This work investigates the properties of Gaussian-smoothed sliced divergences for comparing distributions.
problem Comparing probability distributions while preserving privacy.
method Investigates the theoretical properties of Gaussian-smoothed sliced Wasserstein distance and generalized versions.
result Gaussian smoothed sliced Wasserstein distance converges with a rate of \(O(n^{-1/2})\).
Solves Lp-Gaussian chord Minkowski problem using Gauss curvature flow.
problem Solving the Lp-Gaussian chord Minkowski problem. method Using Gauss curvature flow to obtain smooth even solutions.
result Obtains smooth even solutions to the Lp-Gaussian chord Minkowski problem. We present a general probabilistic perspective on Gaussian filtering and smoothing. This allows us to show that common approaches to Gaussian filtering/smoothing can be distinguished solely by their methods of computing/approximating the means and covariances of joint probabilities. This implies that novel filters and …
Smoothed SGD improves quantile estimation without crossing curves.
problem Estimating quantiles without crossing estimated curves.
method Smoothed SGD algorithm with Bahadur representation and Gaussian approximation.
result Smoothed SGD provides non-asymptotic tail probability bounds and a Gaussian approximation for quantile estimates.
Smooth Schrödinger Bridges improve trajectory inference by smoothing Gaussian processes.
problem Improving trajectory inference in applications like particle tracking.
method Generalizes Schrödinger Bridge problem to smooth Gaussian processes, solving the problem on phase space.
result The method outperforms existing methods on real datasets.
Proves existence and uniqueness of solutions to the Lp Gaussian Minkowski problem.
problem Existence and uniqueness of solutions to the Lp Gaussian Minkowski problem.
method Analyzes existence and uniqueness of solutions for different values of p.
result Existence and uniqueness of smooth solutions for p > n.
New method provides tighter robustness guarantees for adversarial attacks.
problem Ensuring robustness against adversarial attacks in machine learning models.
method Developed a Second-order Smoothing (SoS) robustness certificate using Gaussian random smoothing.
result SoS certificates are tighter and provide improved robustness on high-dimensional datasets.
A comprehensive methodology is provided for smoothing noisy, irregularly sampled data with non-Gaussian noise using smoothing splines. We demonstrate how the spline order and tension parameter can be chosen a priori from physical reasoning. We also show how to allow for non-Gaussian noise and outliers which are typical…
The paper establishes bounds on the smoothness parameter in Gaussian process interpolation.
problem Estimating the smoothness parameter in Gaussian process models.
method Approximation theory in Sobolev spaces and general theorems on parameter estimation.
result Maximum likelihood estimation recovers the true smoothness for certain classes of functions.
Bounds on Gaussian approximation for neural networks with novel smoothing techniques.
problem Approximating the distribution of wide random neural networks.
method Stein's method, Gaussian smoothing, Laplacian operators, Cameron-Martin space.
result First bounds on Gaussian approximation of wide random neural networks.
GPCDL uses Gaussian Processes to learn smooth templates from data.
problem Lack of smoothness in learned templates leads to overfitting and poor predictive performance.
method GPCDL incorporates Gaussian Process priors to enforce smoothness in the learned templates.
result GPCDL outperforms unregularized CDL in accuracy and predictive performance across various SNRs and applications.
Regularization improves robustness of smoothed classifiers.
problem Certifying robustness of smoothed classifiers.
method Regularizing prediction consistency over Gaussian noise.
result Significantly improved certified robustness with less training costs.
New framework improves adversarial robustness certification for various perturbations.
problem Certifying robustness against adversarial attacks in deep learning models.
method Unified functional optimization approach with non-Gaussian smoothing noise for multiple types of attacks.
result Achieves better certification results and identifies key trade-offs between accuracy and robustness.
Randomized smoothing, using just a simple isotropic Gaussian distribution, has been shown to produce good robustness guarantees against ℓ2-norm bounded adversaries. In this work, we show that extending the smoothing technique to defend against other attack models can be challenging, especially in the high-dimensi…
Unified framework for ensemble transport-based smoothing of non-Gaussian time series.
problem Bayesian time series re-analysis with non-Gaussian distributions.
method Measure transport approach to derive consistent prior-to-posterior transformations.
result General ensemble framework for transport-based smoothing of state-space models.
New method for robust fixed-point smoothing without state augmentation.
problem Estimating initial states in Gaussian smoothing algorithms.
method Cholesky-based formulation without state augmentation.
result Matches runtime and robustness of existing methods.
