Differentiates Fréchet mean for hyperbolic space applications.
problem Difficulty in applying Fréchet mean due to lack of closed-form derivative.
method Developed differentiation method and explicit gradient expressions for hyperbolic space.
result Fully integrated Fréchet mean into hyperbolic neural network pipeline.
GEORCE-FM algorithm optimizes Fréchet means and distances efficiently.
problem Computing Fréchet means on Riemannian manifolds efficiently.
method GEORCE-FM algorithm that simultaneously computes Fréchet means and distances in local charts.
result GEORCE-FM algorithm converges globally and locally quadratically, and scales to large datasets.
Study shows zero probability of cut locus for Fréchet mean on Riemannian manifolds.
problem Understanding the cut locus of Fréchet mean on Riemannian manifolds.
method Analytical proof and examples.
result Cut locus of Fréchet mean has zero probability.
Estimates means in metric spaces using quantization.
problem No practical estimator for Fréchet means in all metric spaces.
method Introduced estimators based on random quantization and data-driven partitioning.
result Universal consistency of estimators across separable metric spaces and Banach spaces.
Sharp threshold found for Frechet mean of inhomogeneous graphs.
problem Finding the Frechet mean of inhomogeneous Erdos-Renyi random graphs.
method Thresholding the expected adjacency matrix of the ensemble.
result The Frechet mean graph of inhomogeneous Erdos-Renyi random graphs exhibits a sharp threshold.
New graph properties inherited by Frechet mean and median.
problem Characterizing the average of graph-valued samples.
method Analysis of Frechet mean and median graphs.
result Edge density is hereditary in Frechet mean and median graphs.
Improved method for computing Fréchet means on SPD matrices.
problem Computing Fréchet means on the manifold of SPD matrices.
method Random matrix theory-based approach for estimating Fréchet means.
result Significantly outperforms state-of-the-art methods in experiments.
New method uses CNNs to estimate graph means.
problem Estimating the mean of graph-valued data.
method Convolutional Neural Networks (CNNs) for graph morphology learning.
result CNNs reliably recover the sample Frechet mean.
We compute an approximate Fréchet mean for sets of sparse graphs.
problem Characterizing the location of a set of graphs in a metric space.
method We use the pseudometric defined by the ℓ₂ norm of eigenvalues of adjacency matrices.
result We describe an algorithm to approximate the Fréchet mean of a set of graphs.
A new mechanism for differentially private Fréchet mean on SPD matrices.
problem Privacy-preserving statistical summaries for SPD matrices.
method Tangent Gaussian mechanism for log-Euclidean metric.
result Significantly better utility and computational efficiency.
The paper computes an approximation to the sample Frechet mean of graph sets using spectral information.
problem Characterizing the location of a set of graphs in a metric space.
method The Frechet mean is computed for sets of large graphs using the pseudometric defined by the norm between eigenvalues of adjacency matrices.
result An algorithm to approximate the sample Frechet mean of undirected unweighted graphs is described.
Study on the smoothness of solutions to a specific type of stochastic differential equation.
problem Regularity of solutions to mean-field G G G -SDEs. method Analysis of first and second order Fréchet differentiability in the random initial condition.
result Established the Fréchet differentiability of the solution and specified the corresponding equations.
Paper introduces a medoid-based approach for efficient Fréchet regression.
problem Regression in metric spaces with random objects.
method Adapted random forest algorithm with medoid-based splitting rule.
result Asymptotic equivalence and consistency of the regression estimator.
Decentralized optimization on dynamic manifolds with improved regret bound.
problem Optimizing on nonstationary Riemannian manifolds in decentralized systems.
method Decentralized projected Riemannian gradient descent with weighted Frechet mean consensus.
result Achieved dynamic regret bound of O ( T ( 1 + P T ) / ( 1 − σ 2 ( W ) ) ) {\cal O}(\sqrt{T(1+P_T)}/\sqrt{(1-σ_2(W))}) O ( T ( 1 + P T ) / ( 1 − σ 2 ( W )) ) . Framework detects shape shifts in functional profiles using Fréchet mean and shape invariant model.
problem Detecting shape shifts in functional profiles.
method Combining Fréchet mean and shape invariant model for interpretable parameterization of profile deviations.
result Potential shifts in shape deformation process distinguished by significant shifts in amplitude and/or phase.
The paper develops predictors for functional data on manifolds.
problem Functional data prediction on time-varying manifolds.
method Least-squares local linear Fréchet curve predictor and weighted Fréchet mean approach.
result Asymptotical optimality of the proposed predictors.
