Related papers: Diffusion maps for changing data
Models of network diffusion typically rely on the Laplacian matrix, capturing interactions via direct connections. Beyond direct interactions, information in many systems can also flow via indirect pathways, where influence typically…
Many problems in machine learning can be expressed by means of a graph with nodes representing training samples and edges representing the relationship between samples in terms of similarity, temporal proximity, or label information. Graphs…
We consider a collection of $n$ points in $\mathbb{R}^d$ measured at $m$ times, which are encoded in an $n \times d \times m$ data tensor. Our objective is to define a single embedding of the $n$ points into Euclidean space which summarizes…
A central problem in data analysis is the low dimensional representation of high dimensional data, and the concise description of its underlying geometry and density. In the analysis of large scale simulations of complex dynamical systems,…
In this paper we address the problem of understanding the success of algorithms that organize patches according to graph-based metrics. Algorithms that analyze patches extracted from images or time series have led to state-of-the art…
This work focuses on exploring the potential benefits of introducing a nonlinear Laplacian in Sheaf Neural Networks for graph-related tasks. The primary aim is to understand the impact of such nonlinearity on diffusion dynamics, signal…
Time-dependent conformal maps are used to model a class of growth phenomena limited by coupled non-Laplacian transport processes, such as nonlinear diffusion, advection, and electro-migration. Both continuous and stochastic dynamics are…
In manifold learning, algorithms based on graph Laplacians constructed from data have received considerable attention both in practical applications and theoretical analysis. In particular, the convergence of graph Laplacians obtained from…
Geological parameterization entails the representation of a geomodel using a small set of latent variables and a mapping from these variables to grid-block properties such as porosity and permeability. Parameterization is useful for data…
We introduce the Linearized Diffusion Map (LDM), a novel linear dimensionality reduction method constructed via a linear approximation of the diffusion-map kernel. LDM integrates the geometric intuition of diffusion-based nonlinear methods…
Graph embedding provides a feasible methodology to conduct pattern classification for graph-structured data by mapping each data into the vectorial space. Various pioneering works are essentially coding method that concentrates on a…
Understanding the mutual interdependence between the behavior of dynamical processes on networks and the underlying topologies promises new insight for a large class of empirical networks. We present a generic approach to investigate this…
We introduce multi-frequency vector diffusion maps (MFVDM), a new framework for organizing and analyzing high dimensional datasets. The new method is a mathematical and algorithmic generalization of vector diffusion maps (VDM) and other…
When performing classification tasks, raw high dimensional features often contain redundant information, and lead to increased computational complexity and overfitting. In this paper, we assume the data samples lie on a single underlying…
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…
The metric dimension of a graph is the smallest number of nodes required to identify all other nodes based on shortest path distances uniquely. Applications of metric dimension include discovering the source of a spread in a network,…
A new approach to the modeling of nonfree particle diffusion is presented. The approach uses a general setup based on geometric graphs (networks of curves), which means that particle diffusion in anything from arrays of barriers and pore…
Kernel-based non-linear dimensionality reduction methods, such as Local Linear Embedding (LLE) and Laplacian Eigenmaps, rely heavily upon pairwise distances or similarity scores, with which one can construct and study a weighted graph…
This paper investigates score-based diffusion models when the underlying target distribution is concentrated on or near low-dimensional manifolds within the higher-dimensional space in which they formally reside, a common characteristic of…
Inspired by graph-based methodologies, we introduce a novel graph-spanning algorithm designed to identify changes in both offline and online data across low to high dimensions. This versatile approach is applicable to Euclidean and…