Proposes CC-NMDF for analyzing manifold-valued data.
problem Nonlinear structure in manifold-valued data requires new analysis methods.
method Curvature-corrected nonnegative manifold data factorization (CC-NMDF) with an iterative algorithm.
result Demonstrates CC-NMDF on real-world diffusion tensor MRI data.
New method reconstructs manifolds from data using Gaussian processes.
problem Reconstructing lower-dimensional structure from complex data.
method Local covariance matrices and Gaussian processes for probabilistic manifold reconstruction.
result Probabilistic manifold reconstruction with Gaussian processes.
Proposes generating virtual data points to overcome the curse of dimensionality.
problem Increased intrinsic dimensionality requires large data sets for local sampling.
method Manifold embedding motivated super sampling (MESS) framework.
result Generates virtual data points that faithfully represent the manifold.
New method for manifold topological learning avoids remeshing issues.
problem Persistent homology on manifolds is numerically inconsistent.
method Persistent de Rham-Hodge Laplacians in Eulerian representation.
result Avoids numerical inconsistency over multiscale manifolds.
The paper refutes the manifold hypothesis for image data and proposes the union of manifolds hypothesis.
problem The manifold hypothesis fails to capture the structure of image data.
method Empirical verification of the union of manifolds hypothesis on image datasets.
result Image data lies on a disconnected set with varying intrinsic dimensions.
Proposes a method to cluster multi-aspect data using manifold learning with NMF.
problem Clustering multi-aspect data with diverse features and views.
method Includes inter-manifold learning in NMF framework to handle different data types.
result The method improves clustering accuracy and efficiency on various datasets.
PFM generates novel samples on data manifolds using pullback geometry.
problem Generating novel samples on complex data manifolds.
method Pullback Flow Matching framework leveraging pullback geometry and isometric learning.
result PFM achieves improved manifold learning and generative performance.
Proposes a normalization technique for manifold valued data.
problem Instability in optimization for manifold valued data.
method Develops a general normalization technique for manifold valued data.
result Demonstrates performance gain in synthetic and real datasets.
Paper proposes methods to learn sub-manifolds and estimate densities in normalizing flows.
problem Normalizing flows struggle with finding sub-manifolds in high-dimensional data.
method Introduces per-pixel penalized log-likelihood and hierarchical training approaches.
result Validated superior performance in manifold learning and density estimation.
Study reveals how manifold geometry impacts linear regression solutions.
problem Impact of manifold geometry on linear regression solutions.
method Linear regression applied to manifold-structured data, focusing on extrinsic geometry.
result Linear regression does not have a unique solution on flat manifolds.
New algorithm tackles regression on manifold data using diffusion and semi-supervised learning.
problem Regression on high-dimensional manifold data with complex structures.
method Diffusion-based spectral algorithm using graph Laplacian and heat kernel.
result Algorithm achieves convergence rate dependent on intrinsic manifold dimension, avoiding curse of dimensionality.
New methods implement manifold scattering transform for high-dimensional point cloud data.
problem Classifying data on complex, non-linear manifolds.
method Adapting diffusion maps theory for numerical implementation.
result Effective for signal and manifold classification tasks.
Quantum dynamics algorithm learns manifold from data.
problem Learning manifolds from high-dimensional datasets.
method Simulation of quantum dynamics on a graph embedding of data.
result Algorithm reveals connections between data sampling and quantization.
A diffusion model estimates data manifold dimension by tracking likelihood increases.
problem Estimating the intrinsic dimension of data manifolds.
method Trained diffusion model approximates score function, revealing manifold directionality.
result Diffusion model provides an approximation of the tangent space's dimension.
Given a smooth non-trapping compact manifold with strictly con- vex boundary, we consider an inverse problem of reconstructing the manifold from the scattering data initiated from internal sources. This data consist of the exit directions of geodesics that are emaneted from interior points of the manifold. We show that…
Geometric framework detects outliers in high-dimensional data.
problem Detecting outliers in high-dimensional data.
method Geometric framework exploiting manifold structure.
result Significant improvement in outlier detection in high-dimensional data.
Proposes a scalable framework for extracting data manifold geometry.
problem Efficiently mapping and learning data manifold geometry.
method Score-based pullback Riemannian geometry integrating pullback Riemannian geometry and generative models.
result High-quality geodesics and reliable intrinsic dimension estimation.
