The paper examines smoothness in diffusion algebra.
problem Smoothness in diffusion algebras.
method Not explicitly detailed in the abstract.
result Not explicitly detailed in the abstract.
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48 results for “Diffusion Geometry”
Reconstructs Riemannian geometry from diffusion properties.
problem Recovering Riemannian geometry from diffusion data.
method Intrinsic reconstruction from diffusion semigroup and calculus.
result Reveals Riemannian structure from diffusion properties.
Computing Diffusion Geometrymath.DG
New framework for diffusion geometry simplifies complex calculations.
problem Challenges in applying calculus and geometry to real data.
method Reformulates calculus and geometry via diffusion processes.
result Improves precision, robustness, and computational efficiency.
Cosine schedule is optimal for discrete diffusion models.
problem Choosing the best discretization schedule for diffusion models.
method Optimized using Fisher-Rao geometry.
result Cosine schedule is Fisher-Rao optimal.
New method improves counterfactual distribution learning for high-dimensional outcomes.
problem Counterfactual distribution learning for high-dimensional outcomes with concentrated structure.
method Geometry-adaptive diffusion-guided smoothing estimators combining causal nuisance adjustment and local outcome geometry.
result Geometry-adaptive methods show steeper error decay in semi-synthetic experiments.
Unified framework for multi-view diffusion geometries using intertwined diffusion trajectories.
problem Constructing multi-view diffusion geometries with flexible view interaction and fusion.
method Intertwined multi-view diffusion trajectories (MDTs) as a class of inhomogeneous diffusion processes.
result Established theoretical properties and derived diffusion distances and embeddings.
EntroPath learns manifold geometry from diffusion paths.
problem Learning geodesic geometry from data graphs with spurious shortcuts.
method Maximum Entropy Path Ensemble Embedding (MERW) with k-step diffusion paths.
result EntroPath converges to squared geodesic distance in the short-time limit.
Compressed Diffusioncs.LG
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…
GeoFunFlow: Geometric Function Flow Matching for Inverse Operator Learning over Complex Geometriescs.LG
GeoFunFlow tackles inverse problems on complex geometries with efficient learning.
problem Challenges in inverse problems governed by PDEs, especially on irregular geometries.
method Combines geometric function autoencoder and latent diffusion model trained via rectified flow.
result Achieves state-of-the-art reconstruction accuracy and efficient inference.
Paper analyzes latent space geometry in generative models using Fisher information.
problem Understanding the structure of latent spaces in generative models.
method Reconstructs Fisher information metric from generated samples and posterior distribution.
result Reveals fractal structure and abrupt changes in Fisher metric at phase boundaries.
Study shows how heat leaks from material sets in low diffusivity scenarios.
problem Understanding heat leakage from material sets in low diffusivity limits.
method Generalized leading-order asymptotics for time-dependent diffusion processes.
result Diffusive transport out of a material set is proportional to the surface area of the set boundary.
Unified theory of measure-preserving diffusions on manifolds.
problem Deriving a complete recipe for measure-preserving diffusions on manifolds.
method Developed a geometric theory that unifies and generalizes previous constructions, relying on intrinsic geometry of the target measure.
result The completeness result is a direct consequence of manifold topology and target measure geometry.
We consider Lagrangian coherent structures (LCSs) as the boundaries of material subsets whose advective evolution is metastable under weak diffusion. For their detection, we first transform the Eulerian advection-diffusion equation to Lagrangian coordinates, in which it takes the form of a time-dependent diffusion or h…
New methods estimate curvature, tangent spaces, and dimension of noisy data.
problem Estimating geometric properties of noisy or sparse data.
method Diffusion geometry tools for Riemannian manifold analysis.
result Significantly outperforms existing methods in noisy or sparse data.
The paper shows that energy futures yield curves have an affine geometry.
problem Estimating dynamic behavior of yield curves from data while avoiding arbitrage.
method Finite dimensional models for yield curves, diffusion coefficients, and compatibility conditions.
result The compatibility of yield curves with diffusion coefficients forces an affine geometry.
Finslerian graph neural networks recover nonlinear diffusion geometry
problem Graph neural networks on point clouds
method Estimates of the Finsler Laplacian
result Recovery of Finsler geometry
The paper analyzes the latent geometry of generative diffusion models.
problem The manifold overfitting phenomenon in generative models.
method Statistical physics approach to analyze the spectrum of eigenvalues and singular values of the Jacobian of the score function.
result Three distinct qualitative phases during the generative process: trivial, manifold coverage, and consolidation phases.
