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<p>This study introduces an innovative nonlinear association measure specifically designed for complex data analysis: hyperbolic covariance, inspired by hyperbolic geometry. This measure is crafted using a characteristic covariance kernel within hyperbolic spaces and features a crucial property: independence-zero equivalence. This property guarantees that the hyperbolic covariance between two random vectors is zero if and only if independent. Building on this foundation, we propose a novel test statistic for independence testing, detailing its asymptotic behaviors under null and alternative hypotheses. Furthermore, we establish that our test attains the optimal minimax rate of convergence, which is proportional to <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40304_2025_448_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="32" /> </InlineMediaObject> <EquationSource Format="TEX">\(\sqrt{n}\)</EquationSource> <EquationSource Format="MATHML"><math> <msqrt> <mi>n</mi> </msqrt> </math></EquationSource> </InlineEquation>. Through a series of case studies, we demonstrate the superior testing power of our proposed method in various contexts, effectively handling nonlinear relationships.</p>

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Hyperbolic Covariance and its Applications in Independence Test

  • 期刊论文

  • 发表日期：

    2025年11月10日

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  • Roulin Wang,
  • Zhe Gao,
  • Xueqin Wang

摘要

This study introduces an innovative nonlinear association measure specifically designed for complex data analysis: hyperbolic covariance, inspired by hyperbolic geometry. This measure is crafted using a characteristic covariance kernel within hyperbolic spaces and features a crucial property: independence-zero equivalence. This property guarantees that the hyperbolic covariance between two random vectors is zero if and only if independent. Building on this foundation, we propose a novel test statistic for independence testing, detailing its asymptotic behaviors under null and alternative hypotheses. Furthermore, we establish that our test attains the optimal minimax rate of convergence, which is proportional to \(\sqrt{n}\) n . Through a series of case studies, we demonstrate the superior testing power of our proposed method in various contexts, effectively handling nonlinear relationships.