Related papers: Position: Beyond Euclidean -- Foundation Models Sh…
Humans have the remarkable ability to use held objects as tools to interact with their environment. For this to occur, humans internally estimate how hand movements affect the object's movement. We wish to endow robots with this capability.…
Machine learning often aims to produce latent embeddings of inputs which lie in a larger, abstract mathematical space. For example, in the field of 3D modeling, subsets of Euclidean space can be embedded as vectors using implicit neural…
Natural language text exhibits hierarchical structure in a variety of respects. Ideally, we could incorporate our prior knowledge of this hierarchical structure into unsupervised learning algorithms that work on text data. Recent work by…
Euclidean geometry has historically played a central role in cultivating logical reasoning and abstract thinking within mathematics education, but has experienced waning emphasis in recent curricula. The resurgence of interest, driven by…
Rapid and accurate simulations of fluid dynamics around complicated geometric bodies are critical in a variety of engineering and scientific applications, including aerodynamics and biomedical flows. However, while scientific machine…
Graph-structured data are widespread in real-world applications, such as social networks, recommender systems, knowledge graphs, chemical molecules etc. Despite the success of Euclidean space for graph-related learning tasks, its ability to…
Riemannian geometry provides the fundamental framework for optimization on nonlinear spaces such as matrix manifolds, which arise in machine learning, signal processing, and robotics. While the underlying theory is classical, existing…
This paper investigates the notion of learning user and item representations in non-Euclidean space. Specifically, we study the connection between metric learning in hyperbolic space and collaborative filtering by exploring Mobius…
In this article, I develop an elementary system of axioms for Euclidean geometry. On one hand, the system is based on the symmetry principles which express our a priori ignorant approach to space: all places are the same to us (the…
Deep learning has achieved a remarkable performance breakthrough in several fields, most notably in speech recognition, natural language processing, and computer vision. In particular, convolutional neural network (CNN) architectures…
This thesis investigates how foundation models can be systematically leveraged to enhance robotic capabilities, enabling more effective localization, interaction, and manipulation in unstructured environments. The work is structured around…
The book is designed for a semester-long course in Foundations of Geometry and meant to be rigorous, conservative, elementary and minimalist. List of topics: Euclidean geometry: The Axioms / Half-planes / Congruent triangles / Perpendicular…
Relational representation learning transforms relational data into continuous and low-dimensional vector representations. However, vector-based representations fall short in capturing crucial properties of relational data that are complex…
The correlation matrix is a central representation of functional brain networks in neuroimaging. Traditional analyses often treat pairwise interactions independently in a Euclidean setting, overlooking the intrinsic geometry of correlation…
Non-Euclidean method of the generalized geometry construction is considered. According to this approach any generalized geometry is obtained as a result of deformation of the proper Euclidean geometry. The method may be applied for…
Distance metrics and their nonlinear variant play a crucial role in machine learning based real-world problem solving. We demonstrated how Euclidean and cosine distance measures differ not only theoretically but also in real-world medical…
Recent developments in foundation models, like Large Language Models (LLMs) and Vision-Language Models (VLMs), trained on extensive data, facilitate flexible application across different tasks and modalities. Their impact spans various…
Non-Euclidean constraints are inherent in many kinds of data in computer vision and machine learning, typically as a result of specific invariance requirements that need to be respected during high-level inference. Often, these geometric…
Foundation models, as a mainstream technology in artificial intelligence, have demonstrated immense potential across various domains in recent years, particularly in handling complex tasks and multimodal data. In the field of geophysics,…
Although large language models (LLMs) are highly interactive and extendable, current approaches to ensure reliability in deployments remain mostly limited to rejecting outputs with high uncertainty in order to avoid misinformation. This…