Synthetic Data Generation and Deep Learning for the Topological Analysis of 3D Data

We present a method for inferring dense depth maps from images and sparse depth measurements by leveraging synthetic data to learn the association of sparse point clouds with dense natural shapes, and using the image as evidence to validate the predicted depth map. Our learned prior for natural shapes uses only sparse depth as input, not images, so the method is not affected by the covariate shift when attempting to transfer learned models from synthetic data to real ones. Th...

Generative models aim to learn the distribution of observed data by generating new instances. With the advent of neural networks, deep generative models, including variational autoencoders (VAEs), generative adversarial networks (GANs), and diffusion models (DMs), have progressed remarkably in synthesizing 2D images. Recently, researchers started to shift focus from 2D to 3D space, considering that 3D data is more closely aligned with our physical world and holds immense prac...

Point cloud data have been widely explored due to its superior accuracy and robustness under various adverse situations. Meanwhile, deep neural networks (DNNs) have achieved very impressive success in various applications such as surveillance and autonomous driving. The convergence of point cloud and DNNs has led to many deep point cloud models, largely trained under the supervision of large-scale and densely-labelled point cloud data. Unsupervised point cloud representation ...

Many scientific fields study data with an underlying structure that is a non-Euclidean space. Some examples include social networks in computational social sciences, sensor networks in communications, functional networks in brain imaging, regulatory networks in genetics, and meshed surfaces in computer graphics. In many applications, such geometric data are large and complex (in the case of social networks, on the scale of billions), and are natural targets for machine learni...

Existing deep learning algorithms for point cloud analysis mainly concern discovering semantic patterns from global configuration of local geometries in a supervised learning manner. However, very few explore geometric properties revealing local surface manifolds embedded in 3D Euclidean space to discriminate semantic classes or object parts as additional supervision signals. This paper is the first attempt to propose a unique multi-task geometric learning network to improve ...

We study how the topology of a data set $M = M_a \cup M_b \subseteq \mathbb{R}^d$, representing two classes $a$ and $b$ in a binary classification problem, changes as it passes through the layers of a well-trained neural network, i.e., with perfect accuracy on training set and near-zero generalization error ($\approx 0.01\%$). The goal is to shed light on two mysteries in deep neural networks: (i) a nonsmooth activation function like ReLU outperforms a smooth one like hyperbo...

Geometric deep learning has gained much attention in recent years due to more available data acquired from non-Euclidean domains. Some examples include point clouds for 3D models and wireless sensor networks in communications. Graphs are common models to connect these discrete data points and capture the underlying geometric structure. With the large amount of these geometric data, graphs with arbitrarily large size tend to converge to a limit model -- the manifold. Deep neur...

In this paper, we propose a point cloud classification method based on graph neural network and manifold learning. Different from the conventional point cloud analysis methods, this paper uses manifold learning algorithms to embed point cloud features for better considering the geometric continuity on the surface. Then, the nature of point cloud can be acquired in low dimensional space, and after being concatenated with features in the original three-dimensional (3D)space, bo...

This paper presents an innovative enhancement to the Sphere as Prior Generative Adversarial Network (SP-GAN) model, a state-of-the-art GAN designed for point cloud generation. A novel method is introduced for point cloud generation that elevates the structural integrity and overall quality of the generated point clouds by incorporating topological priors into the training process of the generator. Specifically, this work utilizes the K-means algorithm to segment a point cloud...

Topological deep learning (TDL) is a rapidly evolving field that uses topological features to understand and design deep learning models. This paper posits that TDL may complement graph representation learning and geometric deep learning by incorporating topological concepts, and can thus provide a natural choice for various machine learning settings. To this end, this paper discusses open problems in TDL, ranging from practical benefits to theoretical foundations. For each p...