M-flows learn data manifolds and densities, improving manifold learning and inference.
problem Representing datasets with manifold structure more faithfully.
method Combining normalizing flows, GANs, autoencoders, and energy-based models, with a new training algorithm.
result M-flows learn data manifolds better than standard flows and provide handles for dimensionality reduction.
New method classifies manifold-valued data using Riemannian geometry.
problem Classifying data on curved Riemannian manifolds.
method Probabilistic Learning Vector Quantization on Symmetric Positive Definite Matrices.
result The method outperforms traditional Euclidean methods on manifold-valued data.
Deep Gaussian processes on manifolds improve performance on complex data.
problem Complex data on manifolds that shallow models struggle with.
method Residual deep Gaussian processes on Riemannian manifolds.
result Significant improvement in prediction quality and uncertainty calibration.
DMT enhances deep neural networks to better preserve data structures.
problem Preserving geometric, topological, and distributional structures of data in NLDR.
method Deep manifold transformation (DMT) using cross-layer LGP constraints.
result DMT networks outperform existing NLDR methods in preserving data structures.
Recovering manifold geometry from geodesic intersections.
problem Recovering the geometry of a Riemannian manifold from geodesic intersection lengths.
method Applying stitching data to solve the delayed collision data problem.
result Geometry of the manifold can be recovered from geodesic intersection lengths.
Framework learns data manifold and generative model from corrupted data.
problem Learning from corrupted data with latent manifold structures.
method Riemannian AmbientFlow, incorporating normalizing flows and Riemannian Autoencoders.
result Framework recovers underlying data distribution and smooth manifold parametrization.
When analyzing empirical data, we often find that global linear models overestimate the number of parameters required. In such cases, we may ask whether the data lies on or near a manifold or a set of manifolds (a so-called multi-manifold) of lower dimension than the ambient space. This question can be phrased as a (mu…
Optimizes data-driven design problems on implicit manifolds using score functions.
problem Optimizing over implicit low-dimensional manifolds in high-dimensional data.
method Introduces a link function connecting data distribution to manifold operations, enabling efficient optimization.
result Establishes theoretical guarantees for feasibility and optimality of proposed algorithms.
Proposes a new flow model to better represent data on manifolds.
problem Flow models struggle to represent data on lower-dimensional manifolds accurately.
method Introduces a manifold prior that leverages spread divergence to improve model performance.
result Improves both sample and representation quality, identifies manifold intrinsic dimension.
Normal-bundle bootstrap generates new data preserving geometric structure.
problem Probabilistic models often exhibit salient geometric structure.
method NBB method decomposes probability measure into manifold and normal spaces, estimates manifold as density ridge, and generates new data by bootstrapping projection vectors.
result NBB generates new data that preserves the geometric structure of a given data set.
A new metric-based principal curve method learns 1D manifolds from spatial data.
problem Learning 1D manifolds from spatial data.
method Metric-based Principal Curve (MPC) approach.
result The method effectively learns the shape of 1D manifolds from synthetic and real datasets.
In this paper, we consider the variational regularization of manifold-valued data in the inverse problems setting. In particular, we consider TV and TGV regularization for manifold-valued data with indirect measurement operators. We provide results on the well-posedness and present algorithms for a numerical realizatio…
Neural Manifold ODEs improve manifold data modeling.
problem Adapting deep generative models to non-Euclidean spaces.
method Introducing Neural Manifold ODEs for manifold generalization and continuous probability computation.
result Improves density estimation and downstream tasks on arbitrary manifolds.
R-PCA extends PCA to Riemannian manifolds for structured data.
problem Applying PCA to data on Riemannian manifolds without vector space operations.
method Adapting PCA to Riemannian manifolds by equipping data with local metrics.
result Unified approach for dimensionality reduction and statistical analysis on manifolds.
Kernel test detects manifold data differences with high-dimensional noise.
problem Detecting differences between manifold data samples.
method Kernel-based two-sample test statistic related to MMD for manifold data.
result The test power exceeds a threshold depending on manifold dimensionality, Hölder order, and squared divergence.
VAELLS learns latent manifold structure to improve VAE model accuracy.
problem VAEs struggle with mismatched latent structure and global structure.
method Integrates learnable manifold model into latent space of VAE.
result Improves model accuracy by matching prior to data manifold structure.