The paper analyzes diffusion condensation for data geometry and topology.
problem Understanding the geometry and topology of high-dimensional data.
method Time-inhomogeneous diffusion process with geometric, spectral, and topological analysis.
result The condensation process defines intrinsic condensation homology and ambient persistent homology.
Riemannian stochastic gradient descent approximates a diffusion process called Riemannian stochastic modified flow.
problem Improving convergence rate of Riemannian stochastic gradient descent.
method Using stochastic differential geometry, the paper shows RSGD can be approximated by the Riemannian stochastic modified flow (RSMF).
result RSGD can be approximated by the solution to the RSMF driven by an infinite-dimensional Wiener process, increasing the order of approximation.
New method for sampling diffusion bridges on sub-Riemannian manifolds.
problem Sampling conditioned diffusion processes on sub-Riemannian manifolds is challenging.
method Score matching for machine learning, adapted to non-holonomic frames.
result Demonstrated method works on Heisenberg group and other sub-Riemannian manifolds.
A geometric reformulation of the martingale problem associated with a set of diffusion processes is proposed. This formulation, based on second order geometry and Ito integration on manifolds, allows us to give a natural and effective definition of Lie symmetries for diffusion processes.
A new method uses string method to explore diffusion models.
problem Understanding the geometry of learned distributions in diffusion models.
method String method to compute continuous paths between samples.
result The string method identifies realistic morphing sequences and transition pathways.
New method synthesizes data on curved spaces for better interpolation.
problem Synthesizing data on curved spaces for better interpolation.
method Riemannian Diffusion Schrödinger Bridge
result Generalizes Diffusion Schrödinger Bridge to curved spaces for better interpolation.
A new image completion method inspired by brain cells.
problem Image restoration from corrupted data.
method Biologically-inspired sub-Riemannian model with frequency and phase.
result Completion of two-dimensional images using cortical cell responses.
Geodesic walks converge to Brownian motion on Finsler manifolds.
problem Understanding random walks on Finsler manifolds.
method Analyzing convergence of geodesic random walks to diffusion processes.
result The Brownian motion on a Riemannian metric is a key result.
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…
GCDM generates valid large 3D molecules and optimizes existing molecules.
problem Lack of geometric properties in 3D molecule generation models.
method Introduces Geometry-Complete Diffusion Model (GCDM) using equivariant GNNs.
result Significantly outperforms existing models in 3D molecule generation and optimization.
Develops a smooth operator framework for analyzing neural network representations.
problem Analyzing the geometry of feedforward neural network representations.
method Introduces a smooth operator-theoretic approach based on diffusion Markov operators derived from feature clouds.
result Establishes a stable operator-geometric framework for tracking training, width, and perturbation stability.
Diffusion models adapt to data geometry through log-domain smoothing.
problem Understanding why diffusion models generalize well across diverse domains.
method Investigating the role of score matching and log-domain smoothing in diffusion models.
result Log-domain smoothing adapts the diffusion model to the data manifold.
Study on reliability of latent reuse in diffusion models under distribution shift.
problem When can latent spaces from a source dataset be reused for a target dataset with different distributions?
method Considered a source-target setting with approximately low-dimensional datasets near different subspaces. Analyzed the target-domain score error due to principal-angle misalignment and target ambient noise.
result Latent reuse is reliable only if the source and target subspaces are close and the target ambient noise is not too amplified.
Manifold learning techniques for dynamical systems and time series have shown their utility for a broad spectrum of applications in recent years. While these methods are effective at learning a low-dimensional representation, they are often insufficient for visualizing the global and local structure of the data. In thi…
Paper proposes a new method for supervised manifold learning using random forest proximities.
problem Existing supervised manifold learning methods fail to uncover meaningful embeddings due to using class-conditional distances.
method Proposes a data-geometry-preserving variant of random forest proximities as an initialization for manifold learning methods.
result Local and global structure preservation is near universal across manifold learning approaches using diffusion-based algorithms.
