This survey introduces synthetic timelike Ricci curvature bounds in Lorentzian spaces.
problem Synthetic timelike Ricci curvature bounds in non-smooth Lorentzian spaces.
method Optimal transport and entropy tools.
result Synthetic version of Hawking's singularity theorem and synthetic characterisation of Einstein's vacuum equations.
Proves equivalence of two types of Ricci curvature bounds.
problem Equivalence of distributional and synthetic Ricci curvature bounds.
method Analyzes weighted Riemannian manifolds with specific smoothness conditions.
result Proves equivalence of Ricci curvature bounds under given conditions.
Performing supervised learning from the data synthesized by using Generative Adversarial Networks (GANs), dubbed GAN-synthetic data, has two important applications. First, GANs may generate more labeled training data, which may help improve classification accuracy. Second, in scenarios where real data cannot be release…
Entropy study on synthetic spaces with curvature bounds.
problem Entropy functional on synthetic spaces with curvature bounds.
method Rigorous justification of entropy formula, monotonicity, and rigidity properties; heat kernel bounds.
result Bounds for heat equation solutions on synthetic spaces.
Equivalence found between smooth and synthetic timelike curvature bounds.
problem Understanding timelike sectional curvature bounds in spacetime geometry.
method Established equivalence between sectional curvature bounds on timelike planes and synthetic timelike bounds.
result Equivalence proved for sectional curvature bounds on timelike planes and synthetic timelike bounds on strongly causal spacetimes.
Study compares synthetic and distributional Ricci curvature bounds.
problem Comparing synthetic and distributional approaches to lower Ricci curvature bounds.
method Analyzes synthetic via weak displacement convexity and distributional via non-negativity of Ricci-tensor.
result Distributional bounds imply entropy bounds for C1 metrics and vice versa for C1,1 under convergence condition. Privacy is enhanced by synthetic data release even with unlimited data.
problem Improving privacy guarantees for synthetic data release.
method Analyzing a bounded-parameter assumption to show privacy amplification persists with unlimited synthetic records.
result Privacy amplification is possible even with an unbounded number of synthetic records.
Unified study of Riemannian and sub-Riemannian geometries with synthetic Ricci curvature bounds.
problem Unified framework for Riemannian and sub-Riemannian geometries.
method Study of gauge metric measure spaces.
result Unified synthetic Ricci curvature lower bounds for both Riemannian and sub-Riemannian structures.
Differentially private synthetic control estimates treatment effects while protecting privacy.
problem Estimating treatment effects on sensitive data without revealing individual information.
method Combines non-private synthetic control and differentially private empirical risk minimization.
result Private synthetic control produces accurate predictions with minimal privacy cost.
Survey on curvature bounds and isoperimetric inequalities.
problem Interplay between curvature bounds and isoperimetric problems.
method Survey of classical and recent results.
result Recent developments on curvature bounds and isoperimetric problems.
The paper analyzes SMOTE for imbalanced classification, providing theoretical bounds and guidelines.
problem The challenge of imbalanced classification problems, especially with minority classes.
method Theoretical analysis of SMOTE and related oversampling techniques for minority classes.
result Derives concentration and excess risk bounds for SMOTE and kernel-based classifiers.
Develops optimal transport in Lorentzian spaces with synthetic curvature bounds.
problem Synthetic curvature bounds for Lorentzian spaces.
method Optimal transport, convexity analysis of entropy functionals.
result Synthetic notion of timelike Ricci curvature lower bounds.
We analyze Lorentzian spacetimes subject to curvature-dimension bounds using the Bakry-Émery-Ricci tensor. We extend the Hawking-Penrose type singularity theorem and the Lorentzian timelike splitting theorem to synthetic dimensions N≤1, including all negative synthetic dimensions. The rigidity of the timelike spli…
New relation between curvature bounds and spacetime inextendibility.
problem Inextendibility of spacetimes under low regularity conditions.
method Synthetic curvature and causal character analysis.
result Low-regularity spacetimes with unbounded curvature.
New relation between curvature bounds and spacetime inextendibility.
problem Inextendibility of spacetimes under low regularity conditions.
method Synthetic curvature and causal character maximizers.
result Low-regularity inextendibility linked to unbounded curvature.
Synthetic proof shows globally hyperbolic Lorentzian spaces with specific curvature are warped products.
problem Synthetic proof of rigidity for globally hyperbolic Lorentzian spaces.
method Synthetic geometry and warped product analysis.
result Spaces with specific curvature and distance realizer are warped products.
