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Register a new userOn the Importance of Asymmetry and Monotonicity Constraints in Maximal Correlation Analysis
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Added by
Elad Domanovitz
Publication date
2019
fields
Informatics Engineering
and research's language is
English
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The maximal correlation coefficient is a well-established generalization of the Pearson correlation coefficient for measuring non-linear dependence between random variables. It is appealing from a theoretical standpoint, satisfying Renyis axioms for a measure of dependence. It is also attractive from a computational point of view due to the celebrated alternating conditional expectation algorithm, allowing to compute its empirical version directly from observed data. Nevertheless, from the outset, it was recognized that the maximal correlation coefficient suffers from some fundamental deficiencies, limiting its usefulness as an indicator of estimation quality. Another well-known measure of dependence is the correlation ratio which also suffers from some drawbacks. Specifically, the maximal correlation coefficient equals one too easily whereas the correlation ratio equals zero too easily. The present work recounts some attempts that have been made in the past to alter the definition of the maximal correlation coefficient in order to overcome its weaknesses and then proceeds to suggest a natural variant of the maximal correlation coefficient as well as a modified conditional expectation algorithm to compute it. The proposed dependence measure at the same time resolves the major weakness of the correlation ratio measure and may be viewed as a bridge between these two classical measures.
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Based on the notion of maximal correlation, Kimeldorf, May and Sampson (1980) introduce a measure of correlation between two random variables, called the concordant monotone correlation (CMC). We revisit, generalize and prove new properties of this measure of correlation. It is shown that CMC captures various types of correlation detected in measures of rank correlation like the Kendall tau correlation. We show that the CMC satisfies the data processing and tensorization properties (that make ordinary maximal correlation applicable to problems in information theory). Furthermore, CMC is shown to be intimately related to the FKG inequality. Furthermore, a combinatorical application of CMC is given for which we do not know of another method to derive its result. Finally, we study the problem of the complexity of the computation of the CMC, which is a non-convex optimization problem with local maximas. We give a simple but exponential-time algorithm that is guaranteed to output the exact value of the generalized CMC.
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