Related papers: A roadmap for curvature-based geometric data analy…

In this article, we study curvature-like feature value of data sets in Euclidean spaces. First, we formulate such curvature functions with desirable properties under the manifold hypothesis. Then we make a test property for the validity of…

Topological data analysis (TDA) provides insight into data shape. The summaries obtained by these methods are principled global descriptions of multi-dimensional data whilst exhibiting stable properties such as robustness to deformation and…

Estimation of differential geometric quantities in discrete 3D data representations is one of the crucial steps in the geometry processing pipeline. Specifically, estimating normals and sharp feature lines from raw point cloud helps improve…

Mapping complex input data into suitable lower dimensional manifolds is a common procedure in machine learning. This step is beneficial mainly for two reasons: (1) it reduces the data dimensionality and (2) it provides a new data…

Researchers have now achieved great success on dealing with 2D images using deep learning. In recent years, 3D computer vision and Geometry Deep Learning gain more and more attention. Many advanced techniques for 3D shapes have been…

Geometric, topological and graph theory modeling and analysis of biomolecules are of essential importance in the conceptualization of molecular structure, function, dynamics, and transport. On the one hand, geometric modeling provides…

Despite significant advances in the field of deep learning in applications to various fields, explaining the inner processes of deep learning models remains an important and open question. The purpose of this article is to describe and…

Latent variable models are powerful tools for learning low-dimensional manifolds from high-dimensional data. However, when dealing with constrained data such as unit-norm vectors or symmetric positive-definite matrices, existing approaches…

Unsupervised machine learning lacks ground truth by definition. This poses a major difficulty when designing metrics to evaluate the performance of such algorithms. In sharp contrast with supervised learning, for which plenty of quality…

During the past two decades there has been a lot of interest in developing statistical depth notions that generalize the univariate concept of ranking to multivariate data. The notion of depth has also been extended to regression models and…

We introduce a new intrinsic measure of local curvature on point-cloud data called diffusion curvature. Our measure uses the framework of diffusion maps, including the data diffusion operator, to structure point cloud data and define local…

Diffusion models have become a leading framework in generative modeling, yet their theoretical understanding -- especially for high-dimensional data concentrated on low-dimensional structures -- remains incomplete. This paper investigates…

Data sets tend to live in low-dimensional non-linear subspaces. Ideal data analysis tools for such data sets should therefore account for such non-linear geometry. The symmetric Riemannian geometry setting can be suitable for a variety of…

We propose a deep learning strategy to estimate the mean curvature of two-dimensional implicit interfaces in the level-set method. Our approach is based on fitting feed-forward neural networks to synthetic data sets constructed from…

Graph clustering is a longstanding research topic, and has achieved remarkable success with the deep learning methods in recent years. Nevertheless, we observe that several important issues largely remain open. On the one hand, graph…

Deep learning is the mainstream technique for many machine learning tasks, including image recognition, machine translation, speech recognition, and so on. It has outperformed conventional methods in various fields and achieved great…

In this work, we investigate Riemannian geometry based dimensionality reduction methods that respect the underlying manifold structure of the data. In particular, we focus on Principal Geodesic Analysis (PGA) as a nonlinear generalization…

Much of the success of deep learning is drawn from building architectures that properly respect underlying symmetry and structure in the data on which they operate - a set of considerations that have been united under the banner of…

We present an approach to learning features that represent the local geometry around a point in an unstructured point cloud. Such features play a central role in geometric registration, which supports diverse applications in robotics and 3D…

The purpose of this paper is to employ the language of Cartan moving frames to study the geometry of the data manifolds and its Riemannian structure, via the data information metric and its curvature at data points. Using this framework and…