Quantifies fractional isoperimetric inequality with strong control over boundary oscillation.
problem Fractional isoperimetric inequality and its quantitative aspects.
method Regularization process with a new spirit.
result Stability estimates for fractional Cheeger inequality.
New approach shows data memorization trade-offs in large models.
problem Data memorization in large language models and its privacy implications.
method Developed a new approach using strong data processing inequalities to prove lower bounds on memorization.
result Proved that Ω(d) bits of training data information must be memorized for O(1) examples, decaying with example growth. Paper sets fundamental limits for distributed covariance estimation with constrained communication.
problem Estimating high-dimensional covariance matrices in a feature-split setting with limited communication.
method Developed a Conditional Strong Data Processing Inequality (C-SDPI) to establish minimax lower bounds and an optimal estimation protocol.
result Achieved nearly optimal estimation protocol with sample and communication requirements matching lower bounds up to logarithmic factors.
The paper analyzes convergence rates of Langevin dynamics and Proximal Sampler using Φ-divergence.
problem Analyzing convergence rates of Langevin dynamics and Proximal Sampler.
method Extending mixing time analyses to Φ-divergence, using strong data processing inequalities. result Convergence of Φ-divergence to 0 exponentially fast along Unadjusted Langevin Algorithm and Proximal Sampler. The paper proves strong holomorphic Morse inequalities on complex manifolds with optimal estimates.
problem Holomorphic Morse inequalities on non-compact complex manifolds with optimal fundamental estimates.
method Established strong holomorphic Morse inequalities under optimal fundamental estimates.
result Strong holomorphic Morse inequalities hold true on non-compact complex manifolds with optimal fundamental estimates.
This paper shows how to learn variational inequalities fast with strong monotonicity.
problem Learning variational inequalities efficiently.
method Extending convex optimization techniques to variational inequalities with strong monotonicity.
result Fast generalization rates of Θ(1/ε) for learning variational inequalities. Sharp isoperimetric inequalities for the sine transform of even isotropic measures are established. The corresponding reverse inequalities are obtained in an asymptotically optimal form. These new inequalities have direct applications to strong volume estimates for convex bodies from data about their sections or projec…
Study confirms equivalence in Heisenberg groups between curvature-dimension conditions and strong Brunn-Minkowski inequalities.
problem Equivalence between curvature-dimension conditions and strong Brunn-Minkowski inequalities in Heisenberg groups.
method Optimal transport and approximation techniques in sub-Riemannian Heisenberg group Hn, combined with previous works.
result Confirms the equivalence in Heisenberg groups between curvature-dimension conditions and strong Brunn-Minkowski inequalities.
Proves strong Morse inequalities for area functional in low dimensions.
problem Proving Morse inequalities for area functional in specific dimensions.
method Analyzes area functional in codimension one, proving inequalities under given dimension constraints.
result Strong Morse inequalities for area functional in specified dimensions.
Quantum states can be learned efficiently using gentle measurements.
problem Efficiently learning quantum states with minimal measurements.
method Introducing α-LGM measurements and proving strong quantum DPI.
result The number of states needed for accurate learning is of order 1/(ε^2 α^2).
We investigate the framework of privacy amplification by iteration, recently proposed by Feldman et al., from an information-theoretic lens. We demonstrate that differential privacy guarantees of iterative mappings can be determined by a direct application of contraction coefficients derived from strong data processing…
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.
Characterizes geodesics on spheres with Morse index bounds and inequalities.
problem Understanding geodesics on spheres using Morse theory.
method Morse-theoretic characterization and strong Morse inequalities.
result Existence of geodesics with specific Morse indices on spheres.
We discuss relations among various positivities of knots and links, such as strong quasipositivity and quasipositivity. We give several pieces of supporting evidence for conjectural statements concerning these positivities and the defect of Bennequin inequality. Finally, we determine strong quasipositivity and quasipos…
Data processing inequalities link Fisher information to local differential privacy constraints.
problem Understanding how Fisher information scales with local differential privacy constraints.
method Developed data processing inequalities for Fisher information under local differential privacy.
result Implications for private estimation with optimal bounds and error rates.
The paper improves count data regression models for overdispersed data.
problem Improving regression models for overdispersed count data.
method Double ℓ1-regularized negative binomial regressions. result Oracle inequalities and consistency for Lasso estimators of partial regression coefficients.