Improved sampling from high-dimensional Gaussians using smoothed scores.
problem Sampling from high-dimensional Gaussian distributions with gradient information.
method Using smoothed scores, which are gradients of the logarithms of Gaussian-convolved densities, to overcome approximation barriers.
result Improved sampling efficiency with a complexity of \(O\left(\left(\logκ+\log(e\sqrt d/δ_{
m TV})
ight)\log(e\sqrt d/δ_{
m TV})
ight)\) smoothed-score queries.
Paper provides GOT convergence guarantees for sub-gamma distributions and dependent samples.
problem Estimating GOT distance under general settings.
method Gaussian-smoothed optimal transport (GOT) framework, sub-gamma distributions, dependent samples, kernel MMD distances.
result Convergence guarantees for GOT distance under more general settings.
Improved model for non-smooth signals with complex spectra.
problem Current models struggle with non-smooth signals and complex spectral structures.
method CGPCM and RGPCM models with causality and Bayesian nonparametric interpretations, improved variational inference.
result Proposed models show better performance on synthetic and real-world data.
This paper aims at refined error analysis for binary classification using support vector machine (SVM) with Gaussian kernel and convex loss. Our first result shows that for some loss functions such as the truncated quadratic loss and quadratic loss, SVM with Gaussian kernel can reach the almost optimal learning rate, p…
Smooth orbit equivalence proves metric equivalence for geodesic flows.
problem Proving metric equivalence for geodesic flows under orbit equivalence.
method Proving metric equivalence for geodesic flows under orbit equivalence.
result Smooth orbit equivalence implies conformal equivalence of metrics.
Study on Gaussian random fields' singularities on manifolds.
problem Understanding singularities of Gaussian random fields on manifolds.
method Computed expected values of singularities under various conditions.
result Explicit formulae for singularities under different constraints.
Bayesian optimization tackles non-smooth tuning problems.
problem Optimizing black-box functions with non-smoothness and limited samples.
method Proposed a clustered Gaussian process (cGP) model for non-smooth optimization.
result Improvement of up to 90% in performance for repetitive experiments.
Introduces a new theoretical framework for exponential smoothing.
problem Theoretical foundation and robustness of simple exponential smoothing.
method Stochastic gradient ascent to optimize Gaussian log-likelihood functions.
result Simple exponential smoothing converges to the trend of a trend-stationary process.
We study the problem of isometrically embedding a two-dimensional Riemannian manifold into Euclidean three-space. It is shown that if Gaussian curvature vanishes to finite order and its zero set consists of two smooth curves tangent at a point, then local sufficiently smooth isometric embedding exists.
We propose a principled algorithm for robust Bayesian filtering and smoothing in nonlinear stochastic dynamic systems when both the transition function and the measurement function are described by non-parametric Gaussian process (GP) models. GPs are gaining increasing importance in signal processing, machine learning,…
New findings show learning deeper neural networks is hard even with Gaussian inputs and non-degenerate weights.
problem The computational complexity of learning neural networks, especially deeper ones.
method Smoothed analysis framework and local pseudorandom generators.
result Learning depth-3 ReLU networks under Gaussian input distribution is hard even if weight matrices are non-degenerate.
We show how to turn any classifier that classifies well under Gaussian noise into a new classifier that is certifiably robust to adversarial perturbations under the ℓ2 norm. This "randomized smoothing" technique has been proposed recently in the literature, but existing guarantees are loose. We prove a tight robu…
Unified smoothing for robust classification improves accuracy.
problem Improving robustness of classifiers against adversarial attacks.
method Learned smoothed densities and randomized smoothing.
result Provable robust accuracies higher than state-of-the-art defenses.
Develops state-space deep Gaussian processes for irregular signals.
problem Solving deep Gaussian process regression problems for irregular signals/functions.
method Represent DGPs as SDEs, solve using state-space filtering and smoothing methods.
result Rich class of priors compatible with irregular signals/functions.
Generative models improved with smoothed score functions for better sample quality.
problem Improving generative models for better sample quality.
method Smoothed score functions based on factorial Gaussian kernels.
result Single noise level achieved 14.15 Fréchet inception distance on CIFAR-10.
