Study Finsler metric measure manifolds' concentration properties.
problem Understanding concentration properties in Finsler metric measure manifolds.
method Established relationships with observable diameter, isoperimetric inequalities, and first eigenvalue.
result Derived a Cheng type upper bound estimate for the first closed eigenvalue.
Paper addresses concentration of distances for fractional quasi p-norms, identifying conditions for concentration and anti-concentration.
problem Understanding concentration of distances for fractional quasi p-norms in high dimensions.
method Analyzes conditions for concentration and anti-concentration of distances for fractional quasi p-norms.
result Identifies conditions for concentration and anti-concentration of fractional quasi p-norms, ruling out some approaches and specifying conditions for control.
The paper studies reward concentration in MDPs, covering asymptotic and non-asymptotic settings.
problem Reward concentration in Markov Decision Processes (MDPs).
method Unified approach to reward concentration in MDPs, including asymptotic and non-asymptotic bounds.
result Rate-equivalent definitions of regret for learning policies.
The paper provides concentration inequalities for Markov chain variance estimators.
problem Estimating the variance of Markov chains with concentration properties.
method Martingale decomposition method for uniformly geometrically ergodic Markov chains.
result Explicit control of the p-th moment of the OBM estimator difference and dependence on p and mixing time.
The paper studies Dirac operators and their solutions concentrating near singular sets.
problem Understanding concentration properties of solutions to Dirac equations.
method Analyzes Dirac operators of the form Dε=D+ε−1A and their solutions. result Solutions concentrate exponentially near the locus where the rank of ker(A) jumps. The paper shows how solutions of perturbed Dirac operators concentrate near singular sets.
problem Understanding concentration of solutions for perturbed Dirac operators.
method Analyzing the algebraic criterion on $(c, \A)$ and spectral properties of deformed Laplacians.
result Proves an index localization theorem based on spectral separation properties.
The study uses heat flow to analyze properties of Laplace eigenfunctions on manifolds and domains.
problem Analyzing mass concentration and nodal domains of Laplace eigenfunctions.
method Heat diffusion technique to study eigenfunctions and their nodal sets.
result Discovers new insights into the decay and behavior of Laplace eigenfunctions.
The quantification of diversification benefits due to risk aggregation plays a prominent role in the (regulatory) capital management of large firms within the financial industry. However, the complexity of today's risk landscape makes a quantifiable reduction of risk concentration a challenging task. In the present pap…
SGD converges to an invariant distribution with sub-Gaussian or sub-exponential properties.
problem Optimizing smooth and strongly convex objectives using SGD.
method Analysis through Markov chains, focusing on convergence and concentration properties.
result SGD iterates and their invariant limit distribution inherit sub-Gaussian or sub-exponential concentration properties.
Many recent works have shown that adversarial examples that fool classifiers can be found by minimally perturbing a normal input. Recent theoretical results, starting with Gilmer et al. (2018b), show that if the inputs are drawn from a concentrated metric probability space, then adversarial examples with small perturba…
Paper shows how gradient concentration helps in learning from inexact data.
problem Learning from inexact and stochastic training data.
method Combines probabilistic gradient concentration with inexact optimization techniques.
result Derives sharp test error guarantees for learning.
The Langevin Algorithm's stationary distribution is shown to be sub-exponential or sub-Gaussian under certain conditions.
problem Understanding the properties of the Langevin Algorithm's stationary distribution.
method Analysis using a rotation-invariant moment generating function (Bessel function) to study the stationary dynamics of the Langevin Algorithm.
result Concentration results for the Langevin Algorithm's stationary distribution πη are established, showing it is sub-exponential or sub-Gaussian under convex or strongly convex potential conditions. For graphs with non-negative Ollivier curvature, we prove the Liouville property, i.e., every bounded harmonic function is constant. Moreover, we improve Ollivier's results on concentration of the measure under positive Ollivier curvature.
The measure concentration property of an mm-space X is roughly described as that any 1-Lipschitz map on X to a metric space Y is almost close to a constant map. The target space Y is called the screen. The case of Y=R is widely studied in many literature (see \cite{gromov}, \cite{ledoux}, \cite{mil2}…
Study the averaging estimator on graphs with labeled nodes.
problem Understanding the quality of averaging estimators on graph data.
method Rigorously study concentration properties, variance bounds, and risk bounds.
result Contributes to theoretical understanding of graph learning.
Unified framework for measuring concentration in weighted networks considering both weight distributions and network structure.
problem Traditional indices neglect the topology of relationships among network elements.
method Develops a family of topology-aware concentration indices that jointly account for weight distributions and network structure.
result The proposed indices preserve key properties and allow concentration to be evaluated across different dimensions of dependence.