Study sharp inequalities for perimeter functionals in capillarity and convex cones.
problem Quantitative isoperimetric inequalities for perimeter functionals in capillarity and convex cones.
method Derivation of Fuglede-type estimates and application of selection principle.
result Sharp quantitative isoperimetric inequalities in strong and barycentric forms.
Let X be a compact connected CR manifold of dimension 2n−1,n≥2. We assume that there is a transversal CR locally free S1 action on X. Let Lk be the k-th power of a rigid CR line bundle L over X. Without any assumption on the Levi-form of X, we obtain a scaling upper-bound for the partial Szegő …
Study introduces new Bernstein inequalities for dependent data in Hilbert spaces.
problem Learning from non-independent and non-identically distributed data.
method Data-dependent Bernstein inequalities tailored for vector-valued processes in Hilbert space.
result Achieved novel risk bounds for covariance operator estimation and operator learning.
Paper develops a new inequality for non-causal machine learning.
problem Current concentration inequalities cannot be applied to non-causal machine learning.
method Develops a framework for non-causal random fields and proves a Hoeffding-type inequality.
result Obtains a Hoeffding-type concentration inequality for non-causal random fields.
The paper studies how quickly samples from Langevin dynamics become independent.
problem Understanding the dependence between samples along Langevin dynamics and related algorithms.
method Measures dependence via Φ-mutual information and proves strong data processing inequalities. result The Φ-mutual information between samples decreases exponentially to zero. We show how to reduce the general formulation of the mass-angular momentum inequality, for axisymmetric initial data of the Einstein equations, to the known maximal case whenever a geometrically motivated system of equations admits a solution. This procedure is based on a certain deformation of the initial data which p…
Strong geodesic convex function and strong monotone vector field of order m on Riemannian manifolds have been established. A characterization of strong geodesic convex function of order m for the continuously differentiable functions has been discussed. The relation between the solution of a new variational inequal…
Estimates barycenter in geodesic spaces with finite sample bounds.
problem Estimating the barycenter of a distribution in geodesic spaces.
method Finite sample error bounds, Hoeffding- and Bernstein-type concentration inequalities, efficient algorithms.
result Statistical guarantees for efficient barycenter computation.
Paper proves strong uniqueness of cylindrical tangent flows near singularity in Ricci flow.
problem Proving strong uniqueness of cylindrical tangent flows near singularity in Ricci flow.
method Established Lojasiewicz inequality for pointed W-entropy under cylindrical geometry assumption. result Strong uniqueness of cylindrical tangent flows at first singular time of Ricci flow proved.
New statistical inference method for high-dimensional Hawkes processes.
problem Uncertainty evaluation of network estimates in high-dimensional point process data.
method Develops a new statistical inference procedure using concentration inequalities and martingale central limit theory.
result Characterizes the convergence rate of test statistics for high-dimensional Hawkes processes.
We prove a variant of the Davies-Gaffney-Grigor'yan Lemma for the continuous time heat kernel on graphs. We use it together with the Li-Yau inequality to obtain strong heat kernel estimates for graphs satisfying the exponential curvature dimension inequality.
Adaptive Bayesian learning aggregates experts to improve performance.
problem Bayesian online learning's performance depends on inferential choices.
method Treat Bayesian update rules as experts and aggregate them based on sequential predictive losses.
result The aggregate competes with the best expert in hindsight at a low aggregation cost.
Develops inequalities for high-dimensional linear processes with dependent innovations.
problem Estimating high-dimensional VAR(p) systems and HAC covariance estimation.
method Concentration inequalities for l∞ norm of vector linear processes with sub-Weibull, mixingale innovations. result Obtained concentration bounds for the maximum entrywise norm of lag-h autocovariance matrices. We prove a uniform Sobolev inequality for Ricci flow, which is independent of the number of surgeries. As an application, under less assumptions, a non-collapsing result stronger than Perelman's κ non-collapsing with surgery is derived. The proof is shorter and seems more accessible. The result also improves some ear…
We prove the conjecture of Tian on the strong form of the Moser-Trudinger inequality for Kahler-Einstein manifolds with positive first Chern class, when there are no holomorphic vector fields, and, more generally, when the setting is invariant under a maximal compact subgroup of the automorphism group.
