Related papers: Laplacian-Based Dimensionality Reduction Including…

Laplacian mixture models identify overlapping regions of influence in unlabeled graph and network data in a scalable and computationally efficient way, yielding useful low-dimensional representations. By combining Laplacian eigenspace and…

Spectral clustering and diffusion maps are celebrated dimensionality reduction algorithms built on eigen-elements related to the diffusive structure of the data. The core of these procedures is the approximation of a Laplacian through a…

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…

Graph-Laplacians and their spectral embeddings play an important role in multiple areas of machine learning. This paper is focused on graph-Laplacian dimension reduction for the spectral clustering of data as a primary application. Spectral…

We present multiscale graph-based reduction algorithms for upscaling heterogeneous and anisotropic diffusion problems. The proposed coarsening approaches begin by constructing a partitioning of the computational domain into a set of…

Dynamical Systems (DS) are fundamental to the modeling and understanding time evolving phenomena, and have application in physics, biology and control. As determining an analytical description of the dynamics is often difficult, data-driven…

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…

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…

In this paper we study variants of the widely used spectral clustering that partitions a graph into k clusters by (1) embedding the vertices of a graph into a low-dimensional space using the bottom eigenvectors of the Laplacian matrix, and…

We present clustering methods for multivariate data exploiting the underlying geometry of the graphical structure between variables. As opposed to standard approaches that assume known graph structures, we first estimate the edge structure…

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…

Dimensionality reduction techniques aim at representing high-dimensional data in low-dimensional spaces to extract hidden and useful information or facilitate visual understanding and interpretation of the data. However, few of them take…

Diffusion Map is a spectral dimensionality reduction technique which is able to uncover nonlinear submanifolds in high-dimensional data. And, it is increasingly applied across a wide range of scientific disciplines, such as biology,…

Spectral techniques are popular and robust approaches to data analysis. A prominent example is the use of eigenvectors of a Laplacian, constructed from data affinities, to identify natural data groupings or clusters, or to produce a…

Graph Laplacians and related nonlinear mappings into low dimensional spaces have been shown to be powerful tools for organizing high dimensional data. Here we consider a data set X in which the graph associated with it changes depending on…

Several graph data mining, signal processing, and machine learning downstream tasks rely on information related to the eigenvectors of the associated adjacency or Laplacian matrix. Classical eigendecomposition methods are powerful when the…

The second eigenvalue of the Laplacian matrix and its associated eigenvector are fundamental features of an undirected graph, and as such they have found widespread use in scientific computing, machine learning, and data analysis. In many…

We propose a pair of completely data-driven algorithms for unsupervised classification and dimension reduction, and we empirically study their performance on a number of data sets, both simulated data in three-dimensions and images from the…

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…

Convolutional layers within graph neural networks operate by aggregating information about local neighbourhood structures; one common way to encode such substructures is through random walks. The distribution of these random walks evolves…