Theoretical study of random forests for nonlinear time series.
problem Theoretical justification for using random forests in time series modeling.
method Uniform concentration inequality for regression trees and random forests consistency proof.
result Consistency of random forests for nonlinear autoregressive processes.
New theory for BNNs with Gaussian priors achieves optimal posterior concentration rates.
problem Lack of theoretical results for BNNs with Gaussian priors.
method New approximation theory for non-sparse DNNs with bounded parameters.
result BNNs with non-sparse general priors can achieve near-minimax optimal posterior concentration rates.
The paper analyzes sparse high-dimensional linear regression with random design and unknown error variance, providing adaptiveness and concentration rates.
problem Sparse high-dimensional linear regression with random design and unknown error variance.
method Analysis of posterior concentration rates, employing techniques to address model misspecification.
result Adaptiveness and concentration rates of the posterior for sparse high-dimensional linear regression.
The study proves inequalities and curvature properties for Markov chains.
problem Isoperimetric and concentration inequalities for Markov chains.
method Laplacian separation principle for eikonal equation; modified log-Sobolev constant; Ollivier curvature.
result Affirmative answers to open questions and new inequalities.
Paper analyzes sample complexity for offline f-divergence-regularized contextual bandits.
problem Lack of tight analyses for sample complexity in offline reinforcement learning.
method Novel pessimism-based analysis for reverse KL divergence, establishing ildeO(ε−1) sample complexity. result Achieves ildeO(ε−1) sample complexity for reverse KL divergence, surpassing existing bounds. New methods improve accuracy in detecting concentric objects.
problem Detecting concentric geometric objects in noisy data.
method Developed new estimators and compared performance of existing methods.
result New methods outperform existing non-iterative methods and are robust to noise.
This paper improves capital efficiency in AMM protocols with leverage.
problem Improving capital efficiency in Automated Market Makers (AMM).
method Formalizes leveraged liquidity provisioning, defines margin level, assets, and debt.
result Leveraged liquidity positions are safe and possess desirable properties.
Study non-Gaussian measures' concentration properties in metric spaces.
problem Concentration properties for non-linear Gaussian functionals with non-Gaussian tails.
method Prove generalised Transportation-Cost Inequalities (TCIs) for specific functionals.
result Extended TCIs for rough volatility and Parabolic Anderson Model.
Intrinsic dimensionality (ID) is one of the most fundamental characteristics of multi-dimensional data point clouds. Knowing ID is crucial to choose the appropriate machine learning approach as well as to understand its behavior and validate it. ID can be computed globally for the whole data point distribution, or comp…
Optimizes non-linear outcomes from summed contributions.
problem Maximizing a non-linear function of summed small contributions.
method Derives a scalable descent algorithm leveraging concentration properties.
result Directly optimizes for stated objective, e.g., A/B test success criterion.
Expanding on techniques of concentration of measure, we develop a quantitative framework for modeling liquidity risk using convex risk measures. The fundamental objects of study are curves of the form (ρ(λX))λ≥0, where ρ is a convex risk measure and X a random variable, and we call such a curve a \emph{liqu…
A concentration graph associated with a random vector is an undirected graph where each vertex corresponds to one random variable in the vector. The absence of an edge between any pair of vertices (or variables) is equivalent to full conditional independence between these two variables given all the other variables. In…
Quantum kernel methods can lead to trivial models due to exponential concentration of kernel values.
problem Exponential concentration of quantum kernel values can lead to trivial models in QML.
method Analyzing the resources needed to accurately estimate quantum kernel values and identifying four sources of concentration.
result Quantum kernel values can be exponentially concentrated, leading to trivial models.
This paper gives new concentration inequalities for the spectral norm of a wide class of matrix martingales in continuous time. These results extend previously established Freedman and Bernstein inequalities for series of random matrices to the class of continuous time processes. Our analysis relies on a new supermarti…
Concentration of infinitely exchangeable sequences with bounded-difference constants
problem Quantifying uncertainty in AI benchmarks
method Using a mixture-free Hoeffding-type bound
result Tight, mixture-free Hoeffding-type bound for zero-sum linear contrasts
Positive definite kernels and their associated Reproducing Kernel Hilbert Spaces provide a mathematically compelling and practically competitive framework for learning from data. In this paper we take the approximation theory point of view to explore various aspects of smooth kernels related to their inferential proper…
Let F be a Kähler foliation on a compact Riemannian manifold M. we study the properties of infinitesimal automorphisms on (M,F), and in particular we concentrate on the transversal conformal field, transversal projective field and transversally holomorphic field
Study improves fractional posterior for 1-bit matrix completion.
problem Estimating a binary matrix from observed entries.
method Fractional posterior approach with low-rank factorization and spectral scaled Student priors.
result Concentration results for fractional posterior, demonstrating effectiveness in matrix recovery.
