HilbNets define convolutions via Hilbert bundle connection Laplacians, prove that sampled Hilbert cellular sheaf Laplacians converge to the continuous operator, and show that discretized networks are consistent and transferable across samplings.
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Coifman and Stéphane Lafon
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representative citing papers
VeloTree infers differentiation trees from RNA velocity fields by defining cell dissimilarity as the squared varifold distance between integral curves of the velocity field.
EntroPath defines a free-energy dissimilarity from maximum-entropy random walk path ensembles and proves it converges to squared geodesic distance in the short-time limit via Varadhan's formula.
Introduces finite-lag operator geometry deriving a source-centered transport tensor that decomposes into spread and coherent displacement plus an antisymmetric circulation measure, with proofs of covariance and stability.
GaRA generates task-specific LoRA weight updates conditioned on graph structures to enable better whole-graph encoding in LLMs for zero-shot graph learning.
Neural point-forms are introduced as permutation-invariant neural layers that output learned form-comparison matrices for point clouds, with a claimed consistency proof under sampling and manifold assumptions and competitive results on synthetic and biological data.
NOFE is a neural operator method for continuous dimensionality reduction using Graph Kernel Operators that outperforms PCA, t-SNE and UMAP on local structure preservation and sampling independence in datasets including ERA5 climate reanalysis.
Decoding alignment metrics can remain high and unchanged even when encoding manifold topology is causally altered, so they do not imply similar function or computation across neural populations.
A standardized pipeline converts time series to graphs, computes persistence diagrams, and extracts features that classify UCR benchmarks, with diffusion distance outperforming shortest-path metrics and performance varying by graph type.
Logarithmic centroid method recovers adiabatic Kramers scaling in coherence resonance for SRK model with Feller noise despite bathtub effect, identifies strong-noise breakdown, and demonstrates noise-induced transition to functional synchronization in gap-junction coupled systems.
A spectral framework for nonlinear DR uses spectral bases plus cross-entropy optimization to create multi-scale embeddings that preserve both global manifold geometry and local neighborhoods while supporting graph-frequency analysis.
A data-driven framework reduces particle-based transfer operators via concentration projection, geometric manifold, and finite-state discretization to reproduce clustering transitions and metastable states from simulation data.
Short histories of observations can recover the underlying manifold for transporting discontinuous densities when direct source-target pairs are insufficient due to folds or marginalization.
GTSA-PCA replaces global PCA covariance with curvature-weighted local operators and a geodesic alignment step to produce geometry-aware embeddings that improve on standard PCA and UMAP in small-sample high-curvature settings.
Applying the dynamic centrality index C_CTMC to river network trees on 49 US basins shows top-ranked headwaters reach disproportionately many downstream junctions across transport times.
The MPEX AI Digital Twins project reports that its two phase-I AI milestones for hot-spot control and damage assessment are on track for June 2026 demonstration.
citing papers explorer
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Consistent Geometric Deep Learning via Hilbert Bundles and Cellular Sheaves
HilbNets define convolutions via Hilbert bundle connection Laplacians, prove that sampled Hilbert cellular sheaf Laplacians converge to the continuous operator, and show that discretized networks are consistent and transferable across samplings.
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VeloTree: Inferring single-cell trajectories from RNA velocity fields with varifold distances
VeloTree infers differentiation trees from RNA velocity fields by defining cell dissimilarity as the squared varifold distance between integral curves of the velocity field.
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EntroPath: Maximum Entropy Path Ensemble Embedding for Manifold Learning
EntroPath defines a free-energy dissimilarity from maximum-entropy random walk path ensembles and proves it converges to squared geodesic distance in the short-time limit via Varadhan's formula.
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Finite-Lag Operator Geometry of Recurrent Representations
Introduces finite-lag operator geometry deriving a source-centered transport tensor that decomposes into spread and coherent displacement plus an antisymmetric circulation measure, with proofs of covariance and stability.
-
Enhancing LLMs for Graph Tasks via Graph-aware LoRA Generation
GaRA generates task-specific LoRA weight updates conditioned on graph structures to enable better whole-graph encoding in LLMs for zero-shot graph learning.
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Neural Point-Forms
Neural point-forms are introduced as permutation-invariant neural layers that output learned form-comparison matrices for point clouds, with a claimed consistency proof under sampling and manifold assumptions and competitive results on synthetic and biological data.
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NOFE - Neural Operator Function Embedding
NOFE is a neural operator method for continuous dimensionality reduction using Graph Kernel Operators that outperforms PCA, t-SNE and UMAP on local structure preservation and sampling independence in datasets including ERA5 climate reanalysis.
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Decoding Alignment without Encoding Alignment: A critique of similarity analysis in neuroscience
Decoding alignment metrics can remain high and unchanged even when encoding manifold topology is causally altered, so they do not imply similar function or computation across neural populations.
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Persistent Homology of Time Series through Complex Networks
A standardized pipeline converts time series to graphs, computes persistence diagrams, and extracts features that classify UCR benchmarks, with diffusion distance outperforming shortest-path metrics and performance varying by graph type.
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Breakdown of Adiabatic Scaling and Noise-Induced Functional Synchronization in Deeply Quiescent Excitable Systems
Logarithmic centroid method recovers adiabatic Kramers scaling in coherence resonance for SRK model with Feller noise despite bathtub effect, identifies strong-noise breakdown, and demonstrates noise-induced transition to functional synchronization in gap-junction coupled systems.
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A Spectral Framework for Multi-Scale Nonlinear Dimensionality Reduction
A spectral framework for nonlinear DR uses spectral bases plus cross-entropy optimization to create multi-scale embeddings that preserve both global manifold geometry and local neighborhoods while supporting graph-frequency analysis.
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Data-driven Reduction of Transfer Operators for Particle Clustering Dynamics
A data-driven framework reduces particle-based transfer operators via concentration projection, geometric manifold, and finite-state discretization to reproduce clustering transitions and metastable states from simulation data.
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A geometric approach to the transport of discontinuous densities
Short histories of observations can recover the underlying manifold for transporting discontinuous densities when direct source-target pairs are insufficient due to folds or marginalization.
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Curvature-Aware PCA with Geodesic Tangent Space Aggregation for Semi-Supervised Learning
GTSA-PCA replaces global PCA covariance with curvature-weighted local operators and a geodesic alignment step to produce geometry-aware embeddings that improve on standard PCA and UMAP in small-sample high-curvature settings.
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Dynamic centrality of headwater sources in river networks: a stochastic approach via ultrametric Laplacians
Applying the dynamic centrality index C_CTMC to river network trees on 49 US basins shows top-ranked headwaters reach disproportionately many downstream junctions across transport times.
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MPEX AI Digital Twins Milestone Report
The MPEX AI Digital Twins project reports that its two phase-I AI milestones for hot-spot control and damage assessment are on track for June 2026 demonstration.
- Diffusion Operator Geometry of Feedforward Representations