High-dimensional sample entropy for uncovering rich complex structures in data

With the widespread emergence of high-dimensional data in complex systems and practical applications, existing one-dimensional entropy measures and their two-dimensional extensions exhibit significant limitations in capturing spatial information of high-dimensional data and revealing their complex structures. To address this, we propose high-dimensional sample entropy (HDSE), a novel entropy algorithm designed to systematically quantify the complexity and uncertainty of high-dimensional data while establish a unified dimension-agnostic computational framework. This framework extends entropy analysis to data spaces of arbitrary dimensions, thereby avoiding the structural information distortion caused by data reshaping and dimensionality reduction in traditional methods. The core innovation of HDSE lies in directly constructing template sub-blocks in the raw high-dimensional data space by introducing multi-order phase space reconstruction. These sub-blocks exist as hypercubes in the high-dimensional space, enabling them to precisely capture the spatial distribution characteristics of data within local regions. Experimental results demonstrate that HDSE not only maintains theoretically consistent monotonicity across dimensions in synthetic fractional Brownian motion data but also exhibits superior discriminative sensitivity compared to other methods in tests involving three-dimensional mixed process model, validating its advantages in theoretical rigor and practical efficacy. Furthermore, in the classification of rapeseed varieties using RGB image data, features extracted from HDSE consistently surpass those from two-dimensional methods across multiple evaluation metrics. This result confirms HDSE’s capacity to preserve critical structural information in high-dimensional data, establishing it as a more robust analytical tool for complexity assessment and pattern recognition in complex systems.