On convex conceptual regions in deep network representations

The current study of human-machine alignment aims at understanding the geometry of latent spaces and the correspondence to human representations. G\"ardenfors' conceptual spaces is a prominent framework for understanding human representations. Convexity of object regions in conceptual spaces is argued to promote generalizability, few-shot learning, and intersubject alignment. Based on these insights, we investigate the notion of convexity of concept regions in machine-learned latent spaces. We develop a set of tools for measuring convexity in sampled data and evaluate emergent convexity in layered representations of state-of-the-art deep networks. We show that convexity is robust to basic re-parametrization, hence, meaningful as a quality of machine-learned latent spaces. We find that approximate convexity is pervasive in neural representations in multiple application domains, including models of images, audio, human activity, text, and brain data. We measure convexity separately for labels (i.e., targets for fine-tuning) and other concepts. Generally, we observe that fine-tuning increases the convexity of label regions, while for more general concepts, it depends on the alignment of the concept with the fine-tuning objective. We find evidence that pre-training convexity of class label regions predicts subsequent fine-tuning performance.