Related papers: Multimodal diffusion geometry by joint diagonaliza…
In this paper, we construct multimodal spectral geometry by finding a pair of closest commuting operators (CCO) to a given pair of Laplacians. The CCOs are jointly diagonalizable and hence have the same eigenbasis. Our construction…
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…
Observational data usually comes with a multimodal nature, which means that it can be naturally represented by a multi-layer graph whose layers share the same set of vertices (users) with different edges (pairwise relationships). In this…
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…
Graph learning methods have recently been receiving increasing interest as means to infer structure in datasets. Most of the recent approaches focus on different relationships between a graph and data sample distributions, mostly in…
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…
We analyze the spectral clustering procedure for identifying coarse structure in a data set $x_1, \dots, x_n$, and in particular study the geometry of graph Laplacian embeddings which form the basis for spectral clustering algorithms. More…
This paper proposes and analyzes a novel clustering algorithm that combines graph-based diffusion geometry with techniques based on density and mode estimation. The proposed method is suitable for data generated from mixtures of…
In this paper, we focus on graph learning from multi-view data of shared entities for spectral clustering. We can explain interactions between the entities in multi-view data using a multi-layer graph with a common vertex set, which…
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…
Spectral clustering uses a graph Laplacian spectral embedding to enhance the cluster structure of some data sets. When the embedding is one dimensional, it can be used to sort the items (spectral ordering). A number of empirical results…
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…
Data often are formed of multiple modalities, which jointly describe the observed phenomena. Modeling the joint distribution of multimodal data requires larger expressive power to capture high-level concepts and provide better data…
Multi-view spectral clustering can effectively reveal the intrinsic cluster structure among data by performing clustering on the learned optimal embedding across views. Though demonstrating promising performance in various applications,…
Calculus and geometry are ubiquitous in the theoretical modelling of scientific phenomena, but have historically been very challenging to apply directly to real data as statistics. Diffusion geometry is a new theory that reformulates…
The diffusion maps embedding of data lying on a manifold has shown success in tasks such as dimensionality reduction, clustering, and data visualization. In this work, we consider embedding data sets that were sampled from a manifold which…
In this work we study statistical properties of graph-based algorithms for multi-manifold clustering (MMC). In MMC the goal is to retrieve the multi-manifold structure underlying a given Euclidean data set when this one is assumed to be…
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…
We introduce Hodge Diffusion Maps, a novel manifold learning algorithm designed to analyze and extract topological information from high-dimensional data-sets. This method approximates the exterior derivative acting on differential forms,…
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…