arXiv research

LA-VDM accelerates VDM using landmarks to improve data analysis.

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 non-linear dimensionality reduction methods. MFVDM combines different nonlinear emb…

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 other non-linear dimensionality reduction methods, such as LLE, ISOMAP and Laplacian…

We introduce the concept of Hypoelliptic Diffusion Maps (HDM), a framework generalizing Diffusion Maps in the context of manifold learning and dimensionality reduction. Standard non-linear dimensionality reduction methods (e.g., LLE, ISOMAP, Laplacian Eigenmaps, Diffusion Maps) focus on mining massive data sets using w…

VDWs enhance graph neural networks for analyzing complex data.

In this paper, we propose a novel approach for manifold learning that combines the Earthmover's distance (EMD) with the diffusion maps method for dimensionality reduction. We demonstrate the potential benefits of this approach for learning shape spaces of proteins and other flexible macromolecules using a simulated dat…

A new model trains prior and encoder/decoder networks simultaneously for efficient generation.

CADD improves generative quality by augmenting discrete diffusion with continuous latent space.

PolyNSD improves Neural Sheaf Diffusion with polynomial operators and spectral rescaling.

Enhanced probabilistic sampling on manifolds using Double Diffusion Maps and Geometric Harmonics.

DMPS uses diffusion maps and LAWGD for efficient generative modeling.

Diffusion maps are an emerging data-driven technique for non-linear dimensionality reduction, which are especially useful for the analysis of coherent structures and nonlinear embeddings of dynamical systems. However, the computational complexity of the diffusion maps algorithm scales with the number of observations. T…

Sharp Lipschitz bounds for flow-matching and diffusion models with optimal sampling rates.

DM uses semigroup property to tune diffusion time for better data analysis.

Neumann eigenmaps improve landmark-based diffusion map embeddings.

New guarantees for uniquely identifying transport maps and vector fields from finite measure-valued data.

New framework for diffusion geometry simplifies complex calculations.

We address the problem of influence maximization when the social network is accompanied by diffusion cascades. In prior works, such information is used to compute influence probabilities, which is utilized by stochastic diffusion models in influence maximization. Motivated by the recent criticism on the effectiveness o…

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 computational load, reaching up to cubic complexity in the number of data points. This l…

This letter proposes a sparse diffusion steepest-descent algorithm for one bit compressed sensing in wireless sensor networks. The approach exploits the diffusion strategy from distributed learning in the one bit compressed sensing framework. To estimate a common sparse vector cooperatively from only the sign of measur…

Stabilizes complex systems using diffusion models trained on Lyapunov functions.

Method differentiates diffusion model training to predict sample sensitivity.

Improved spectral convergence bounds for diffusion maps on tori.

The author has previously constructed a class of admissible vector fields on the path space of an elliptic diffusion process $x$ taking values in a closed compact manifold. In this Note the existence of flows for this class of vector fields is established and it is shown that the law of $x$ is quasi-invariant under the…

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 shown by Belkin and Niyogi \cite{belkin_niyogi:2007} that the eigenvectors and eige…

Diffusion maps are a nonlinear manifold learning technique based on harmonic analysis of a diffusion process over the data. Out-of-sample extensions with computational complexity $\mathcal{O}(N)$, where $N$ is the number of points comprising the manifold, frustrate applications to online learning applications requiring…

We introduce, test and discuss a method for classifying and clustering data modeled as directed graphs. The idea is to start diffusion processes from any subset of a data collection, generating corresponding distributions for reaching points in the network. These distributions take the form of high-dimensional numerica…

Riemannian stochastic gradient descent approximates a diffusion process called Riemannian stochastic modified flow.

High order discretization schemes of SDEs by using free Lie algebra valued random variables are introduced by Kusuoka, Lyons-Victoir, Ninomiya-Victoir and Ninomiya-Ninomiya. These schemes are called KLNV methods. They involve solving the flows of vector fields associated with SDEs and it is usually done by numerical me…

Functional magnetic resonance imaging (fMRI) produces data about activity inside the brain, from which spatial maps can be extracted by independent component analysis (ICA). In datasets, there are n spatial maps that contain p voxels. The number of voxels is very high compared to the number of analyzed spatial maps. Cl…

New method uses Diffusion Maps for latent space modeling of dynamical systems.

The paper proves Liouville theorems for $V$-harmonic maps under specific curvature conditions.

A new diffusion method approximates Schrödinger bridge with improved convergence.

New method models covariates and responses without parametric assumptions using manifold learning.

Survey of diffusion and optimal transport methods in machine learning.

Two adaptive kernel selection methods improve the accuracy of Kernelized Diffusion Maps.

Let $x$ denote a diffusion process defined on a closed compact manifold. In an earlier article, the author introduced a new approach to constructing admissible vector fields on the associated space of paths, under the assumption of ellipticity of $x$. In this article, this method is extended to yield similar results fo…

A new method quantifies uncertainty in brain injury simulations.

Diffusion Maps improves on Functional PCA for non-linear functional data.

New method circumvents curse of dimensionality in Laplacian estimation.

New model generates data on constrained sets without losing tractability.

DDPM encoder matches optimal transport for natural images.

A new model designs molecular latent vectors for drug discovery.

Distributed adaptive networks achieve better estimation performance by exploiting temporal and as well spatial diversity while consuming few resources. Recent works have studied the single task distributed estimation problem, in which the nodes estimate a single optimum parameter vector collaboratively. However, there …

Develops a surrogate model for predicting system responses using GDMaps and geometric harmonics.

Complex biological systems have been successfully modeled by biochemical and genetic interaction networks, typically gathered from high-throughput (HTP) data. These networks can be used to infer functional relationships between genes or proteins. Using the intuition that the topological role of a gene in a network rela…

Studying SGD on deep neural networks using diffusion maps.

Diffusion maps help learn complex quantum phase transitions from data.