An unsupervised learning algorithm to cluster hyperspectral image (HSI) data is proposed that exploits spatially-regularized random walks. Markov diffusions are defined on the space of HSI spectra with transitions constrained to near spatial neighbors. The explicit incorporation of spatial regularity into the diffusion…
Adaptive sampling improves graph diffusion models by maintaining uniform information speed.
problem Standard diffusion models overlook non-homogeneous dynamics on complex manifolds.
method Information-geometric framework using Fisher-Rao metric and Drift Variation Score (DVS).
result DVS solver ensures uniform rate of distributional change, improving structural fidelity and efficiency.
New method uses weighted SDEs to improve sampling from complex distributions.
problem Sampling from highly non-log-concave distributions.
method Introduces weighted stochastic differential equations to augment diffusion-based samplers.
result Demonstrates improved exploration of nonconvex or multimodal landscapes.
We consider the problem of constructing diffusion operators high dimensional data to address counterfactual functions , such as individualized treatment effectiveness. We propose and construct a new diffusion metric that captures both the local geometry of and the directions of variance of . The res…
Interpolates mean shift and spectral clustering on graphs.
problem Data clustering algorithms.
method Fokker-Planck equations on data graphs.
result New theoretical insights on diffusion maps and mean shift dynamics.
Quantum field theory connects Riemannian geometry to quantum fluctuations.
problem Generating Riemannian structures from quantum fluctuations.
method QFT approach to Riemannian Geometry, focusing on Ricci curvature.
result Ricci curvature is crucial in generating Riemannian structures.
Establishes a link between heat diffusion and manifold distances in data.
problem No theoretical link between diffusion-based manifold learning and geodesic distances.
method Formulates heat geodesic embeddings based on Riemannian geometry.
result Method outperforms state-of-the-art in preserving manifold distances and cluster structure.
This article proposes an active learning method for high dimensional data, based on intrinsic data geometries learned through diffusion processes on graphs. Diffusion distances are used to parametrize low-dimensional structures on the dataset, which allow for high-accuracy labelings of the dataset with only a small num…
Riemannian metric matching learns the geometry of high-dimensional datasets using neural networks.
problem Estimating the geometry of high-dimensional datasets from samples
method Riemannian metric matching using neural networks
result Riemannian metric matching rivals or improves -NN-based diffusion geometry estimators
We use tools from -dimensional Brownian motion in conjunction with the Feynman-Kac formulation of heat diffusion to study nodal geometry on a compact Riemannian manifold . On one hand we extend a theorem of Lieb and prove that any nodal domain almost fully contains a ball of radius . …
Harmonic Alignmentcs.LG
We propose a novel framework for combining datasets via alignment of their intrinsic geometry. This alignment can be used to fuse data originating from disparate modalities, or to correct batch effects while preserving intrinsic data structure. Importantly, we do not assume any pointwise correspondence between datasets…
Study quantum diffusion on spectral triples and spinor bundles.
problem Characterize quantum diffusion on almost commutative spectral triples.
method Spin geometry, C *-Dirichlet forms, quantum stochastic flows.
result Existence of covariant quantum stochastic flows on spinor bundles.
Developed a diffusion model on spherical data, addressing geometric and stochastic challenges.
problem Diffusion models on spherical data face unique geometric and stochastic issues.
method Extended spectral diffusion to spherical harmonics, introducing modified stochastic differential equations.
result Introduced a geometry-dependent inductive bias in spectral diffusion models.
Generalized Robust Adaptive-Bandwidth Multi-View Manifold Learning in High Dimensions with Noisestat.ML
Proposes GRAB-MDM for robust multiview data fusion.
problem Limited theoretical guarantees for multiview fusion methods in noisy high-dimensional data.
method Generalized Robust Adaptive-Bandwidth Multiview Diffusion Maps (GRAB-MDM) with adaptive bandwidth selection.
result Adaptive bandwidths lead to robust recovery of shared intrinsic structure in noisy multiview data.
New algorithm clusters hyperspectral images at multiple scales.
problem Clustering hyperspectral images at various scales.
method M-SRDL algorithm using spectral-spatial diffusion distances.
result More accurate clustering labels achieved with spatial regularization.
Diffusion models can generalize well even with coarse scores, thanks to the manifold hypothesis.
problem Understanding why diffusion models generate novel samples with coarse scores.
method Exploring the manifold hypothesis to explain diffusion model behavior.
result Diffusion models trained with coarse scores can achieve near-parametric rates of generalization, faster than estimating the full data distribution.