Related papers: Geodesic Distance Function Learning via Heat Flow …
Diffusion-based manifold learning methods have proven useful in representation learning and dimensionality reduction of modern high dimensional, high throughput, noisy datasets. Such datasets are especially present in fields like biology…
Manifold learning is a fundamental problem in machine learning with numerous applications. Most of the existing methods directly learn the low-dimensional embedding of the data in some high-dimensional space, and usually lack the…
Geodesic distances on manifolds have numerous applications in image processing, computer graphics and computer vision. In this work, we introduce an approach called `LGGD' (Learned Generalized Geodesic Distances). This method involves…
Unsigned Distance Functions (UDFs) can be used to represent non-watertight surfaces in a deep learning framework. However, UDFs tend to be brittle and difficult to learn, in part because the surface is located exactly where the UDF is…
Recent literature has shown that symbolic data, such as text and graphs, is often better represented by points on a curved manifold, rather than in Euclidean space. However, geometrical operations on manifolds are generally more complicated…
In this paper, we develop a novel method for fast geodesic distance queries. The key idea is to embed the mesh into a high-dimensional space, such that the Euclidean distance in the high-dimensional space can induce the geodesic distance in…
Many statistical and machine learning approaches rely on pairwise distances between data points. The choice of distance metric has a fundamental impact on performance of these procedures, raising questions about how to appropriately…
We introduce the heat method for computing the shortest geodesic distance to a specified subset (e.g., point or curve) of a given domain. The heat method is robust, efficient, and simple to implement since it is based on solving a pair of…
Geodesic distance serves as a reliable means of measuring distance in nonlinear spaces, and such nonlinear manifolds are prevalent in the current multimodal learning. In these scenarios, some samples may exhibit high similarity, yet they…
Manifolds discovered by machine learning models provide a compact representation of the underlying data. Geodesics on these manifolds define locally length-minimising curves and provide a notion of distance, which are key for reduced-order…
Function approximation based on data drawn randomly from an unknown distribution is an important problem in machine learning. The manifold hypothesis assumes that the data is sampled from an unknown submanifold of a high dimensional…
Euclidean distance geometry is the study of Euclidean geometry based on the concept of distance. This is useful in several applications where the input data consists of an incomplete set of distances, and the output is a set of points in…
Distance metric learning can be viewed as one of the fundamental interests in pattern recognition and machine learning, which plays a pivotal role in the performance of many learning methods. One of the effective methods in learning such a…
Laplacian-based methods are popular for the dimensionality reduction of data lying in $\mathbb{R}^N$. Several theoretical results for these algorithms depend on the fact that the Euclidean distance locally approximates the geodesic distance…
We propose a manifold matching approach to generative models which includes a distribution generator (or data generator) and a metric generator. In our framework, we view the real data set as some manifold embedded in a high-dimensional…
Single-view 3D object reconstruction is a challenging fundamental problem in computer vision, largely due to the morphological diversity of objects in the natural world. In particular, high curvature regions are not always captured…
We present GeGnn, a learning-based method for computing the approximate geodesic distance between two arbitrary points on discrete polyhedra surfaces with constant time complexity after fast precomputation. Previous relevant methods either…
Finding meaningful representations and distances of hierarchical data is important in many fields. This paper presents a new method for hierarchical data embedding and distance. Our method relies on combining diffusion geometry, a central…
We investigate the dependence of the approximation capacity of deep residual networks on its depth in a continuous dynamical systems setting. This can be formulated as the general problem of quantifying the minimal time-horizon required to…
We present a learning-based method, namely GeoUDF,to tackle the long-standing and challenging problem of reconstructing a discrete surface from a sparse point cloud.To be specific, we propose a geometry-guided learning method for UDF and…