In today’s data-driven world, the complexity of the information we seek to analyze is escalating rapidly. Traditional statistical tools rooted in Euclidean geometry sometimes fall short when applied to the increasingly prevalent non-Euclidean data structures. Such data arises naturally in many cutting-edge fields ranging from three-dimensional imaging to robotics and artificial intelligence. Addressing this critical challenge, researchers from South Korea have developed an innovative method known as the Huber mean, designed specifically to operate robustly on curved geometric spaces called Riemannian manifolds. This breakthrough redefines how averages are computed when data does not lie flat but instead inhabits complex curved domains.
Riemannian manifolds are mathematical spaces that generalize curved surfaces, extending the concept of curves and surfaces to higher dimensions. Unlike the familiar flat planes of Euclidean geometry, these manifolds curve and bend in intricate ways. Data points on such manifolds—such as rotations in 3D space, shape configurations, or diffusion tensors in medical images—cannot be meaningfully averaged using conventional arithmetic means. Averages must instead respect the manifold’s geometry to preserve meaningful interpretations and insights. This requirement creates a pressing demand for statistical approaches that are both geometrically sound and resilient to noise.
Traditional methods such as the Fréchet mean have served as a foundation for summarizing manifold-valued data by minimizing the sum of squared distances on the manifold. While mathematically elegant, the Fréchet mean is notoriously sensitive to outliers and extreme data points. Given the nature of real-world data, which is often noisy and subject to contamination, this sensitivity undermines the reliability of statistical conclusions. To overcome this limitation, Professor Jongmin Lee of Pusan National University and Professor Sungkyu Jung of Seoul National University have extended the Fréchet framework by integrating robust statistical principles, culminating in what they term the Huber mean.
The Huber mean adapts the robust Huber loss function — a hybrid between the least-squares (L₂) and least-absolute-deviation (L₁) loss functions that scholars have long valued for balancing efficiency and robustness. For data points close to the central trend, the Huber mean behaves like a least-squares estimator, offering high efficiency and sensitivity to subtle variations. Yet, when faced with large discrepancies or potential outliers, it switches to a least-absolute-deviation regime, which limits the undue influence of extreme points. This elegant fusion empowers the estimator to maintain stability even under significant data contamination.
What sets the Huber mean apart is its automatic adaptability to the manifold environment. Unlike previous robust methods that were constrained to flat spaces or had limited generalization, this estimator operates intrinsically within the curved geometry, honouring the manifold’s structure. The researchers rigorously demonstrate that the Huber mean satisfies important statistical properties such as existence and uniqueness under broad conditions. Moreover, they provide theoretical guarantees addressing convergence rates and unbiasedness, all critical for building trust in its application to practical problems.
Robustness, a hallmark for any statistical method deployed in the wild, receives a quantifiable boost here: the Huber mean achieves a breakdown point of 0.5. Put simply, this means that the estimator can tolerate up to 50% of the data being outliers without losing its reliability. This level of robustness is a significant advancement for manifold statistics, where previously available methods could be easily skewed or made unstable by just a handful of irregular observations.
The implications of this development ripple across many high-impact fields. In medical imaging, the accurate averaging of anatomical shapes such as brain structures or organs can enhance both diagnosis and treatment monitoring. The Huber mean offers a way to mitigate the effects of artifacts and noise inherent to imaging technologies, potentially leading to more reliable biomarkers. Robot navigation and control, domains deeply reliant on 3D orientation data that naturally resides on special manifolds like rotation groups, can be improved by robustly averaging directional measurements and trajectories, especially in environments fraught with uncertainties.
Artificial intelligence and machine learning also stand to gain significantly from this innovation. Increasingly, modern algorithms apply geometric and topological methods to understand data that reflect transformations, networks, or continuous deformations. Statistical estimators, such as the Huber mean which respect geometric structures and resist outliers, can enhance the robustness and fairness of these models. These developments are indispensable as AI systems grow more complex and interact closely with real-world environments, where noise and anomalies are unavoidable.
Computational tractability, often a bottleneck for manifold-based statistics, was also addressed in this research. The team developed an efficient algorithm to compute the Huber mean, which converges rapidly in practice. This makes the method viable for the large-scale datasets typical in contemporary scientific and engineering applications. By bridging sophisticated mathematical insights with practical algorithms, the Huber mean stands ready for integration into software tools that researchers and practitioners use daily.
Through the Huber mean, Professor Jongmin Lee and colleagues present a paradigm shift in the statistical treatment of complex geometric data. By combining rigorous theoretical foundations with algorithmic innovation, they provide a robust, geometry-aware statistical framework tailored for the era of big and complex data. This aligns closely with the broader scientific movement toward trustworthy AI and precision medicine, where reliable data summarization underpins critical decisions.
The significance of this work extends beyond academia. As autonomous systems proliferate, and as medical diagnostics move toward personalized and precision approaches, the ability to accurately summarize non-linear, manifold-valued data becomes imperative. The Huber mean offers the robustness and adaptability required to handle real-world imperfections, empowering better, more resilient decision-making processes in technology and healthcare.
Ultimately, the Huber mean exemplifies the fruitful intersection of abstract geometry, robust statistics, and modern computation. As data scientists and engineers grapple with the challenges posed by manifold-valued data, this method offers a robust and practical tool, unlocking new layers of insight and reliability. The future of data on curved spaces looks brighter, more stable, and primed for breakthroughs across disciplines.
Subject of Research: Not applicable
Article Title: Huber means on Riemannian manifolds
News Publication Date: 25-Aug-2025
References:
DOI: 10.1093/jrsssb/qkaf054
Image Credits: Pusan National University
Keywords: Artificial intelligence, Applied mathematics, Computer modeling, Machine learning, Robotics, Medical imaging, Statistics, Engineering, Computational science, Artificial neural networks
