Discrete Gaussian noise preserves privacy and accuracy in differential privacy.
problem Finite computers cannot represent continuous Gaussian noise, leading to privacy breaches and loss of interpretability.
method Introduced and analyzed discrete Gaussian noise, providing privacy and accuracy guarantees similar to continuous Gaussian noise.
result Discrete Gaussian noise offers the same privacy and accuracy as continuous Gaussian noise, with efficient sampling algorithms.
A system for federated learning with private data, adding discrete Gaussian noise and secure aggregation.
problem Training models on private data distributed across devices while ensuring privacy.
method Discretizes data, adds discrete Gaussian noise, and uses secure aggregation to protect privacy.
result Matches the accuracy of central differential privacy with less than 16 bits of precision per value.
Simplified analysis of diffusion models using discrete random variables.
problem Theoretical analysis of diffusion models is complex and requires rigorous proofs.
method Simplified framework for analyzing Euler--Maruyama discretization of VP-SDEs using Grönwall's inequality.
result Standard Gaussian noise can be replaced by discrete random variables without sacrificing convergence guarantee.
Safety filter for unknown discrete-time systems with learned models and noise covariance.
problem Ensuring safety for unknown discrete-time linear systems with Gaussian noise.
method Develops a learning-based safety filter using empirical model and noise covariance, optimizing control actions to stay within safety constraints.
result Minimally modifies nominal control actions to ensure safety with high probability, tightening constraints as more data is collected.
New geometric SDEs and discretizations on Riemannian manifolds with error bounds.
problem Modeling diffusion processes on Riemannian manifolds with geometric SDEs.
method Introduced a new construction of geometric SDEs and provided non-asymptotic error bounds.
result First non-asymptotic error bound for geometric Euler-Murayama discretization.
Discrete noise improves graph generation quality and speed.
problem Generating high-quality discrete graph samples.
method Using discrete noise in diffusion models for graph generation.
result Discrete noise leads to 1.5x better MMDs and 30x faster sampling.
D2P-Fed improves privacy and communication in federated learning.
problem Achieving both differential privacy and communication efficiency in federated learning.
method Applying discrete Gaussian noise to private data transmission.
result D2P-Fed outperforms state-of-the-art by 4.7% to 13.0% in model accuracy with one-third less communication cost.
A new RG approach connects discrete and continuous time descriptions of Gaussian processes.
problem Discretization of continuous stochastic processes for accurate simulation or model inference.
method Renormalization Group (RG) approach for Gaussian time series generated by auto-regressive models.
result RG fixed points correspond to discretizations of linear SDEs, providing insights into process accuracy.
Neural Ordinary Differential Equation (Neural ODE) has been proposed as a continuous approximation to the ResNet architecture. Some commonly used regularization mechanisms in discrete neural networks (e.g. dropout, Gaussian noise) are missing in current Neural ODE networks. In this paper, we propose a new continuous ne…
CANDI solves the gap between continuous and discrete diffusion models for text generation.
problem Underperformance of continuous diffusion models in discrete data domains.
method Introduces token identifiability and a hybrid framework (CANDI) to decouple discrete and continuous corruption.
result CANDI successfully avoids temporal dissonance, enabling continuous diffusion benefits for discrete spaces.
Estimating causal models from observational data is a crucial task in data analysis. For continuous-valued data, Shimizu et al. have proposed a linear acyclic non-Gaussian model to understand the data generating process, and have shown that their model is identifiable when the number of data is sufficiently large. Howe…
Method estimates mixture components without discretizing parameters.
problem Learning from mixtures of continuous features with noise.
method Off-the-grid optimization method for continuous parameter space.
result Prediction error bound similar to Lasso predictor.
The paper improves privacy accounting for discrete-valued mechanisms and the subsampled Gaussian mechanism.
problem Improving the accuracy and efficiency of differential privacy accounting for discrete outputs.
method Uses fast Fourier transform (FFT) for rigorous error analysis and accounting of privacy loss.
result Provides strict lower and upper bounds for (ε,δ)-values, demonstrating up to 75% reduction in noise variance. Efficiently calculates privacy guarantees for 2020 Census data.
problem Evaluate privacy guarantees for 2020 U.S. Census data releases.
method Sieve-accelerated quadrature method to evaluate tail probabilities of high-dimensional convolutions.
result Achieves 1,824-fold speedup over prior methods while maintaining error tolerances.
Improved privacy-preserving statistical estimates with customizable noise reduction.
problem Balancing privacy and accuracy in statistical estimation.
method Introducing the Brownian mechanism, which adds Gaussian noise to a sequence of estimates, gradually reducing it based on the practitioner's needs.
result The Brownian mechanism produces more accurate estimates while maintaining strong privacy guarantees, outperforming existing methods.