PGPCA improves PCA for nonlinear data in neuroscience.
problem Nonlinear data distribution in neuroscience.
method Developed PGPCA for nonlinear manifolds, incorporating EM algorithm.
result PGPCA outperforms PPCA in modeling data around nonlinear manifolds.
New approach combines geometric and probabilistic methods to estimate manifold dimension in high-dimensional data.
problem Estimating the dimension of manifolds in high-dimensional data.
method Combines a modified box-counting algorithm (geometric) and a new probabilistic method (nearest neighbor distance analysis).
result The combined method is robust, fast, and effective in estimating manifold dimension.
This research develops prediction sets for regression on manifolds using conformal inference.
problem Prediction sets for regression on manifolds, especially in non-Euclidean spaces.
method Conformal inference principles extended to manifolds, proving asymptotic almost sure convergence.
result Empirical prediction sets on manifolds converge to population counterparts.
Paper investigates hardness of learning neural networks under manifold hypothesis.
problem Hardness of learning neural networks under the manifold hypothesis.
method Extending proofs of hardness in the SQ and cryptographic settings to the geometric setting.
result Learning is hard under input manifolds of bounded curvature but learnable with additional assumptions on manifold volume.
Sparse coding has been popularly used as an effective data representation method in various applications, such as computer vision, medical imaging and bioinformatics, etc. However, the conventional sparse coding algorithms and its manifold regularized variants (graph sparse coding and Laplacian sparse coding), learn th…
Manifold learning based methods have been widely used for non-linear dimensionality reduction (NLDR). However, in many practical settings, the need to process streaming data is a challenge for such methods, owing to the high computational complexity involved. Moreover, most methods operate under the assumption that the…
Develops intrinsic Gaussian process regression for manifold-valued data.
problem Lack of intrinsic Gaussian process methods for manifold-valued response variables.
method Proposes an intrinsic covariance structure and a novel intrinsic Gaussian process regression model.
result Establishes asymptotic properties and shows posterior consistency.
Study on hyperbolic manifolds and their boundary data, focusing on volume functions.
problem Determining the hyperbolic metric from boundary data of convex co-compact hyperbolic manifolds.
method Analysis of volume functions and their relation to boundary data, using first variations.
result New connections with physics and probability theory, with open questions remaining.
A new Gaussian process regression method infers implicit manifold structure from data.
problem Scaling Gaussian process regression to high-dimensional data.
method Proposes a fully differentiable Gaussian process regression technique that infers implicit manifold structure from data.
result Improves predictive performance and calibration of standard Gaussian process regression in high-dimensional settings.
Enhances data augmentation for regression tasks.
problem Limited effectiveness of data augmentation in regression.
method Curvature-Enhanced Manifold Sampling (CEMS).
result CEMS improves performance in regression tasks.
Conformal-DP improves differential privacy on manifold data by calibrating perturbations based on local densities.
problem Lack of density-awareness in existing differential privacy mechanisms for manifold data leads to biased and suboptimal privacy-utility trade-offs.
method Proposes Conformal-DP, a density-aware differential privacy mechanism using conformal transformations to calibrate perturbations based on local densities.
result Demonstrates improved privacy-utility trade-off in heterogeneous data distribution settings compared to state-of-the-art mechanisms.
Improves latent space structure for better data representation.
problem Limited ability of conventional priors to encode data manifold structure.
method Introduces an Encoded Prior Sliced Wasserstein AutoEncoder with iterative training and geodesic interpolation.
result Learned manifold encoding preserves topological and geometric properties of data.
A method for learning distributions on complex manifolds using normalizing flows.
problem Learning distributions on non-Euclidean manifolds with high efficiency and accuracy.
method Learning a distribution on a manifold by combining local models that form an open cover.
result The method achieves better sample efficiency and competitive performance on manifolds of unknown topology.
Develops Shapley explainability solutions respecting data manifold.
problem Tenable assumption of uncorrelated features in Shapley explainability.
method Two solutions: generative modelling and direct learning of Shapley value-function.
result On-manifold Shapley explainability overcomes drawbacks of 'off-manifold' values.
New neural networks flatten and reconstruct manifolds from samples.
problem Learning from high-dimensional data embedded in submanifolds.
method Flattening Networks (FlatNet) that linearize and reconstruct embedded submanifolds.
result FlatNet achieves balance of interpretability, feasibility, and generalization.