Study local invariants of divergence-free webs in geometry.
problem Characterize triviality of divergence-free webs.
method Introduce two local invariants: differential and geometric.
result Triviality of either invariant characterizes trivial divergence-free web-germs.
Efficiently visualizes uncertainty in local divergence of 2D vector fields.
problem Uncertainty in vector field data leads to inaccurate divergence computations.
method Closed-form approach for highly efficient and accurate uncertainty visualization of local divergence, assuming independently Gaussian-distributed vector uncertainties.
result Significantly enhanced efficiency and accuracy of our algorithms over classical MC approach.
Locally private mechanisms' output divergence bounds derived.
problem Bounding divergence between locally private mechanisms' outputs.
method Sharp upper bounds on divergence between input and output distributions.
result Established locally private versions of estimation risk bounds.
Local mass perspective on Bayesian inference
problem Measuring distributional discrepancy in Bayesian inference
method Introducing Mass Index and Regularised Extended KL
result Proving inequalities for comparing local small-ball masses
LDP is equivalent to contraction of E_γ-divergence, impacting privacy and utility.
problem Analyzing trade-offs between privacy and utility in estimation problems.
method Equivalence of LDP constraints to contraction coefficients of E_γ-divergence, using f-divergences and estimation-theoretic tools.
result LDP guarantees can be expressed in terms of contraction coefficients of arbitrary f-divergences.
We study Eγ-divergence contraction and its privacy implications.
problem Analyzing privacy in data processing and algorithms.
method Generalizing Dobrushin's coefficient to Eγ-divergence and deriving contraction coefficients. result Local differential privacy can be expressed in terms of Eγ-divergence contraction, leading to precise sample size reductions. This note proves that any locally extremal non-self-conjugate geodesic loop in a Riemannian manifold is a closed geodesic. As a consequence, any complete and non-contractible Riemannian manifold with diverging injectivity radii along diverging sequences and without points conjugate to themselves, possesses a minimizing…
Replacing MSE with f-divergence in diffusion models improves robustness under data contamination.
problem Improving robustness of diffusion models under data contamination.
method Replacing MSE with f-divergence in diffusion models.
result Empirical improvement in performance under data contamination.
Electrostatic systems with specific tensors are locally conformally flat.
problem Understanding the geometry of electrostatic systems with special tensors.
method Proving local conformal flatness for electrostatic manifolds with divergence-free Bach tensor.
result Three-dimensional electrostatic manifolds with divergence-free Bach tensor are locally conformally flat.
Develops a contraction framework for MCMC mixing rates.
problem Proving mixing-time bounds for MCMC algorithms.
method Global and local contraction coefficients under Eγ-divergence. result Explicit global contraction coefficients for Gaussian smoothing.
Paper proposes f-EBM for training deep EBMs using various f-divergences.
problem Training deep EBMs with intractable partition functions.
method Introduces f-EBM framework and optimization algorithm for any f-divergence.
result f-EBM outperforms contrastive divergence and other f-divergences.
Proposes practical kernel tests for f-divergences with theoretical guarantees.
problem Two-sample testing and machine unlearning evaluation.
method Regularized f-divergence kernel tests, adaptive to hyperparameters. result Different f-divergences highlight localized differences. Study geodesics in curved spaces, counts ambiguous paths, confirms number theory conjectures.
problem Counting ambiguous geodesics in curved spaces.
method Asymptotic formula for common perpendiculars in negatively curved spaces, applying to modular orbifolds and number fields.
result Confirms and extends Motohashi's conjecture on binary additive divisor problem.
Unified analysis of KL divergence using shifted composition for sampling.
problem Sampling from target distributions with KL divergence guarantees.
method Shifted composition rule applied to KL divergence, combining local error analysis and Girsanov's theorem.
result Unified KL guarantees for strongly log-concave, weakly log-concave, and log-Sobolev distributions.
New divergence identity for scalar curvature helps prove rigidity of tensors.
problem Proving rigidity of Codazzi tensors under curvature and invariant conditions.
method Derived a divergence identity for a vector field and applied it to tensor rigidity.
result New proof of Tang-Yan theorem on constant eigenvalues for tensors.