Computing accurate estimates of the Fourier transform of analog signals from discrete data points is important in many fields of science and engineering. The conventional approach of performing the discrete Fourier transform of the data implicitly assumes periodicity and bandlimitedness of the signal. In this paper, we…
Neural networks estimate time-varying parameters in AR(p) models with different noise types.
problem Forecasting time-dependent parameters in AR(p) processes with varying noise.
method Deep learning for time-varying coefficients, Gaussian and Laplace noise models.
result Simple model with time-varying parameters can effectively forecast complex dynamics.
SGD handles label noise with bounds improving over SGLD.
problem Label noise in non-convex optimization.
method Stochastic gradient descent with uniform dissipativity and smoothness conditions, using Wasserstein distance and algorithmic stability.
result Generalization error bounds with a rate of n−2/3, better than SGLD's n−1/2. In this manuscript we introduce numerical Gaussian process Kalman filtering (GPKF). Numerical Gaussian processes have recently been developed to simulate spatiotemporal models. The contribution of this paper is to embed numerical Gaussian processes into the recursive Kalman filter equations. This embedding enables us t…
We prove quantitative convergence rates at which discrete Langevin-like processes converge to the invariant distribution of a related stochastic differential equation. We study the setup where the additive noise can be non-Gaussian and state-dependent and the potential function can be non-convex. We show that the key p…
Adversarial perturbations dramatically decrease the accuracy of state-of-the-art image classifiers. In this paper, we propose and analyze a simple and computationally efficient defense strategy: inject random Gaussian noise, discretize each pixel, and then feed the result into any pre-trained classifier. Theoretically,…
Study on discrete Gaussian curvature for polyhedral surfaces.
problem Discretization of Gaussian curvature for polyhedral surfaces.
method Generalization of discrete conformal equivalence to define discrete Gaussian curvature and classify polyhedral surfaces.
result Existence of polyhedral surfaces with constant discrete Gaussian curvature in every discrete conformal class.
Unified framework for discrete diffusion modeling with flexible noising processes.
problem Efficient modeling of large discrete state spaces with arbitrary corruption dynamics.
method Generalized Discrete Diffusion from Snapshots (GDDS) framework that supports uniformization for fast noising and snapshot-based ELBO for reverse process.
result GDDS outperforms existing discrete diffusion methods in training efficiency and generation quality.
This paper approximates SA iterates using Gaussian distributions for tail bounds.
problem Characterizing the distribution of stochastic approximation iterates in finite time.
method Approximating pre-limit distributions of SA iterates by Gaussian sequences with recursively defined covariances.
result Explicit bounds on the Wasserstein-1 distance between rescaled iterates and Gaussians.
Study of Ricci flow on discrete surfaces of revolution with constant Gaussian curvature.
problem Understanding Ricci flow on discrete surfaces of revolution.
method Explicit parametrizations and Ricci flow analysis for discrete surfaces of revolution.
result Discrete surfaces of revolution approach constant Gaussian curvature under Ricci flow.
New method approximates Gaussian curvature on discrete surfaces.
problem Approximating solutions to the prescribed Gaussian curvature problem.
method Discrete conformality and convex functional minimization.
result Efficient numerical method to compute solutions.
The Kalman filter is extensively used for state estimation for linear systems under Gaussian noise. When non-Gaussian Lévy noise is present, the conventional Kalman filter may fail to be effective due to the fact that the non-Gaussian Lévy noise may have infinite variance. A modified Kalman filter for linear systems wi…
Discretizations of Langevin diffusions provide a powerful method for sampling and Bayesian inference. However, such discretizations require evaluation of the gradient of the potential function. In several real-world scenarios, obtaining gradient evaluations might either be computationally expensive, or simply impossibl…
FLDD improves discrete diffusion models by learning a non-Markovian noising process.
problem Efficiency and quality of discrete diffusion models in few-step generation.
method Introduces a learnable non-Markovian forward (noising) process to match the target distribution.
result FLDD produces higher quality samples in fewer steps compared to conventional discrete diffusion models.
Differential privacy is a framework for privately releasing summaries of a database. Previous work has focused mainly on methods for which the output is a finite dimensional vector, or an element of some discrete set. We develop methods for releasing functions while preserving differential privacy. Specifically, we sho…
Improved analysis for diffusion models reduces KL divergence error dependence on data dimension and discretization step size.
problem Analyze the convergence of diffusion-based generative models under minimal assumptions.
method Model the generation process as a composition of reverse ODE and noising steps, leveraging Wasserstein-type error control and noise addition.
result Achieved a linear dependence on data dimension and improved dependence on discretization step size for KL divergence error.
