The paper introduces methods to analyze community data using diffusion Frechet functions.
problem Analyzing complex ecosystems and their interactions.
method Developed methods for modeling and analyzing the organization of complex data across spatial scales.
result Introduced diffusion Frechet functions and vectors for robustly describing the shapes of probability distributions.
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.
Proves Euler-Lagrange equations for complex functionals on Fréchet manifolds.
problem Variational calculus on Fréchet manifolds.
method Proves Euler-Lagrange equations for functionals on Fréchet manifolds.
result Validates Euler-Lagrange equations for specific functionals.
Global diffeomorphism theorem for Fréchet spaces established.
problem Establishing global diffeomorphisms in Fréchet spaces.
method Extending local diffeomorphisms to global ones, using generalized gradients and Lipschitz functions.
result Global diffeomorphism theorem proved for Fréchet spaces.
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.
A new method learns continuous guidance weights to improve diffusion model quality and distributional alignment.
problem Improving perceptual quality and distributional alignment of samples from conditional diffusion models.
method Learned continuous guidance weights ω c , ( s , t ) ω_{c,(s,t)} ω c , ( s , t ) are used to minimize distributional mismatch and reward guided sampling. result Improvements in Fréchet inception distance (FID) for image generation and better image-prompt alignment in text-to-image applications.
New method improves sampling from score-based models by correcting bias.
problem Bias in sampling from score-based diffusion models.
method Metropolis-Hastings or Barker's accept-reject steps to correct bias, using the score function.
result Improves sample quality on synthetic and image datasets, yielding consistent gains in FID.
Global implicit function theorem for Fréchet spaces, solving derivative loss problems.
problem Solving initial value problems with derivative loss in Fréchet spaces.
method Global implicit function theorems for Keller's C c 1 C_c^1 C c 1 -mappings in Fréchet spaces, applied through submersions and transversality. result Global existence and uniqueness of solutions to initial value problems with derivative loss.
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.
FDBM models use fractional Brownian motion to model complex stochastic processes.
problem Capturing memory effects and long-range dependencies in stochastic processes.
method Developed a generative diffusion bridge framework using a Markovian approximation of fractional Brownian motion.
result FDBM outperforms standard models in predicting future states and unpaired data translation.
The paper improves Fréchet bounds and sharpens their applicability.
problem Uncertainty in joint distribution and marginals in multivariate distributions.
method Optimal transport duality, representation of increasing convex functionals, explicit computation of conjugates.
result Improved Fréchet-Hoeffding bounds are pointwise sharp under uncertainty in marginals but not in absence.
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.
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.
The aim of this article is to present the category of bounded Frechet manifolds in respect to which we will review the geometry of Frechet manifolds with a stronger accent on its metric aspect. An inverse function theorem in the sense of Nash and Moser in this category is proved, and some applications to Riemannian geo…
We extend the Palais-Smale condition to Keller's C c 1 C_c^1 C c 1 -functionals on Fréchet spaces. Using this condition together with Ekeland's variational principle, we obtain some results regarding the existence of minima. In this setting, we prove that the Palais-Smale condition for functionals bounded below implies the coerci…
SiD distills pretrained diffusion models into a fast one-step generator.
problem Efficiently distilling pretrained diffusion models into a fast generator.
method Reformulates forward diffusion processes as semi-implicit distributions and uses three score-related identities to create a loss mechanism.
result Achieves high FID performance and significantly reduces generation time.
A new algorithm solves signed Fréchet regression on manifolds with bounded curvature.
problem Signed Fréchet regression on Riemannian manifolds with bounded curvature.
method Proximal DC algorithm (FRIDA) for computing signed Fréchet regression fits.
result Existence and interiority of minimizers, strong convexity of proximal subproblems, and convergence to stationary points.
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.
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.
A method for reducing dimensions in Fréchet regression models.
problem Complex data objects in metric space-valued responses.
method Mapping metric-space valued random objects to real-valued variables and applying classical SDR.
result Consistent and asymptotically convergent method for Fréchet SDR.