EpiMer merges models by solving Fréchet mean on a Riemannian manifold.
problem Integrating knowledge from multiple models without retraining.
method EpiMer casts model merging as solving the Fréchet mean on a Riemannian manifold, restricting computation to a low-rank subspace.
result EpiMer outperforms flat-geometry methods on image classification tasks.
Study shows effective resistance distance yields more accurate network barycenter than Hamming distance.
problem Identifying the best metric for computing the Fréchet mean network.
method Compared the effectiveness of Hamming distance and effective resistance distance in capturing network topology.
result Effective resistance distance produces a more accurate Fréchet mean network.
We obtain a Central Limit Theorem for closed Riemannian manifolds, clarifying along the way the geometric meaning of some of the hypotheses in Bhattacharya and Lin's Omnibus Central Limit Theorem for Fréchet means. We obtain our CLT assuming certain stability hypothesis for the cut locus, which always holds when the ma…
A new method for network regression using optimal transport.
problem How network topology changes with Euclidean covariates.
method Optimal transport approach based on Wasserstein metric.
result The method improves prediction accuracy in real-world data.
This work analyzes Fréchet regression using comparison geometry, providing theoretical and practical insights.
problem Analyzing data on complex structures like manifolds and graphs.
method Theoretical analysis through comparison geometry, focusing on existence, uniqueness, and stability of the Fréchet mean.
result Key results on the existence, uniqueness, and stability of the Fréchet mean, along with statistical guarantees for nonparametric regression.
DFNNs predict non-Euclidean responses from Euclidean predictors.
problem Regression with non-Euclidean responses.
method Deep Fréchet neural networks (DFNNs) approximating conditional Fréchet means.
result DFNNs consistently outperform existing methods in empirical studies.
DFR models dynamic distributional data with weighted Fréchet means.
problem Regression of distribution-valued responses over time.
method Dynamic Fréchet Regression (DFR) with index-aware weighting and feature selection.
result Improved predictive accuracy and feature recovery over existing methods.
Deep single-index Fréchet regression for metric space-valued outputs
problem Predicting outputs in non-Euclidean spaces
method DeSI (Deep Single-Index Fréchet Regression)
result Interpretable index direction for inputs
Efficiently clusters data on manifolds using Fréchet maps.
problem Clustering on high-dimensional, non-Euclidean manifolds is computationally challenging.
method Introduces p p p -Fréchet map to embed manifold data into Euclidean space for k-means clustering. result Significant performance gains in runtime and accuracy compared to existing methods.
This paper explores the impact of metric choice on Fréchet regression.
problem Choosing the right metric for Fréchet regression in complex data.
method Review and extensive numerical studies of existing dimension reduction methods.
result Different metrics significantly affect the estimation of central and central mean space.
Proposes a new random forest weighted local Fréchet regression method.
problem Complex metric space valued responses and curse of dimensionality in Fréchet regression.
method Locally adaptive kernel generated by random forests for local average and local linear Fréchet regression.
result Significantly improves existing Fréchet regression methods with theoretical guarantees.
Develop a framework for barycentric projections of optimal transport plans on Riemannian manifolds.
problem Optimal transport couplings are probabilistic objects, while many learning pipelines require deterministic maps.
method Develop a framework for barycentric projections of transport couplings on Riemannian manifolds.
result The intrinsic projection maps each source point to the conditional Fréchet mean of its destination law and is shown to be the best deterministic representative under squared geodesic loss.
Paper optimizes FTPL for adversarial and stochastic bandits with specific tail distributions.
problem Optimizing Follow-the-Perturbed-Leader (FTPL) policy for bandit problems.
method Analyzes FTPL with Fréchet-type tail distributions in adversarial and stochastic settings.
result FTPL with certain Fréchet-type tail distributions achieves O ( K T ) \mathcal{O}(\sqrt{KT}) O ( K T ) regrets in adversarial bandits. This paper is a short summary of our recent work on the medians and means of probability measures in Riemannian manifolds. Firstly, the existence and uniqueness results of local medians are given. In order to compute medians in practical cases, we propose a subgradient algorithm and prove its convergence. After that, F…
A new method for averaging data on manifolds is proposed, offering simplicity and efficiency.
problem The difficulty of computing Fréchet means on manifolds, especially Stiefel and Grassmann.
method Proposed RL-barycenters, simpler arithmetic means projected onto the manifold.
result RL-barycenters yield simple yet effective means on Stiefel and Grassmann manifolds.
