Related papers: Geographic Space as Manifolds
Historically, there have been many attempts to produce an appropriate mathematical formalism for modeling the nature of physical space, such as Euclid's geometry, Descartes' system of Cartesian coordinates, the Argand plane, Hamilton's…
Inverse analysis has been utilized to understand unknown underground geological properties by matching the observational data with simulators. To overcome the underconstrained nature of inverse problems and achieve good performance, an…
This book introduces the new research area of Geometric Data Science, where data can represent any real objects through geometric measurements. The first part of the book focuses on finite point sets. The most important result is a complete…
Neural representations of 3D data have been widely adopted across various applications, particularly in recent work leveraging coordinate-based networks to model scalar or vector fields. However, these approaches face inherent challenges,…
Convex optimization is a vibrant and successful area due to the existence of a variety of efficient algorithms that leverage the rich structure provided by convexity. Convexity of a smooth set or a function in a Euclidean space is defined…
Geometric torsions are torsions of acyclic complexes of vector spaces which consist of differentials of geometric quantities assigned to the elements of a manifold triangulation. We use geometric torsions to construct invariants for a…
An increasingly common viewpoint is that protein dynamics data sets reside in a non-linear subspace of low conformational energy. Ideal data analysis tools for such data sets should therefore account for such non-linear geometry. The…
The standard approach to representation learning on attributed graphs -- i.e., simultaneously reconstructing node attributes and graph structure -- is geometrically flawed, as it merges two potentially incompatible metric spaces. This…
Many scientific problems require to process data in the form of geometric graphs. Unlike generic graph data, geometric graphs exhibit symmetries of translations, rotations, and/or reflections. Researchers have leveraged such inductive bias…
Statistical analysis of a graph often starts with embedding, the process of representing its nodes as points in space. How to choose the embedding dimension is a nuanced decision in practice, but in theory a notion of true dimension is…
There has been a growing interest in statistical inference from data satisfying the so-called manifold hypothesis, assuming data points in the high-dimensional ambient space to lie in close vicinity of a submanifold of much lower dimension.…
Among plenty of applications, low-dimensional homogeneous spaces appear in cosmological models as both, classical factor spaces of multidimensional geometry and minisuperspaces in canonical quantization. Here a new tool to restrict their…
Polygon representation learning is essential for diverse applications, encompassing tasks such as shape coding, building pattern classification, and geographic question answering. While recent years have seen considerable advancements in…
We present a new technique that enables manifold learning to accurately embed data manifolds that contain holes, without discarding any topological information. Manifold learning aims to embed high dimensional data into a lower dimensional…
Manifold learning methods are an invaluable tool in today's world of increasingly huge datasets. Manifold learning algorithms can discover a much lower-dimensional representation (embedding) of a high-dimensional dataset through non-linear…
The last decade has witnessed an experimental revolution in data science and machine learning, epitomised by deep learning methods. Indeed, many high-dimensional learning tasks previously thought to be beyond reach -- such as computer…
We study the base distribution in chart-based generative models on Riemannian manifolds. Standard methods sample in Euclidean tangent space and then map the sample to the manifold with a chart. This is convenient, but it changes the meaning…
We develop a new method for visualizing and refining the invariances of learned representations. Specifically, we test for a general form of invariance, linearization, in which the action of a transformation is confined to a low-dimensional…
Many models for spatial and spatio-temporal data assume that "near things are more related than distant things," which is known as the first law of geography. While geography may be important, it may not be all-important, for at least two…
Spatial statistics is an area of study devoted to the statistical analysis of data that have a spatial label associated with them. Geographers often refer to the "location information" associated with the "attribute information," whose…