The paper analyzes how synthetic data training degrades diffusion models, providing bounds and characterizing different drift regimes.
problem The degradation of performance in diffusion models trained on synthetic data.
method Theoretical analysis of score-based diffusion models, focusing on the accumulated divergence between generated and target distributions.
result Upper and lower bounds on the accumulated divergence, providing the first lower bound for diffusion models.
New method generates private synthetic data with optimal utility for smooth queries.
problem Achieving strong utility guarantees for meaningful downstream analysis of sensitive datasets.
method Proposes a polynomial-time algorithm for generating (ε,δ)-differentially private synthetic data with minimax optimal error rates for smooth queries. result Achieves a minimax error rate of Ok,d(n−min{1,dk}) for k-smooth queries, up to a log(n) factor. The study proposes conjectures on limit spaces of Riemannian manifolds with Ricci curvature.
problem Understanding the regularity of limit spaces of Riemannian manifolds with Ricci curvature.
method Synthetic treatment of lower bounds on Ricci curvature for metric measure spaces.
result Several conjectures on the regularity of limit spaces.
We will study metric measure spaces (X,d,m) beyond the scope of spaces with synthetic lower Ricci bounds. In particular, we introduce distribution-valued lower Ricci bounds BE1(κ,∞) ∙ for which we prove the equivalence with sharp gradient estimates, ∙ the class of which will be preserved under…
In this paper we investigate Lott-Sturm-Villani's synthetic lower Ricci curvature bound on Riemannian manifolds with boundary. We prove several measure rigidity results for some important functional and geometric inequalities, which completely characterize CD(K,∞) condition and non-collapsed ${\rm CD}(K, …
Synthetic data augmentation can improve imbalanced classification metrics.
problem Improving imbalanced classification metrics
method Developing a framework for analyzing the effects of synthetic data augmentation on score-based classification
result Augmentation can improve AUROC, AUPRC, balanced accuracy, and F1 score
We study the sample complexity of private synthetic data generation over an unbounded sized class of statistical queries, and show that any class that is privately proper PAC learnable admits a private synthetic data generator (perhaps non-efficient). Previous work on synthetic data generators focused on the case that …
Synthetic data can amplify privacy in linear regression models.
problem Understanding how synthetic data can enhance privacy in linear regression models.
method Investigated through the linear regression framework, analyzing synthetic data generated from random inputs and controlled inputs.
result Releasing a limited number of synthetic data points amplifies privacy beyond the model's inherent guarantees when inputs are random, but not when inputs are controlled by an adversary.
A privacy-preserving synthetic data generation framework that distinguishes between true and phantom data disclosures.
problem Detecting and explaining data disclosures in synthetic datasets.
method Customizable empirical auditing framework with statistical hypothesis testing.
result Demonstrated tighter privacy leakage bounds than prior methods.
Stability of timelike Ricci bounds in low-regularity spacetimes.
problem Stability of synthetic timelike Ricci curvature bounds under C0-limits. method Constructing smooth approximations and analyzing limiting behavior via Lorentzian optimal transport.
result Impulsive gravitational waves satisfy synthetic timelike Ricci curvature lower bounds.
Enhances statistical inference using synthetic data.
problem Limited labeled data for statistical inference.
method GESPI framework that combines synthetic and real data.
result Error rate remains below a user-specified bound and decreases with synthetic data quality.
This paper considers the quantification of the prediction performance in Gaussian process regression. The standard approach is to base the prediction error bars on the theoretical predictive variance, which is a lower bound on the mean square-error (MSE). This approach, however, does not take into account that the stat…
The study improves harmonic map theory for metric spaces with curvature bounds.
problem Harmonic maps between specific metric spaces with curvature constraints.
method Synthetic geometry, Optimal Transport, Heat Flow, viscosity theory.
result Established Lipschitz continuity and Bochner-Eells-Sampson inequality.
We prove that in two dimensions the synthetic notions of lower bounds on sectional and on Ricci curvature coincide.
We discuss various characterizations of synthetic upper Ricci bounds for metric measure spaces in terms of heat flow, entropy and optimal transport. In particular, we present a characterization in terms of semiconcavity of the entropy along certain Wasserstein geodesics which is stable under convergence of mm-spaces. A…
Study inradius collapsed manifolds with lower Ricci curvature bounds, proving properties of their limits.
problem Characterizing limits of inradius collapsed manifolds with lower Ricci curvature bounds.
method Analyzing families of manifolds with specific curvature and boundary conditions, proving properties of the limits.
result Limits of inradius collapsed manifolds have at most two boundary components and a lower Ricci curvature bound.
Paper establishes utility theory for synthetic data generation.
problem Lack of theoretical understanding in synthetic data utility.
method Statistical learning framework with two utility metrics: generalization and model ranking.
result Theoretical bounds for synthetic data utility metrics ensure comparable generalization and consistent model comparison.