We define Lie and Courant algebroids on Fréchet manifolds. Moreover, we construct a Dirac structure on the generalized tangent bundle of a Fréchet manifold and show that it inherits a Fréchet Lie algebroid structure. We show that the Lie algebroid cohomology of the $\bb$ -cotangent bundle Lie algebroid of a weakly sympl…
We introduce the new class of submanifolds of co-Banach type in tame Fréchet manifolds and construct tame Fréchet submanifolds as inverse images of regular values of certain tame maps. Our method furnishes an easy way to construct tame Fréchet manifolds. The results presented are key ingredients in the construction of …
Paper develops statistical tests for covariance matrix regression on manifold.
problem Regression with random covariance matrices in Fréchet space.
method Develops Wasserstein F-tests for Bures-Wasserstein manifold.
result Asymptotic null distribution and power of the test.
Insurance benefits risk sharing for finite mean risks but not for infinite mean risks.
problem The effect of risk sharing and diversification for infinite mean risks.
method Investigation of risk sharing and diversification for infinite mean models, including stable, Pareto, and Fréchet distributions.
result Risk sharing can have a negative effect for infinite mean models, a phenomenon known as the nondiversification trap.
Some recent work in Frechet geometry is briefly reviewed. In particular an earlier result on the structure of second tangent bundles in the finite dimensional case was extended to infinite dimensional Banach manifolds and Frechet manifolds that could be represented as projective limits of Banach manifolds. This led to …
Paper develops methods for semi-supervised Fréchet regression.
problem High costs of obtaining non-Euclidean labels.
method Proposes semi-supervised NW Fréchet regression and semi-supervised kNN Fréchet regression.
result Demonstrates superior performance over supervised methods.
Novel convex risk measures aggregate multiple uncertain sources for insurance firms.
problem Managing risk from multiple uncertain sources in insurance.
method Proposes convex risk measures based on Fréchet mean.
result Allows for robust risk characterization and closed-form expressions.
A new metric compares dynamical systems using operator eigenvalues.
problem Comparing and interpolating nonlinear dynamical systems from trajectory data.
method Representing systems as distributions of operator eigenvalues and projectors, defining a spectral-Grassmann Wasserstein metric.
result The proposed metric outperforms standard operator-based distances in machine learning applications.
A new model for curves on manifolds using rolling operations.
problem Modeling curves on manifolds without explicit parametrization.
method Using rolling operations to construct Gaussian processes on manifolds.
result Conditions for the rolling of mean to equal Fréchet mean and estimators of parameters.
Extends differential privacy to Riemannian manifolds, improving utility.
problem Releasing private statistical summaries on Riemannian manifolds.
method Extended Laplace or K-norm mechanism using intrinsic distances and volumes.
result Demonstrates rate optimality and utility improvement over ambient spaces.
The paper proves critical point results for Frechet manifolds.
problem Finding critical points in the context of Frechet manifolds.
method Using a deformation result and sufficient conditions for the Palais-Smale condition.
result Proves a mountain pass theorem and three critical points theorem.
Study shows continuous evolution of curves in Fréchet distance.
problem Continuous evolution of curves under curvature flow.
method Curvature flow and level-set flow, analyzed in Fréchet distance.
result Evolution of curves depends continuously on initial curve.
Paper studies weighted Fermat-Frechet problem for simplex edge lengths.
problem Finding optimal edge lengths for simplex deformations.
method Isometric embedding techniques for K K K -Space. result New variational method to solve weighted Fermat-Frechet problem.
The consistency of Fréchet medians is proved for probability measures in proper metric spaces. In the context of Riemannian manifolds, assuming that the probability measure has more than a half mass lying in a convex ball and verifies some concentration conditions, the positions of its Fréchet medians are estimated. It…
The study finds the minimum number of critical points for functionals on Frechet spaces and Finsler manifolds.
problem Finding the minimum number of critical points for functionals on Frechet spaces and Finsler manifolds.
method Applying the Lusternik-Schnirelmann category to evaluate the minimal number of critical points for Keller C c 1 C_c^1 C c 1 -functionals on Frechet spaces and Finsler manifolds. result The minimal number of critical points is determined by the Lusternik-Schnirelmann category.
In this paper we develop the geometry of bounded Fréchet manifolds. We prove that a bounded Fréchet tangent bundle admits a vector bundle structure. But the second order tangent bundle T 2 M T^2M T 2 M of a bounded Fréchet manifold M M M , becomes a vector bundle over M M M if and only if M M M is endowed with a linear connection. As a…
Paper develops novel privacy mechanism for Riemannian manifold data using geometric analysis and heat diffusion.
problem Privacy-preserving estimation of generalized Frechet mean on Riemannian manifolds.
method Characterizes Renyi divergence via Harnack inequalities, introduces mechanisms based on heat diffusion and Langevin process.
result Proposes mechanisms for nonnegative and general Riemannian manifolds with detailed utility analyses.