Interaction of Statistics and Geometry
Theory and Methodology
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Principal Flows
Curves on the manifold serve as an analogous of principal components.
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Manifold Fitting
Fit smooth low-dim manifolds from high-dim noisy data points.
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Principal Submanifolds
Multi-dimensional generalization of principal flows.
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Fixed Boundary Flows
Conditional extension of principal flows with fixed boundary points.
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Principal Boundary
Principal decision boundary on Riemannian manifolds.
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Principal Nested Submanifolds
A multi-scale decomposition of data via nested submanifolds.
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Differentially Private Manifold Denoising
Privacy-preserving manifold denoising for noisy query points.
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Curvature-driven Manifold Fitting
Curvature-aware recovery of low-dimensional manifolds under isotropic noise.
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Estimating Riemannian Quantities
Recover geometric quantities from noisy data via density derivatives.
Applications
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MFCGAN
Combination of manifold fitting with CycleGAN.
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scAMF
Single-Cell Analysis via Manifold Fitting.
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CellScope
Cellular analyzing tool via Manifold Fitting.
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Metabolic Biomarkers
Using manifold fitting to analysis metabolomic heterogeneity and disease risks in UK Biobank.
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Pancreatic Cancer
Using manifold fitting to analyze pancreatic cancer risk with UK Biobank.
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Missing Data
Analyze the effect of missing data in UK Biobank.
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RNA Clash Correction
RNA clustering and clash correction via Principal Submanifold.
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Multiple Myeloma
Biomarker-guided preclinical risk stratification for myeloma in UK Biobank.
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Dementia
Multidimensional physical fitness and dementia risk in UK Biobank.