In this paper, we study Lipschitz-Fredholm vector fields on Bounded-Fréchet-Finsler manifolds. In this context we generalize the Morse-Sard-Brown theorem, asserting that if M M M is a connected smooth bounded-Fréchet-Finsler manifold endowed with a strengthened connection K \mathcal{K} K and if ξ ξ ξ is a smooth Lipschitz-Fr…
Extends Fredholm theory to Frechet spaces.
problem Extending Fredholm theory to Frechet spaces.
method Extending linear Fredholm maps to Frechet spaces using Nash-Moser inverse function theorem.
result Existence of a constant rank theorem for nonlinear Fredholm maps in Frechet spaces.
FastFID efficiently trains generative models with FID loss.
problem Efficiently training generative models with FID loss.
method Introduces FastFID to train generative models with FID as a loss function.
result Improves FID for GANs using FID as an additional loss.
This work tackles regression on non-Euclidean spaces, specifically positive-definite matrices with the Bures-Wasserstein metric.
problem Regression on non-Euclidean spaces, specifically positive-definite matrices with the Bures-Wasserstein metric.
method Developed a sufficient condition for the existence of a minimizer of the conditional barycenter problem, characterized the optimization landscape, and developed a projection-free algorithm for approximate computation of first-order stationary points.
result The objective is free of local maxima under the sufficient condition, and the algorithm enables the use of stochastic Riemannian optimization methods for large-scale setups.
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…
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. 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 …
Improves risk and variability measures continuity and consistency.
problem Improving the continuity and consistency of risk and variability measures.
method Analyzes convex and order bounded above functionals on Frechet lattices and Orlicz spaces.
result Order-continuous, law-invariant functionals on Orlicz spaces are strongly consistent everywhere.
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.
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.
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 …
Global diffeomorphism proof between tame Fréchet spaces.
problem Existence of global diffeomorphism between Fréchet spaces.
method Mountain Pass Theorem and sufficient conditions for diffeomorphism.
result Existence of global diffeomorphism between tame Fréchet spaces.
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.
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.
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.
Study shows essential properties of Z 2 n \mathbb{Z}_2^n Z 2 n -manifolds.
problem Existence of categorical products in Z 2 n \mathbb{Z}_2^n Z 2 n -manifolds. method Detailed analysis of function sheaves and morphisms.
result Existence of categorical products in Z 2 n \mathbb{Z}_2^n Z 2 n -manifolds. 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.
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.
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…
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…
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.
Slice theorem in infinite dimensions for Lie groups.
problem Generalizing slice theorem to infinite-dimensional settings.
method Developed slice theorem for locally convex Lie groups on locally convex manifolds using advanced theorems.
result Existence of orbit type stratification under slice condition.
Study on Frechet distance properties for paths and graphs.
problem Understanding topological properties of Frechet distance spaces.
method Proving path-connectedness of Frechet distance spaces and metric balls.
result Spaces of paths and graphs under Frechet distance are path-connected.
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.
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.
Integrability criterion for projective limits of Banach distributions on Fréchet manifolds.
problem Integrability of projective limits of involutive bundles on Banach manifolds.
method An integrability criterion for a projective limit of Banach distributions.
result Result of integrability of projective limit of involutive bundles on a projective sequence of Banach manifolds.
Study Yang-Mills connections on surfaces via Fréchet reduction.
problem Characterize Yang-Mills connections on orientable closed surfaces.
method Fréchet reduction to define and constrain gauge equivalence classes of connections.
result Deduced stratified symplectic structure on unbased Yang-Mills connections.
Transformer model pretrains on synthetic graphs for AD detection.
problem Limited labeled data and class imbalance in AD diagnosis.
method Diffusion-generated synthetic graphs, Graph Transformers, transfer learning.
result Framework outperforms baselines in AD diagnosis metrics.