorithm with a slight loss of accuracy.
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