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 ℓ 1 \ell_1 ℓ 1 -regularized negative binomial regressions. result Oracle inequalities and consistency for Lasso estimators of partial regression coefficients.
Extends Gaussian process approach to handle linear inequality constraints.
problem Real-world problems with inequality constraints.
method Finite-dimensional Gaussian approach with linear inequality constraints, MCMC techniques.
result Efficient results on data fitting and uncertainty quantification.
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.
Novel GP-modulated Cox process framework with linear inequality constraints.
problem Modeling point patterns with positiveness and inequality constraints.
method Directly impose positiveness and inequality constraints on the Gaussian process without restrictions on covariance functions.
result Accurate inference of intensity functions with improved results for monotonic processes.
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.
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 ∞ l_\infty l ∞ norm of vector linear processes with sub-Weibull, mixingale innovations. result Obtained concentration bounds for the maximum entrywise norm of lag- h h h autocovariance matrices. This paper approximates Gaussian process emulators with constraints and noisy data.
problem Realistic stochastic emulators with inequality constraints and noisy observations.
method Monte Carlo and Markov Chain Monte Carlo methods with noise term.
result Improved performance of MC and MCMC samplers with noisy observations and constraints.
Enhances Doob's inequality for sub-martingales.
problem Fundamental importance of Doob's inequality in stochastic process theory.
method Generalization of Doob's L p L^p L p inequality for sub-martingales. result A tighter estimate on Doob's inequality for sub-martingales.
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.
Paper connects tensor regression and Gaussian processes for multi-way data analysis.
problem Learning high-order correlations from multi-way data.
method Demonstrates connections between low-rank tensor regression and Gaussian processes, proving oracle inequality and learning curve.
result Low-rank tensor regression is equivalent to constrained Bayesian inference in Gaussian processes, with learning dependent on eigenvalues and variable correlations.
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…
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…
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 Φ Φ -divergence.
problem Analyzing convergence rates of Langevin dynamics and Proximal Sampler.
method Extending mixing time analyses to Φ Φ Φ -divergence, using strong data processing inequalities. result Convergence of Φ Φ Φ -divergence to 0 exponentially fast along Unadjusted Langevin Algorithm and Proximal Sampler. 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.
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 ) Ω(d) Ω ( d ) bits of training data information must be memorized for O ( 1 ) O(1) O ( 1 ) examples, decaying with example growth. New measures generalize existing ones, linking information and risk.
problem Linking information measures and risk in statistical decision problems.
method Introducing new families of divergence measures and deriving an information processing equality.
result Extension of variational φ φ φ -divergence representation to multiple distributions. Improved analysis of UCRL2 with empirical Bernstein inequality reduces exploration-exploitation regret.
problem Exploration-exploitation in communicating Markov Decision Processes.
method Analysis of UCRL2 with Empirical Bernstein inequalities (UCRL2B).
result Regret bound of O ~ ( D Γ S A T ) \widetilde{O}(\sqrt{DΓS A T}) O ( D Γ S A T ) for UCRL2B. We study the tradeoff between the statistical error and communication cost of distributed statistical estimation problems in high dimensions. In the distributed sparse Gaussian mean estimation problem, each of the m m m machines receives n n n data points from a d d d -dimensional Gaussian distribution with unknown mean θ θ θ w…
Paper improves risk bound for MTL with graph-dependent data.
problem Sub-optimal risk bound in multi-task learning with graph-dependent data.
method Proposes a new Bennett-type inequality and develops new Talagrand-type inequality and local fractional Rademacher complexity.
result Derives a sharper risk bound of O ( log n n ) O(\frac{\log n}{n}) O ( n l o g n ) . Gradual training and gradient clipping improve RNN performance.
problem RNNs are hard to train and prone to overfitting.
method Formulated RNN as a Markov chain, gradually trained, and used layer-wise gradient clipping.
result Improvements in language modeling tasks.
Asymptotically consistent clustering algorithms for ergodic stochastic processes are developed.
problem Clustering stochastic processes with consistency guarantees.
method Review and development of clustering algorithms for ergodic stochastic processes.
result Asymptotically consistent clustering algorithms can be obtained for ergodic stochastic processes.
New sparse Gaussian process method tackles unconstrained regression problems.
problem Dealing with physical systems that satisfy inequality constraints.
method Extends constrained Gaussian process by redefining hat basis functions.
result Reduces computational complexity from O ( n 3 ) O(n^{3}) O ( n 3 ) to O ( n m 2 ) O(nm^{2}) O ( n m 2 ) . This paper investigates the supervised learning problem with observations drawn from certain general stationary stochastic processes. Here by \emph{general}, we mean that many stationary stochastic processes can be included. We show that when the stochastic processes satisfy a generalized Bernstein-type inequality, a u…
The paper improves NBR for count data using elastic-net regularization, achieving consistency and weak signal detection.
problem Sparse negative binomial regression for count data with non-asymptotic advantages.
method Elastic-net estimator with oracle inequalities derived under Compatibility Factor Condition and Stabil Condition.
result Sign consistency and grouping effect with high probability, and true variable set recovery under certain conditions.
