Related papers: G-invariant diffusion maps
Graph Laplacian based algorithms for data lying on a manifold have been proven effective for tasks such as dimensionality reduction, clustering, and denoising. In this work, we consider data sets whose data points lie on a manifold that is…
In this article, we consider the manifold learning problem when the data set is invariant under the action of a compact Lie group $K$. Our approach consists in augmenting the data-induced graph Laplacian by integrating over the $K$-orbits…
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,…
Nonlinear manifold learning algorithms, such as diffusion maps, have been fruitfully applied in recent years to the analysis of large and complex data sets. However, such algorithms still encounter challenges when faced with real data. One…
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…
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…
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…
This is a tutorial and survey paper for nonlinear dimensionality and feature extraction methods which are based on the Laplacian of graph of data. We first introduce adjacency matrix, definition of Laplacian matrix, and the interpretation…
We introduce a data-driven dynamic factor framework for modeling the joint evolution of high-dimensional covariates and responses without parametric assumptions. Standard factor models applied to covariates alone often lose explanatory…
This paper presents a diffusion based probabilistic interpretation of spectral clustering and dimensionality reduction algorithms that use the eigenvectors of the normalized graph Laplacian. Given the pairwise adjacency matrix of all…
Consider a finite dimensional real vector space and a finite group acting unitarily on it. We study the general problem of constructing Euclidean stable embeddings of the quotient space of orbits. Our embedding is based on subsets of sorted…
We construct an extension of diffusion geometry to multiple modalities through joint approximate diagonalization of Laplacian matrices. This naturally extends classical data analysis tools based on spectral geometry, such as diffusion maps…
Gaussian graphical models have become a well-recognized tool for the analysis of conditional independencies within a set of continuous random variables. From an inferential point of view, it is important to realize that they are composite…
Laplacian Eigenvectors of the graph constructed from a data set are used in many spectral manifold learning algorithms such as diffusion maps and spectral clustering. Given a graph constructed from a random sample of a $d$-dimensional…
Data living on manifolds commonly appear in many applications. Often this results from an inherently latent low-dimensional system being observed through higher dimensional measurements. We show that under certain conditions, it is possible…
We study the problem asking if one can embed manifolds into finite dimensional Euclidean spaces by taking finite number of eigenvector fields of the connection Laplacian. This problem is essential for the dimension reduction problem in…
We present a comprehensive extension of the latent position network model known as the random dot product graph to accommodate multiple graphs -- both undirected and directed -- which share a common subset of nodes, and propose a method for…
In this paper, we present GGSD, a novel graph generative model based on 1) the spectral decomposition of the graph Laplacian matrix and 2) a diffusion process. Specifically, we propose to use a denoising model to sample eigenvectors and…
This paper considers the problem of embedding directed graphs in Euclidean space while retaining directional information. We model a directed graph as a finite set of observations from a diffusion on a manifold endowed with a vector field.…