Related papers: Sparse Dowker Nerves
Given a point cloud $P$ in Euclidean space and a positive parameter $t$ we can consider the $t$-neighborhood $P^{t}$ of $P$ consisting of points at distance less than $t$ to $P$. Homology of $P^{t}$ gives information about components,…
Topological data analysis combines machine learning with methods from algebraic topology. Persistent homology, a method to characterize topological features occurring in data at multiple scales is of particular interest. A major obstacle to…
Deep neural networks have emerged as powerful tools for learning operators defined over infinite-dimensional function spaces. However, existing theories frequently encounter difficulties related to dimensionality and limited…
The aim of this paper is to present a method for computation of persistent homology that performs well at large filtration values. To this end we introduce the concept of filtered covers. We show that the persistent homology of a bounded…
We present a network architecture for processing point clouds that directly operates on a collection of points represented as a sparse set of samples in a high-dimensional lattice. Naively applying convolutions on this lattice scales…
The \v{C}ech complex is one of the most widely used tools in applied algebraic topology. Unfortunately, due to the inclusive nature of the \v{C}ech filtration, the number of simplices grows exponentially in the number of input points. A…
Current learning-based methods predict NeRF or 3D Gaussians from point clouds to achieve photo-realistic rendering but still depend on categorical priors, dense point clouds, or additional refinements. Hence, we introduce a novel point…
We propose Sparse Neural Network architectures that are based on random or structured bipartite graph topologies. Sparse architectures provide compression of the models learned and speed-ups of computations, they can also surpass their…
Convolutional networks are the de-facto standard for analyzing spatio-temporal data such as images, videos, and 3D shapes. Whilst some of this data is naturally dense (e.g., photos), many other data sources are inherently sparse. Examples…
In recent years, with the development of computing resources and LiDAR, point cloud semantic segmentation has attracted many researchers. For the sparsity of point clouds, although there is already a way to deal with sparse convolution,…
The rapid progress of Artificial Intelligence research came with the development of increasingly complex deep learning models, leading to growing challenges in terms of computational complexity, energy efficiency and interpretability. In…
Coded recurrent neural networks with three levels of sparsity are introduced. The first level is related to the size of messages, much smaller than the number of available neurons. The second one is provided by a particular coding rule,…
Persistent homology of the Rips filtration allows to track topological features of a point cloud over scales, and is a foundational tool of topological data analysis. Unfortunately, the Rips-filtration is exponentially sized, when…
Neural networks and other machine learning models compute continuous representations, while humans communicate with discrete symbols. Reconciling these two forms of communication is desirable to generate human-readable interpretations or to…
This paper addresses the problem of generating dense point clouds from given sparse point clouds to model the underlying geometric structures of objects/scenes. To tackle this challenging issue, we propose a novel end-to-end learning-based…
Neural networks have seen limited use in prediction for high-dimensional data with small sample sizes, because they tend to overfit and require tuning many more hyperparameters than existing off-the-shelf machine learning methods. With…
We outline a novel clustering scheme for simplicial complexes that produces clusters of simplices in a way that is sensitive to the homology of the complex. The method is inspired by, and can be seen as a higher-dimensional version of,…
Deep neural networks have made remarkable progresses on various computer vision tasks. Recent works have shown that depth, width and shortcut connections of networks are all vital to their performances. In this paper, we introduce a method…
Point clouds are unstructured and unordered data, as opposed to images. Thus, most machine learning approach developed for image cannot be directly transferred to point clouds. In this paper, we propose a generalization of discrete…
Non-uniformed 3D sparse data, e.g., point clouds or voxels in different spatial positions, make contribution to the task of 3D object detection in different ways. Existing basic components in sparse convolutional networks (Sparse CNNs)…