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Deep learning methods have played a more and more important role in hyperspectral image classification. However, the general deep learning methods mainly take advantage of the information of sample itself or the pairwise information between…
We introduce a novel diffusion-based spectral algorithm to tackle regression analysis on high-dimensional data, particularly data embedded within lower-dimensional manifolds. Traditional spectral algorithms often fall short in such…
Diffusion models (DMs) have emerged as the new state-of-the-art family of deep generative models. To gain deeper insights into the limitations of diffusion models in recommender systems, we investigate the fundamental structural disparities…
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…
The Diffusion Map is a nonlinear dimensionality reduction technique used to analyze high-dimensional data, with recent applications extending to datasets from the social sciences. Previous research has given little attention to how the…
Accurately capturing the full-range response of structures is crucial in structural health monitoring (SHM) for ensuring safety and operational integrity. However, limited sensor deployment due to cost, accessibility, or scale often hinders…
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…
In recent years, manifold learning has become increasingly popular as a tool for performing non-linear dimensionality reduction. This has led to the development of numerous algorithms of varying degrees of complexity that aim to recover man…
In this work, we propose a simple kernel ridge regression (KRR) framework with a dynamic-aware validation strategy for long-term prediction of complex dynamical systems. By employing a data-driven kernel derived from diffusion maps, the…
Diffusion maps (DM) constitute a classic dimension reduction technique, for data lying on or close to a (relatively) low-dimensional manifold embedded in a much larger dimensional space. The DM procedure consists in constructing a spectral…
Diffusion magnetic resonance imaging, a non-invasive tool to infer white matter fiber connections, produces a large number of streamlines containing a wealth of information on structural connectivity. The size of these tractography outputs…
Manifold learning approaches seek the intrinsic, low-dimensional data structure within a high-dimensional space. Mainstream manifold learning algorithms, such as Isomap, UMAP, $t$-SNE, Diffusion Map, and Laplacian Eigenmaps do not use data…
Various Graph Neural Networks (GNNs) have been successful in analyzing data in non-Euclidean spaces, however, they have limitations such as oversmoothing, i.e., information becomes excessively averaged as the number of hidden layers…
Estimating correspondences between pairs of deformable shapes remains a challenging problem. Despite substantial progress, existing methods lack broad generalization capabilities and require category-specific training data. To address these…
We introduce diffusion geometry as a new framework for geometric and topological data analysis. Diffusion geometry uses the Bakry-Emery $\Gamma$-calculus of Markov diffusion operators to define objects from Riemannian geometry on a wide…
Hyperspectral imaging is a powerful technology that is plagued by large dimensionality. Herein, we explore a way to combat that hindrance via non-contiguous and contiguous (simpler to realize sensor) band grouping for dimensionality…
Learning the distribution of data on Riemannian manifolds is crucial for modeling data from non-Euclidean space, which is required by many applications in diverse scientific fields. Yet, existing generative models on manifolds suffer from…
The absence of intrinsic adjacency relations and orientation systems in hypergraphs creates fundamental challenges for constructing sheaf Laplacians of arbitrary degrees. We resolve these limitations through symmetric simplicial sets…
Selecting an appropriate kernel is a central challenge in kernel-based spectral methods. In \emph{Kernelized Diffusion Maps} (KDM), the kernel determines the accuracy of the RKHS estimator of a diffusion-type operator and hence the quality…
Spectral methods that are based on eigenvectors and eigenvalues of discrete graph Laplacians, such as Diffusion Maps and Laplacian Eigenmaps are often used for manifold learning and non-linear dimensionality reduction. It was previously…