We present a novel notion of outlier, called the Concentration Free Outlier Factor, or CFOF. As a main contribution, we formalize the notion of concentration of outlier scores and theoretically prove that CFOF does not concentrate in the Euclidean space for any arbitrary large dimensionality. To the best of our knowled…
Develops a fast variational approximation for high-dimensional empirical Bayes posteriors.
problem Optimal posterior computation in high-dimensional settings with prior tails effect.
method Variational approximation of empirical Bayes posterior with data-driven centers and thin-tailed conjugate priors.
result Retains optimal concentration rate properties and superior performance compared to existing methods.
We study the isoperimetric, functional and concentration properties of n-dimensional weighted Riemannian manifolds satisfying the Curvature-Dimension condition, when the generalized dimension N is negative, and more generally, is in the range N∈(−∞,1), extending the scope from the traditional range $N \i…
New analysis proves sketching operators' RIP guarantees for mixture models without importance sampling.
problem Proving sketching operators' Restricted Isometry Property (RIP) for mixture models without assuming importance sampling.
method Proposed alternative analysis based on new deterministic bounds and concentration inequalities.
result Theoretical guarantees for sketching operators without importance sampling.
This paper formalizes Uniswap v3 using PTA and FST for rigorous analysis.
problem Formal modeling of Uniswap v3's concentrated liquidity for rigorous analysis.
method Formal state machine models using PTA and FST, proving rounding bounds.
result Formal justification of Uniswap v3's ε-slack and rounding safety. NUTS mixing time scales as d^(1/4) for Gaussian distributions.
problem Improving the efficiency of the No-U-Turn Sampler (NUTS) for Gaussian distributions.
method Coupling argument leveraging geometric structure of Gaussian concentration, uniformity analysis of NUTS transitions.
result The mixing time of NUTS scales as d^(1/4) for Gaussian distributions, up to logarithmic factors.
Random SNNs are stable and simple, with low-frequency Fourier spectra.
problem Stability and robustness of spiking neural networks.
method Boolean function analysis and Fourier spectrum concentration.
result Random LIF-SNNs are stable and biased towards simple functions.
New property ensures neural networks generalize well with limited data.
problem Limited training data limits model generalization in neural networks.
method Introduces NeuRIP, a uniform concentration event for ReLU networks.
result All shallow ReLU networks generalize uniformly if they achieve NeuRIP.
New algorithm for countable bandits with optimal regret.
problem Stochastic bandit problem with countably many arms.
method Fully adaptive online learning algorithm with O(log n) expected cumulative regret.
result Achieves optimal regret of O(log n) after any number of plays n.
Many modern machine learning classifiers are shown to be vulnerable to adversarial perturbations of the instances. Despite a massive amount of work focusing on making classifiers robust, the task seems quite challenging. In this work, through a theoretical study, we investigate the adversarial risk and robustness of cl…
Theory for algebraic data on categories via concentration structures.
problem Defining algebraic structures on categories.
method Introducing concentration structures and concentration monoids.
result Every group can be represented as a concentration monoid of a trivial category.
The special linear groups, the mapping class groups of surfaces, the outer autormorphism groups of free groups appear in numerous domains. Their analogies, developped in particular in K. Vogtmann's work, have been written about a lot. In this report, we concentrate on the contractible spaces on which these groups act i…
We find a deterministic equivalent for random feature regression's test error, independent of feature map dimension.
problem Understanding the generalization performance of random feature ridge regression.
method We derive a deterministic equivalent for the test error of RFRR under a concentration property, showing it can be approximated by a closed-form expression dependent on feature map eigenvalues.
result Our approximation guarantee is non-asymptotic, multiplicative, and independent of the feature map dimension, providing a tight result for the smallest number of features achieving optimal minimax error rate.
The paper develops new inequalities for Markov chain sums, linking them to mixing time.
problem Establishing concentration inequalities for Markov chain sums.
method Developed novel concentration inequalities for geometrically ergodic Markov chains, linking bounds to mixing time constants.
result Explicit bounds for additive functionals of Markov chains, linked to Rosenthal inequality constants and mixing properties.