A groundbreaking study published in Nature Communications by researchers at the Technical University of Denmark unveils a remarkable geometric principle connecting human and machine learning paradigms. The study elucidates how a mathematical property known as convexity underpins the formation of conceptual knowledge both in the human brain and in artificial intelligence (AI) systems, promising to transform our understanding of how learning occurs across these disparate domains.
Convexity, at its core, is a geometric concept that has historically been applied in mathematics and cognitive science to describe how ideas cluster together forming coherent conceptual spaces. For humans, concepts such as “cat” or “wheel” do not exist as isolated points but rather as regions where diverse instances cluster cohesively. This coherency is characterized by convexity—the idea that if two points belong to the concept, every point along the shortest path connecting them also resides within that concept’s region. Imagine a rubber band stretched around all the examples of a concept, encapsulating its essential variations without gaps or outliers. This principle facilitates robust abstraction, enabling effortless generalization from limited examples.
The study probes whether AI systems, particularly deep neural networks, mimic this inherently human property in their internal representations. Despite AI’s complexity and learning mechanisms that vastly differ from biological brains, models trained on vast datasets develop internal ‘latent spaces’—abstract multidimensional maps—where knowledge is organized. The critical question addressed is whether these latent spaces also exhibit convexity, implying a shared structural principle between human cognition and machine intelligence.
To measure this, researchers introduced innovative metrics to examine two distinct types of convexity within AI latent spaces: Euclidean convexity and graph convexity. Euclidean convexity assesses whether the straight line between any two points in a given conceptual region lies completely within that region—akin to traditional geometric convexity. Graph convexity extends this concept to non-Euclidean, curved spaces common in neural networks where straight lines give way to minimal or geodesic paths across a network of data points. This nuanced approach recognizes that AI’s internal landscapes are often highly complex and nonlinear.
The team applied these convexity metrics across a diverse spectrum of AI models handling different data modalities—images, text, audio, human activity patterns, and medical datasets. Remarkably, they discovered that convexity is not a rare artifact but a pervasive property emerging during training, regardless of data type or task. This suggests convexity may be a fundamental and universal organizing principle in deep learning, mirroring its importance in human concept formation.
Moreover, the study delved into how convexity evolves through the AI training pipeline. Deep models commonly undergo two stages: pretraining on broad datasets to learn generalizable features and fine-tuning on specific tasks to refine their abilities. Results indicated that pretraining already establishes convex conceptual regions. Fine-tuning amplifies this property, sharpening the boundaries of classifications and increasing the convexity of the decision regions. This refinement mirrors how humans start with broad categories and progressively specialize with experience and practice.
Intriguingly, the researchers identified a predictive relationship between the convexity of pretraining representations and the models’ performance after fine-tuning. Models exhibiting more convex concept regions early on perform better in specialized tasks later—indicating that convexity could serve as a reliable indicator of a model’s learning potential. This insight opens exciting possibilities for AI development by evaluating models based on the geometry of their internal representations before task-specific training.
The implications of these findings extend beyond academic insight. Convexity may offer a new lens to design AI systems that generalize more efficiently from limited data, a longstanding challenge in machine learning. If AI architectures can be endowed or guided to cultivate convex decision regions during training, they could achieve greater accuracy and reliability even when examples are scarce. This is transformative for real-world applications where data collection is costly or sensitive, such as in healthcare diagnostics or personalized education technologies.
Furthermore, the identification of convexity as a shared structural feature bridges the conceptual divide between biological and artificial intelligence. It suggests that despite evolutionary and mechanistic differences, learning systems converge towards similar organizational principles to process and interpret information. This connection enhances the interpretability and explainability of AI systems—key concerns as these technologies increasingly influence critical societal functions.
The study settles crucial groundwork for future interdisciplinary research integrating cognitive science, geometry, and AI development. By formalizing and quantifying convexity in artificial neural representations, it provides tools to explore how machines can be made to ‘think’ more like humans in a rigorous, mathematically grounded sense. This convergence could usher in a new generation of explainable AI systems where decision-making processes are transparent and intuitively aligned with human conceptual understanding.
As AI continues to permeate diverse sectors, from autonomous vehicles to conversational agents, the ability to reliably quantify and cultivate convexity in latent spaces promises to make these technologies safer, more trustworthy, and easier to collaborate with. The potential for convexity-focused training protocols invites a paradigm shift from purely performance-driven model optimization to geometrically principled design, fostering AI whose internal logic resonates with human thought processes.
While much remains to be explored, including the mechanistic origins of convexity during learning and how it interacts with other geometric and topological features of latent spaces, this pioneering work lays the foundation for demystifying the deep learning “black box.” It points to an elegant and universal principle that not only connects how humans and machines conceptualize the world but also suggests practical pathways for crafting AI systems that are more intelligent, adaptable, and aligned with human values.
This breakthrough is part of the broader “Cognitive Spaces – Next Generation Explainable AI” project funded by the Novo Nordisk Foundation, which aims to develop transparent and user-interpretable AI systems. By shining a light on the geometric secrets of AI’s internal representations, this research is poised to influence both theoretical understanding and applied innovation across artificial intelligence and cognitive neuroscience for years to come.
Subject of Research: Convex decision regions in deep network representations bridging human and machine learning
Article Title: On convex decision regions in deep network representations
News Publication Date: 2-Jul-2025
Web References: https://www.nature.com/articles/s41467-025-60809-y
Image Credits: DTU
Keywords: Convexity, deep learning, latent spaces, human cognition, explainable AI, neural networks, conceptual spaces, machine learning, fine-tuning, pretraining