The paper proves strong uniqueness and rectifiability of generalized cylindrical singularities in Ricci flow.
problem Proving strong uniqueness and rectifiability of generalized cylindrical singularities in Ricci flow.
method Establishing a Lojasiewicz inequality for the pointed W-entropy in Ricci flow under the assumption of geometry near the base point being close to a generalized cylinder. result Proves strong uniqueness of generalized cylindrical tangent flows and shows that the subset of points with rectifiable Sqck(N) is horizontally parabolic. In 1981 Edward Witten proved a remarkable result where he derived the classical Morse Inequalities using ideas from Supersymmetric (SUSY) Quantum Mechanics. In this regard, one has an example where a Physical Theory has something to say about the underlying Mathematical Structure. The objective of this essay is to unde…
This manuscript served as lecture notes for a mini-course in the 2016 Southern California Geometric Analysis Seminar Winter School. The goal is to give a quick introduction to Kahler geometry by describing the recent resolution of Tian's three influential properness conjectures in joint work with T. Darvas. These resul…
We consider an abstract compact orientable Cauchy-Riemann manifold endowed with a Cauchy-Riemann complex line bundle. We assume that the manifold satisfies condition Y(q) everywhere. In this paper we obtain a scaling upper-bound for the Szegö kernel on (0, q)-forms with values in the high tensor powers of the line bund…
Unified framework for deriving generalization bounds in supervised learning.
problem Generalization error bounds in supervised learning.
method Data Processing Inequality PAC-Bayesian framework.
result Unified bounds on binary Kullback-Leibler generalization gap for various divergences.
The data processing inequality doesn't always hold in practice, showing benefits in low-level tasks.
problem The data processing inequality suggests no benefit in pre-processing for classification.
method Theoretical and empirical study of binary classification setup with deep neural networks.
result Pre-classification processing can improve classification accuracy for any finite number of training samples.
We give a slope equality for fibered surfaces whose general fiber is a smooth plane curve. As a corollary, we prove a "strong" Durfee-type inequality for isolated hypersurface surface singularities, which implies Durfee's strong conjecture for such singularities with non-negative topological Euler number of the excepti…
New test for point processes without strong model assumptions.
problem Testing local independence in point processes without strong model assumptions.
method Expansion similar to Volterra expansions to represent marginalized intensities.
result Approximation of true marginalized intensity arbitrarily well.
Study hypothesis testing under quantized samples with communication constraints, achieving near-optimal sample complexity.
problem Optimizing hypothesis testing with quantized samples and communication constraints.
method Developed a polynomial-time algorithm achieving near-optimal sample complexity under communication constraints.
result Achieved near-optimal sample complexity under communication constraints, with a logarithmic factor increase over unconstrained setting.
We consider the Harnack inequality for harmonic functions with respect to three types of infinite dimensional operators. For the infinite dimensional Laplacian, we show no Harnack inequality is possible. We also show that the Harnack inequality fails for a large class of Ornstein-Uhlenbeck processes, although functions…
Unified analysis of efficient local training methods for distributed variational inequalities.
problem Efficient distributed/federated learning for variational inequality problems.
method Unified convergence analysis of communication-efficient local training methods.
result First local gradient descent-accent algorithms with improved communication complexity.
From concentration inequalities for the suprema of Gaussian or Rademacher processes an inequality is derived. It is applied to sharpen existing and to derive novel bounds on the empirical Rademacher complexities of unit balls in various norms appearing in the context of structured sparsity and multitask dictionary lear…
Sharp Hardy and spectral gap inequalities found on special irreversible Finsler manifolds.
problem Understanding Hardy and spectral gap inequalities on irreversible Finsler manifolds.
method Finslerian extension of the method of Riccati pairs.
result Sharpness of Hardy and spectral gap inequalities on specific Finsler manifolds.
In this short note, we will strengthen the classic Doob's Lp inequality for sub-martingale processes. Because this inequality is of fundamental importance to the theory of stochastic process, we believe this generalization will find many interesting applications.
The paper improves machine learning for heavy-tailed panel data.
problem Improving estimates for financial and economic data with fat tails.
method Sparse-group LASSO regularization and Fuk-Nagaev concentration inequality.
result Oracle inequalities for panel data estimators.
Study of diffusion annealed Langevin dynamics for generative models.
problem Theoretical efficiency of score-based diffusion processes.
method Rigorous construction and analysis of diffusion processes with Poincaré and logarithmic Sobolev inequalities.
result Improvement in efficiency of diffusion processes through Poincaré and logarithmic Sobolev inequalities.
The paper proves concentration inequalities for diffusion processes.
problem Proving concentration inequalities for diffusion processes.
method Analysis via the Poisson equation for a broad class of subexponentially ergodic processes.
result Demonstrates power of concentration inequalities in validating conditions for Lasso estimation and sampling algorithms.