Stochastic gradient descent (SGD) has been widely used in machine learning due to its computational efficiency and favorable generalization properties. Recently, it has been empirically demonstrated that the gradient noise in several deep learning settings admits a non-Gaussian, heavy-tailed behavior. This suggests tha…
The paper introduces a new discretization of Gaussian curvature on surfaces.
problem Discretizing Gaussian curvature on surfaces with conic singularities.
method Discrete conformal theory and variational principles with constraints.
result Established a discrete uniformization theorem for surfaces with non-positive Euler number.
Study provides convergence guarantees for discrete diffusion models on finite and infinite state spaces.
problem Challenges in understanding discrete diffusion models on combinatorial state spaces.
method Established convergence bounds for three discrete diffusion models using Euler approximations.
result Optimal non-asymptotic convergence guarantees for discrete diffusion models without boundedness assumptions.
When stock prices are observed at high frequencies, more information can be utilized in estimation of parameters of the price process. However, high-frequency data are contaminated by the market microstructure noise which causes significant bias in parameter estimation when not taken into account. We propose an estimat…
PBM mechanism improves privacy and accuracy in federated learning.
problem Secure and private federated learning with limited privacy budget.
method Poisson Binomial mechanism for discrete differential privacy.
result Achieves same privacy-accuracy trade-offs as Gaussian mechanism.
Noise in SGD affects overparameterized models, favoring sparse solutions.
problem Understanding and mitigating implicit bias in SGD with parameter-dependent noise.
method Theoretical analysis of a quadratically-parameterized model with label noise and Gaussian noise.
result SGD with label noise recovers sparse ground-truth solutions, while SGD with Gaussian noise overfits dense solutions.
New bounds for kernel regression under non-Gaussian noise.
problem Uncertainty quantification for function estimates from noisy observations.
method Novel non-asymptotic probabilistic uniform error bounds for kernel-based regression.
result Proposed bounds apply to a broad class of non-Gaussian noise distributions.
We revisit the Bayesian online inference problems for the linear dynamic systems (LDS) under non- Gaussian environment. The noises can naturally be non-Gaussian (skewed and/or heavy tailed) or to accommodate spurious observations, noises can be modeled as heavy tailed. However, at the cost of such noise robustness, the…
Generalized R2R handles non-Gaussian noise for deep network training.
problem Training deep networks from noisy data alone.
method Extending R2R to handle various noise distributions.
result GR2R loss is an unbiased estimator of supervised loss.
The chapter compares Gaussian process models for stochastic simulators with varying noise.
problem Modeling stochastic simulators with varying noise.
method Various Gaussian process models are compared, including input varying noise variance, non-Gaussian noise, and quantile modeling.
result Sequential design procedures are adapted for these models.
Unified discrete diffusion for categorical data simplifies training and sampling.
problem Training and sampling in discrete diffusion models for categorical data.
method Mathematical simplifications and elegant unification of discrete-time and continuous-time discrete diffusion.
result Unified Simplified Discrete Denoising Diffusion (USD3) outperforms SOTA baselines.
In recent years, correntropy and its applications in machine learning have been drawing continuous attention owing to its merits in dealing with non-Gaussian noise and outliers. However, theoretical understanding of correntropy, especially in the statistical learning context, is still limited. In this study, within the…
Conservation laws improve diffusion model training by optimizing likelihood.
problem Training diffusion models with denoising objectives.
method Developed conservation laws based on GEXIT functions for memoryless noise processes.
result Unified characterization of diffusion model likelihood, reducing training to learning marginal posteriors.
DSPM models control noise volatility, improving financial data analysis.
problem Financial returns exhibit volatility clustering, challenging traditional models.
method DSPM uses a tempered-stable subordinator to control noise volatility, preserving kurtosis and autocorrelation.
result DSPM models accurately capture volatility clustering and noise mechanisms.
WS diffusion models handle anisotropic Gaussian noise better than conventional methods.
problem Handling anisotropic Gaussian noise in imaging inverse problems.
method Whitened Score (WS) diffusion models based on stochastic differential equations.
result WS DMs outperform conventional DMs on anisotropic Gaussian noise.
We focus on variational inference in dynamical systems where the discrete time transition function (or evolution rule) is modelled by a Gaussian process. The dominant approach so far has been to use a factorised posterior distribution, decoupling the transition function from the system states. This is not exact in gene…
Real-world measurement noise in applications like robotics is often correlated in time, but we typically assume i.i.d. Gaussian noise for filtering. We propose general Gaussian Processes as a non-parametric model for correlated measurement noise that is flexible enough to accurately reflect correlation in time, yet sim…