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We introduce novel estimators for computing the curvature, tangent spaces, and dimension of data from manifolds, using tools from diffusion geometry. Although classical Riemannian geometry is a rich source of inspiration for geometric data…
Despite the popularity of the manifold hypothesis, current manifold-learning methods do not support machine learning directly on the latent $d$-dimensional data manifold, as they primarily aim to perform dimensionality reduction into…
Nonlinear dimensionality reduction methods provide a valuable means to visualize and interpret high-dimensional data. However, many popular methods can fail dramatically, even on simple two-dimensional manifolds, due to problems such as…
Manifold learning aims to discover and represent low-dimensional structures underlying high-dimensional data while preserving critical topological and geometric properties. Existing methods often fail to capture local details with global…
An evolutionary multi-objective aerodynamic design optimization method using the computational fluid dynamics (CFD) simulations incorporating deep neural network (DNN) to reduce the required computational time is proposed. In this approach,…
Random geometric graphs are random graph models defined on metric measure spaces. A random geometric graph is generated by first sampling points from a metric space and then connecting each pair of sampled points independently with a…
Representing a manifold of very high-dimensional data with generative models has been shown to be computationally efficient in practice. However, this requires that the data manifold admits a global parameterization. In order to represent…
Diffusion maps are a commonly used kernel-based method for manifold learning, which can reveal intrinsic structures in data and embed them in low dimensions. However, as with most kernel methods, its implementation requires a heavy…
In the field of computer vision, the numerical encoding of 3D surfaces is crucial. It is classical to represent surfaces with their Signed Distance Functions (SDFs) or Unsigned Distance Functions (UDFs). For tasks like representation…
Filaments are ubiquitous in astronomical data sets. Be it in particle simulations or observations, filaments are always tracers of a perturbation in the equilibrium of the studied system and hold essential information on its history and…
The morphology and distribution of airway tree abnormalities enables diagnosis and disease characterisation across a variety of chronic respiratory conditions. In this regard, airway segmentation plays a critical role in the production of…
Under the data manifold hypothesis, high-dimensional data are concentrated near a low-dimensional manifold. We study the problem of Riemannian optimization over such manifolds when they are given only implicitly through the data…
A method for extracting multiscale geometric features from a data cloud is proposed and analyzed. The basic idea is to map each pair of data points into a real-valued feature function defined on $[0,1]$. The construction of these feature…
A new machine learning (ML) technique termed as federated learning (FL) aims to preserve data at the edge devices and to only exchange ML model parameters in the learning process. FL not only reduces the communication needs but also helps…
Aerial manipulation (AM) expands UAV capabilities beyond passive observation to contact-based operations at high altitudes and in otherwise inaccessible environments. Although recent advances show promise, most AM systems are developed in…
Denoising Diffusion Probabilistic Models (DDPM) are powerful state-of-the-art methods used to generate synthetic data from high-dimensional data distributions and are widely used for image, audio, and video generation as well as many more…
Despite high-dimensionality of images, the sets of images of 3D objects have long been hypothesized to form low-dimensional manifolds. What is the nature of such manifolds? How do they differ across objects and object classes? Answering…
Analyzing air pollution data is challenging as there are various analysis focuses from different aspects: feature (what), space (where), and time (when). As in most geospatial analysis problems, besides high-dimensional features, the…
Decision trees (DTs) and their random forest (RF) extensions are workhorses of classification and regression in Euclidean spaces. However, algorithms for learning in non-Euclidean spaces are still limited. We extend DT and RF algorithms to…
Physics-informed machine learning (PIML) integrates partial differential equations (PDEs) into machine learning models to solve inverse problems, such as estimating coefficient functions (e.g., the Hamiltonian function) that characterize…