相关论文: Diffusion maps, spectral clustering and reaction c…

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 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…

Inspired by random walk on graphs, diffusion map (DM) is a class of unsupervised machine learning that offers automatic identification of low-dimensional data structure hidden in a high-dimensional dataset. In recent years, among its many…

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…

This paper focuses on obtaining clustering information about a distribution from its i.i.d. samples. We develop theoretical results to understand and use clustering information contained in the eigenvectors of data adjacency matrices based…

Diffusion condensation is a dynamic process that yields a sequence of multiscale data representations that aim to encode meaningful abstractions. It has proven effective for manifold learning, denoising, clustering, and visualization of…

Many complex systems have natural representations as multi-layer networks. While these formulations retain more information than standard single-layer network models, there is not yet a fully developed theory for computing network metrics…

Because of the significant increase in size and complexity of the networks, the distributed computation of eigenvalues and eigenvectors of graph matrices has become very challenging and yet it remains as important as before. In this paper…

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…

This PhD thesis lays out algebraic and topological structures relevant for the study of probabilistic graphical models. Marginal estimation algorithms are introduced as diffusion equations of the form $\dot u = \delta \varphi$. They…

We study spectral properties of the Fokker-Planck operator that represents particles moving via a combination of diffusion and advection in a time-independent random velocity field, presenting in detail work outlined elsewhere [J. T.…

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…

Random feature maps are ubiquitous in modern statistical machine learning, where they generalize random projections by means of powerful, yet often difficult to analyze nonlinear operators. In this paper, we leverage the "concentration"…

We introduce {\em vector diffusion maps} (VDM), a new mathematical framework for organizing and analyzing massive high dimensional data sets, images and shapes. VDM is a mathematical and algorithmic generalization of diffusion maps and…

Consistency is a key property of all statistical procedures analyzing randomly sampled data. Surprisingly, despite decades of work, little is known about consistency of most clustering algorithms. In this paper we investigate consistency of…

Statisticians increasingly face the problem to reconsider the adaptability of classical inference techniques. In particular, divers types of high-dimensional data structures are observed in various research areas; disclosing the boundaries…

Adaptive networks are suitable for decentralized inference tasks, e.g., to monitor complex natural phenomena. Recent research works have intensively studied distributed optimization problems in the case where the nodes have to estimate a…

Real-world data generation often involves complex inter-dependencies among instances, violating the IID-data hypothesis of standard learning paradigms and posing a challenge for uncovering the geometric structures for learning desired…

The existing research on spectral algorithms, applied within a Reproducing Kernel Hilbert Space (RKHS), has primarily focused on general kernel functions, often neglecting the inherent structure of the input feature space. Our paper…

Here we introduce connectivity operators, namely, diffusion operators, general Laplacian operators, and general adjacency operators for hypergraphs. These operators are generalisations of some conventional notions of apparently different…