Paper studies statistical properties of DP data synthesis algorithms based on Bayesian networks.
problem Ensuring differential privacy in synthetic data generation for high-dimensional data.
method Introduces random noise to low-dimensional marginals of a probabilistic graphical model (BN) to achieve differential privacy.
result Establishes a rigorous accuracy guarantee for BN-based DP synthetic data generators using total variation (TV) distance.
Globalisation theorem for Lorentzian spaces with curvature bounds.
problem Synthetic geometric analysis of Lorentzian length spaces.
method Cat's cradle construction and synthetic geometry.
result An analogue of Toponogov's Globalisation Theorem for Lorentzian length spaces.
A new method uses randomized trials to estimate the strength of unobserved confounding.
problem Unobserved confounding compromises causal conclusions from non-randomized studies.
method Designs a statistical test to detect unobserved confounding strength and estimates a lower bound.
result Estimates an asymptotically valid lower bound on unobserved confounding strength.
Survey on gluing constructions under lower curvature bounds.
problem Understanding lower curvature bounds in various geometric contexts.
method Analyzes gluing constructions in smooth and non-smooth settings.
result Provides conjectures and theorems on synthetic lower Ricci curvature bounds.
Sharp eigenvalue bounds on metric measure spaces extend Cheng's theorem.
problem Extending Cheng's eigenvalue comparison theorem to non-smooth spaces.
method Localization technique, synthetic Ricci curvature bounds via optimal transport.
result Sharp upper bounds on eigenvalues in metric measure spaces.
Study solves Gel'fand's inverse problem in non-smooth spaces with Ricci curvature bounds.
problem Determining a Riemannian manifold from heat kernel on subsets.
method Analyzes mRCD(K,N) spaces with synthetic Ricci curvature bounds. result Unique solvability of Gel'fand's inverse problem for compact mRCD(K,N) spaces. The paper presents a framework to quantify the trade-off between synthetic and real data.
problem Improving generalization with synthetic data when real data is scarce.
method Learning-theoretic framework leveraging algorithmic stability to derive generalization error bounds.
result Optimal synthetic-to-real data ratio minimizing expected test error as a function of Wasserstein distance.
The paper corrects bias in synthetic data for imbalanced learning.
problem Challenges in balancing false positive and negative rates in imbalanced data.
method Proposes a bias correction procedure to generate synthetic data for minority groups.
result Enhances prediction accuracy while avoiding overfitting.
Proves a synthetic Lorentzian Cartan-Hadamard theorem.
problem Formulates and proves a theorem for Lorentzian geometry.
method Uses an appropriate notion of local concavity for Lorentzian (pre-)length spaces.
result Establishes existence and uniqueness of timelike geodesics.
New sub-Riemannian structures fail synthetic curvature bounds.
problem Failure of synthetic curvature bounds in sub-Riemannian geometry.
method New stability results for local MCP under quotients, applied to specific sub-Riemannian structures.
result Ideal sub-Riemannian structures can fail the MCP, generically for high dimensions and rank > 3.
Bounds on treatment effect sensitivity in causal reasoning using Hölder's inequality.
problem Estimating treatment effects in presence of unobserved confounders.
method Using Hölder's inequality, derived bounds on confounding bias based on unmeasured confounding strength.
result Bounds are tight under specific conditions of independence between U and T/Y.
Uniform heat kernel bounds lead to synthetic Ricci curvature conditions for Lipschitz manifolds.
problem Establishing synthetic Ricci curvature conditions for Lipschitz manifolds.
method Uniform heat kernel bounds and synthetic Ricci curvature conditions.
result Uniform heat kernel bounds lead to synthetic Ricci curvature conditions for Lipschitz manifolds.
Novel approach integrates Multivariate Square-root Lasso into Synthetic Control for high-dimensional data.
problem Challenges in practical implementation and computational efficiency of Synthetic Control method for high-dimensional disaggregated data.
method Integrates Multivariate Square-root Lasso into Synthetic Control framework.
result Demonstrates superior computational efficiency without compromising estimation accuracy.
Upper bounds on revised first Betti number and torus stability for RCD spaces.
problem Bounding the revised first Betti number and stability of RCD spaces.
method Proving an upper bound on the rank of the abelianised revised fundamental group and establishing torus stability.
result Spaces with saturated upper bound on revised first Betti number are mGH-close to flat tori.
Almost-Riemannian manifolds fail to meet a synthetic curvature condition.
problem Proving almost-Riemannian manifolds do not satisfy the CD condition. method Developed a new strategy to contradict the 1-dimensional CD condition. result 2D and strongly regular almost-Riemannian manifolds do not satisfy CD(K,N) for any K and N.