Dynamic fairness tackles long-term inequalities in decision making processes.
problem Machine learning models can reproduce and exacerbate human bias, leading to discriminatory outcomes.
method Theoretical model considering time dynamics of decision making processes and fairness definitions.
result Demographic parity is the only fairness notion that avoids long-term inequalities and leads to accurate and fair classification.
New entropy measures reveal information flow in CNNs without approximations.
problem Understanding information flow in convolutional neural networks (CNNs).
method Developed new entropy estimators based on Renyi's α-entropy and applied PID framework.
result Validated fundamental data processing inequalities and revealed properties of CNN training.
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.
The paper proves concentration inequalities for two-sample rank processes and applies them to ranking performance criteria.
problem Measuring the performance of ranking statistics between two populations.
method Proves concentration inequalities for two-sample rank processes indexed by VC classes of scoring functions.
result Generalization capacity of empirical maximizers of ranking performance criteria is investigated.
The paper provides a finite-sample deviation bound for stable autoregressive processes.
problem Deviation bounds for least squares estimators in Gaussian AR(n) processes.
method Utilizes martingale concentration inequalities and tail-bound for χ² distributed variables.
result Problem-dependent finite-time bound on the deviation probability of AR(n) process parameters.
In this paper, we propose a novel framework to analyze the theoretical properties of the learning process for a representative type of domain adaptation, which combines data from multiple sources and one target (or briefly called representative domain adaptation). In particular, we use the integral probability metric t…
Let $({\M}, g(t))$ be a Kähler Ricci flow with positive first Chern class. We prove a uniform isoperimetric inequality for all time. In the process we also prove a Cheng-Yau type log gradient bound for positive harmonic functions on $({\M}, g(t))$ , and a Poincaré inequality without assuming the Ricci curvature is bound…
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).
The paper proves a diastolic inequality linking surface area and loop length.
problem Finding short geodesics on Riemannian surfaces.
method Proving a universal inequality between diastole and area of closed surfaces.
result Every Riemannian surface can be decomposed into two domains with bounded boundary length.
Paper outlines a new mathematical language for experiments.
problem Formalizing the scientific process for automation.
method Formulates the scientific process in precise mathematical language.
result Novel contributions in data processing, bias variance, and deficiency.
Microstructure of market dynamics is studied through analysis of tick price data. Linear trend is introduced as a tool for such analysis. Trend arbitrage inequality is developed and tested. The inequality sets limiting relationship between trend, bid-ask spread, market reaction and average update frequency of price inf…
Study improves self-normalized bounds for vector-valued processes beyond sub-Gaussianity.
problem Limited understanding of self-normalized concentration for vector-valued processes outside sub-Gaussian frameworks.
method Developed concentration inequalities for self-normalized processes with light tails (e.g., Bennett, Bernstein bounds) for vector-valued data.
result Provided new insights and bounds for self-normalized processes with non-sub-Gaussian distributions.
We prove structure theorems for complete manifolds satisfying both the Ricci curvature lower bound and the weighted Poincaré inequality. In the process, a sharp decay estimate for the minimal positive Green's function is obtained. This estimate only depends on the weight function of the Poincaré inequality, and yields …
Develops robust MDPs for unknown disturbances with performance guarantees.
problem Unknown disturbance distribution in MDPs.
method Empirical distribution, sublevel set of distance function, weak convergence, concentration inequality.
result Robust optimal value function converges to true optimal value function with increasing sample sizes.
Financial investment returns lead to growing wealth inequality.
problem Recent rise in wealth inequality in active financial markets.
method Minimalist modelling strategy combining financial markets, wealth accumulation, and compound interest.
result Accumulated financial investment returns cause ever-increasing wealth concentration and inequality.
Probability distributions of money, income, and energy consumption per capita are studied for ensembles of economic agents. The principle of entropy maximization for partitioning of a limited resource gives exponential distributions for the investigated variables. A non-equilibrium difference of money temperatures betw…
We improve bounds for stochastic processes, especially those with heavy tails.
problem Bounding the concentration of sub- ψ ψ ψ processes with heavy tails. method Variational approach to concentration, focusing on sub-Gaussian and other tail conditions.
result First dimension-free self-normalized empirical Bernstein inequality.