We investigate the use of alternative divergences to Kullback-Leibler (KL) in variational inference(VI), based on the Variational Dropout \cite{kingma2015}. Stochastic gradient variational Bayes (SGVB) \cite{aevb} is a general framework for estimating the evidence lower bound (ELBO) in Variational Bayes. In this work, …
The t-distributed Stochastic Neighbor Embedding (t-SNE) is a powerful and popular method for visualizing high-dimensional data. It minimizes the Kullback-Leibler (KL) divergence between the original and embedded data distributions. In this work, we propose extending this method to other f-divergences. We analytically a…
Proves Sard conjecture for specific distributions, controlling divergence of vector fields.
problem Proving the Sard conjecture for certain types of distributions.
method Constructs a singular distribution capturing essential abnormal lifts, proving the conjecture for rank 3 distributions in dimension 4 and generic corank 1 distributions.
result Proves the Sard conjecture for generic co-rank one distributions.
This is the second in a series of papers where we prove a conjecture of Deser and Schwimmer regarding the algebraic structure of ``global conformal invariants''; these are defined to be conformally invariant integrals of geometric scalars. The conjecture asserts that the integrand of any such integral can be expressed …
Paper proposes a new strategy to improve initial performance of federated models.
problem Weight divergence in Federated Averaging (FedAvg) leads to poor initial performance in federated models.
method Local continual training with importance weights evaluated on a proxy dataset.
result The method significantly improves the initial performance of federated models with minimal extra communication costs.
Computing approximate nearest neighbors in high dimensional spaces is a central problem in large-scale data mining with a wide range of applications in machine learning and data science. A popular and effective technique in computing nearest neighbors approximately is the locality-sensitive hashing (LSH) scheme. In thi…
CO2 algorithm creates coresets for generic smooth divergences efficiently.
problem Efficiently creating coresets for generic smooth divergences.
method CO2 algorithm using functional Taylor expansion and maximum mean discrepancy minimization.
result Poly-logarithmically many data points suffice for Sinkhorn divergence approximation.
A new differentiable divergence for time series comparison.
problem Computing discrepancies between time series of varying lengths.
method Proposed a new divergence, soft-DTW divergence, addressing issues of differentiability and positivity.
result Showed that the new divergence is a valid divergence: non-negative and minimized when time series are equal.
The paper proves rigidity of certain solitons with specific properties.
problem Classifying and understanding Bach-flat gradient Schouten solitons.
method Analyzing the properties of Schouten solitons and their Ricci tensors.
result Rigidity of certain Schouten solitons under specific conditions.
Modified K-means ensures local optimality with same complexity.
problem Lack of rigorous analysis on local optimality guarantees of K-means.
method Proposed modifications to K-means ensuring local optimality.
result Proposed methods provide improved locally optimal solutions.
TMDA aligns subdomain data distribution discrepancies across domains using manifold representations.
problem Transfer learning challenges due to domain divergence.
method TMDA uses low-dimensional manifolds to represent subdomains and aligns local data distribution discrepancies across domains using M3D.
result TMDA is a promising method for various transfer learning tasks.
The aim of this paper is to provide new theoretical and computational understanding on two loss regularizations employed in deep learning, known as local entropy and heat regularization. For both regularized losses we introduce variational characterizations that naturally suggest a two-step scheme for their optimizatio…
The paper proves rigidity results for manifolds with special holonomy.
problem Proving rigidity results for compact Riemannian manifolds with special holonomy.
method Using divergence free Weyl tensors and curvature operators, the paper proves similar results for manifolds with special holonomy.
result The paper proves that manifolds with special holonomy are locally symmetric or conformally equivalent to a quotient of the sphere.
The paper analyzes the statistical properties of GANs using f-divergence.
problem Understanding the statistical behavior of GANs and comparing different f-divergences. method Asymptotic analysis of f-divergence GANs, including Kullback-Leibler divergence. result Asymptotically equivalent GANs with the same discriminator classes for correctly specified models.
Robust VB framework handles contamination using min-max median aggregation.
problem Handling contamination and outliers in datasets.
method Partition data into subsets, formulate robust optimization problem, use min-max median KL divergence.
result Min-max median formulation improves robustness and statistical rates.
Private KL distribution estimation improved with instance-optimality.
problem Minimizing KL divergence between true and estimated distributions.
method Construct minimax optimal private estimators, then focus on instance-optimality.
result Achieved instance-optimality up to constant factors for KL estimation.
