Cross-covariance based affinity for graphs
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Cross-covariance based affinity for graphs Rakesh Kumar Yadav1
· Abhishek1 · Shekhar Verma1 · S Venkatesan1
Accepted: 28 September 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract The accuracy of graph based learning techniques relies on the underlying topological structure and affinity between data points, which are assumed to lie on a smooth Riemannian manifold. However, the assumption of local linearity in a neighborhood does not always hold true. Hence, the Euclidean distance based affinity that determines the graph edges may fail to represent the true connectivity strength between data points. Moreover, the affinity between data points is influenced by the distribution of the data around them and must be considered in the affinity measure. In this paper, we propose two techniques, CCGAL and CCGAN that use cross-covariance based graph affinity (CCGA) to represent the relation between data points in a local region. CCGAL also explores the additional connectivity between data points which share a common local neighborhood. CCGAN considers the influence of respective neighborhoods of the two immediately connected data points, which further enhance the affinity measure. Experimental results of manifold learning on synthetic datasets show that CCGA is able to represent the affinity measure between data points more accurately. This results in better low dimensional representation. Manifold regularization experiments on standard image dataset further indicate that the proposed CCGA based affinity is able to accurately identify and include the influence of the data points and its common neighborhood that increase the classification accuracy. The proposed method outperforms the existing state-of-the-art manifold regularization methods by a significant margin. Keywords Graph · Affinity · Euclidean distance · Cross-Covariance · Neighborhoods · Manifold regularization
1 Introduction Manifold learning and regularization methods have been widely used for data representation and processing respectively. A given data is represented as a weighted graph where the weights represent the similarity between data points. In an ideal case, similar data points should have Rakesh Kumar Yadav
[email protected] Abhishek [email protected] Shekhar Verma [email protected] S Venkatesan [email protected] 1
Department of IT Deoghat, Indian Institute of Information Technology Allahabad, Jhalwa, Prayagraj, U.P. India, India
a higher affinity, which satisfies the manifold assumption. In manifold learning, clusters of data points having larger affinity are kept spatially close when projected to lower dimensional space. Similarly, in manifold regularization, the labels of data points in a large affinity neighborhood are assumed to be similar and, hence, the approximated function is penalized of it changes label in such a neighborhood. This shows that affinity fundamental to both manifold learning and regularization methods, which also influence the accuracy of the underlying model. Any manifold le
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