Related papers: Position: Beyond Euclidean -- Foundation Models Sh…

Real-world graphs naturally exhibit hierarchical or cyclical structures that are unfit for the typical Euclidean space. While there exist graph neural networks that leverage hyperbolic or spherical spaces to learn representations that embed…

Recently, large models, or foundation models, have exhibited remarkable performance, profoundly impacting research paradigms in diverse domains. Foundation models, trained on extensive and diverse datasets, provide exceptional…

Large language models (LLMs) represent a new paradigm for processing unstructured data, with applications across an unprecedented range of domains. In this paper, we address, through two arguments, whether the development and application of…

The realization of universal robots is an ultimate goal of researchers. However, a key hurdle in achieving this goal lies in the robots' ability to manipulate objects in their unstructured surrounding environments according to different…

The high computational complexity and increasing parameter counts of deep neural networks pose significant challenges for deployment in resource-constrained environments, such as edge devices or real-time systems. To address this, we…

Beginning the study of non-Euclidean geometries, physical models or representations, such as crochet ones, provide a tangible portrayal of these advanced mathematical concepts. However, their connection to local Euclidean surfaces still…

Modeling environmental ecosystems is essential for effective resource management, sustainable development, and understanding complex ecological processes. However, traditional data-driven methods face challenges in capturing inherently…

Modern vision-language models (VLMs) develop patch embedding and convolution backbone within vector space, especially Euclidean ones, at the very founding. When expanding VLMs to a galaxy scale for understanding astronomical phenomena, the…

Hyperbolic spaces have recently gained momentum in the context of machine learning due to their high capacity and tree-likeliness properties. However, the representational power of hyperbolic geometry is not yet on par with Euclidean…

Advancements in modern science have led to the increasing availability of non-Euclidean data in metric spaces. This paper addresses the challenge of modeling relationships between non-Euclidean responses and multivariate Euclidean…

This thesis introduces "Embodied Spatial Intelligence" to address the challenge of creating robots that can perceive and act in the real world based on natural language instructions. To bridge the gap between Large Language Models (LLMs)…

I introduce a new geometrical approach to thermo--statistical mechanics. Here I highlight the main physical ideas, and how do they translate into geometrical language. I contrast the present approach with previous…

Geometric deep learning has recently achieved great success in non-Euclidean domains, and learning on 3D structures of large biomolecules is emerging as a distinct research area. However, its efficacy is largely constrained due to the…

Large language models (LLMs) have shown great success in text modeling tasks across domains. However, natural language exhibits inherent semantic hierarchies and nuanced geometric structure, which current LLMs do not capture completely…

Vision systems to see and reason about the compositional nature of visual scenes are fundamental to understanding our world. The complex relations between objects and their locations, ambiguities, and variations in the real-world…

Latent space geometry provides a rigorous and empirically valuable framework for interacting with the latent variables of deep generative models. This approach reinterprets Euclidean latent spaces as Riemannian through a pull-back metric,…

Deep neural networks have shown great potential in image reconstruction problems in Euclidean space. However, many reconstruction problems involve imaging physics that are dependent on the underlying non-Euclidean geometry. In this paper,…

One considers geometry with the intransitive equaivalence relation. Such a geometry is a physical geometry, i.e. it is described completely by the world function, which is a half of the squared distance function. The physical geometry…

Graphs provide a natural description of the complex relationships among objects, and play a pivotal role in communications, transportation, social computing, the life sciences, etc. Currently, there is strong agreement that Graph Foundation…

Automating activities through robots in unstructured environments, such as construction sites, has been a long-standing desire. However, the high degree of unpredictable events in these settings has resulted in far less adoption compared to…