We consider (locally) energy finite coordinates associated with a strongly local regular Dirichlet form on a metric measure space. We give coordinate formulas for substitutes of tangent spaces, for gradient and divergence operators and for the infinitesimal generator. As examples we discuss Euclidean spaces, Riemannian…
New principle controls graph-informed adversarial discrepancies.
problem Graph-informed adversarial learning for interpolative divergences.
method Proves infimal subadditivity for interpolative divergences.
result Graph-informed adversarial learning is justified for interpolative divergences.
We study the Hadamard finite part of divergent integrals of differential forms with singularities on submanifolds. We give formulae for the dependence of the finite part on the choice of regularization and express them in terms of a suitable local residue map. The cases where the submanifold is a complex hypersurface i…
Improved UDA framework using f-divergence measures.
problem Addressing distribution shifts in machine learning.
method Refined f-divergence-based discrepancy and f-domain discrepancy. result Novel target error and sample complexity bounds.
DM framework improves robustness and efficiency in latent-mixture models.
problem Efficient and robust inference in latent-mixture models.
method Divergence-minimization framework with monotonic convergence and robustness guarantees.
result DM yields consistent and asymptotically normal estimators under correct specification.
Locally private methods detect changes in time series data.
problem Detecting distributional changes in time series data under local differential privacy.
method Proposed locally differentially private algorithms based on randomized response and binary mechanisms.
result Theoretical performance bounds and empirical validation of detection accuracy.
SRFE clarifies KL divergences without unifying learning frameworks.
problem Inductive biases of KL divergences and their limitations.
method Introducing SRFE, a log-moment-based functional of the likelihood ratio.
result SRFE recovers KL divergences as limits and reveals a mean-variance tradeoff.
A new associative memory uses Sinkhorn divergence for efficient pattern retrieval.
problem Efficiently retrieving patterns from large datasets of weighted point clouds.
method Derived retrieval dynamics as a SHK gradient flow, discretized for a deterministic algorithm.
result Proved basin invariance, geometric convergence, and robust recovery from perturbations.
Global solutions found for a wave-Klein-Gordon system with strong couplings in divergence form.
problem Global well-posedness of a wave-Klein-Gordon system with strong couplings in divergence form.
method Constructed an auxiliary system with shifted primitives to handle the strong couplings.
result Established global well-posedness theorem for the wave-Klein-Gordon system.
We consider rough metrics on smooth manifolds and corresponding Laplacians induced by such metrics. We demonstrate that globally continuous heat kernels exist and are Hölder continuous locally in space and time. This is done via local parabolic Harnack estimates for weak solutions of operators in divergence form with b…
Federated learning is a distributed, privacy-aware learning scenario which trains a single model on data belonging to several clients. Each client trains a local model on its data and the local models are then aggregated by a central party. Current federated learning methods struggle in cases with heterogeneous client-…
We provide a sufficient condition for the local stability of closed Einstein manifolds of positive Ricci curvature under the Ricci iteration in terms of the spectrum of the Lichnerowicz Laplacian acting on divergence-free tensor fields. We use this result to consider the stability of several Einstein manifolds under th…
Decentralized Bayesian learning reduces KL-divergence exponentially.
problem Efficiently learning posterior distributions in a decentralized setting.
method Decentralized Langevin dynamics in a non-convex setting.
result The algorithm converges to the target posterior distribution with exponential decrease in KL-divergence and polynomial decrease in error contributions.
We establish a method for giving lower bounds for the fundamental tone of elliptic operators in divergence form in terms of the divergence of vector fields. We then apply this method to the Lr operator associated to immersed hypersurfaces with locally bounded (r+1)-th mean curvature Hr+1 of the space forms …
EM algorithm converges in KL divergence for exponential families via mirror descent.
problem Lack of understanding of EM's non-asymptotic convergence properties.
method Viewing EM as a mirror descent algorithm, showing convergence rates in KL divergence.
result KL divergence rates for EM in exponential families, invariant to parametrization.
New schemes improve error estimates for sampling from non-log-concave distributions.
problem Improving sampling from non-log-concave distributions with super-linear drift growth.
method Developed tamed Euler and randomized Euler schemes with error estimates.
result Near-optimal error bounds for sampling and optimization problems.
New method proves dimension-free convergence for ULD in KL divergence.
problem Polynomial scaling of existing convergence guarantees in high dimensions.
method Refined KL local error framework, focusing on tr(H) instead of d.
result First dimension-free KL divergence bounds for discretized ULD.