Nonparametric tests via kernel embedding of distributions have witnessed a great deal of practical successes in recent years. However, statistical properties of these tests are largely unknown beyond consistency against a fixed alternative. To fill in this void, we study here the asymptotic properties of goodness-of-fi…
Generative models use kernel smoothing for conditioning on small example sets.
problem Improving generative models' performance with limited conditioning examples.
method Showed that cross-attention conditioning is equivalent to kernel smoothing, specifically a Nadaraya--Watson kernel smoother.
result The approach predicts and confirms three failure regimes for kernel-based conditioning.
The paper solves a geometric problem using curvature flow and variational methods.
problem The Lp-Gaussian Minkowski problem in the Euclidean space. method Gauss curvature flow and Aleksandrov's variational method with Lagrange multipliers.
result The flow converges to a smooth solution of the Lp-Gaussian Minkowski problem. We study stable smooth solutions to the isoperimetric type problem for a Gaussian weight on Euclidean Space. That is, we study hypersurfaces Σn⊂Rn+1 that are second order stable critical points of compact variations that minimize Gaussian weighted area and preserve Gaussian weighted volume. We sho…
The paper proposes a method to model non-smooth functions using clustering, classification, and Gaussian process modeling.
problem Modeling discontinuities and non-smoothness in expensive computational models.
method Three-stage approach combining clustering, classification, and Gaussian process modeling.
result The approach successfully models discontinuities and non-smoothness in various functions.
Algorithm learns polynomial transformations of Gaussian distributions.
problem Learning high-dimensional polynomial transformations of Gaussian distributions.
method Polynomial-time algorithms for smoothed settings, tensor ring decomposition.
result First end-to-end guarantees for learning pushforwards under neural networks.
This paper assesses Gaussian and Exponential mechanisms for certifying adversarial robustness.
problem Certifying adversarial robustness using randomized smoothing mechanisms.
method Proposes a generic framework to assess the appropriateness of randomized smoothing mechanisms.
result Gaussian mechanism is an appropriate option for certifying both ℓ2-norm and ℓ∞-norm robustness. Improved regret bounds for structured linear contextual bandits with Gaussian noise.
problem Optimizing bandit learning algorithms for structured contexts with Gaussian perturbations.
method Proposed simple greedy algorithms for structured linear contextual bandits with Gaussian noise.
result Unified regret analysis for structured parameters with geometric quantities as bounds.
The paper is concerned with non-linear Gaussian filtering and smoothing in continuous-discrete state-space models, where the dynamic model is formulated as an Itô stochastic differential equation (SDE), and the measurements are obtained at discrete time instants. We propose novel Taylor moment expansion (TME) Gaussian …
The Kazdan-Warner problem is solved for Riemann surfaces with smooth boundaries.
problem Realizing smooth functions as Gaussian and geodesic curvatures on compact Riemann surfaces.
method Existence results of Brezis-Merle type equations and uniformization theorem extension.
result Any smooth function on compact Riemann surface with smooth boundary can be realized as a Gaussian curvature function and any on the boundary as a geodesic curvature function.
Unified framework for efficient Gaussian process inference.
problem Efficient inference in non-conjugate Gaussian process models.
method Combines expectation propagation with linearization for improved efficiency.
result Unified view of various inference schemes, including classical smoothers and EP.
Paper improves SDR estimation speed and conditions.
problem Improving sufficient dimension reduction for multi-index models.
method Estimating expected smoothed gradient outer product.
result Achieves fast parametric convergence rate of Cd⋅n−1/2. Gaussian processes are ubiquitous in machine learning, statistics, and applied mathematics. They provide a flexible modelling framework for approximating functions, whilst simultaneously quantifying uncertainty. However, this is only true when the model is well-specified, which is often not the case in practice. In thi…
Machine learning (ML) models trained by differentially private stochastic gradient descent (DP-SGD) have much lower utility than the non-private ones. To mitigate this degradation, we propose a DP Laplacian smoothing SGD (DP-LSSGD) to train ML models with differential privacy (DP) guarantees. At the core of DP-LSSGD is…
Off-the-shelf Gaussian Process (GP) covariance functions encode smoothness assumptions on the structure of the function to be modeled. To model complex and non-differentiable functions, these smoothness assumptions are often too restrictive. One way to alleviate this limitation is to find a different representation of …
The paper strengthens the classical result of MLE convergence to a Gaussian distribution.
problem The classical result of MLE convergence to a Gaussian distribution.
method Sub-Gaussian concentration and entropic normality of the normalized MLE.
result Entropic central limit theorem for a smoothed version of the estimator.
This paper investigates the nonparametric regression problem using SVMs with anisotropic Gaussian RBF kernels. Under the assumption that the target functions are resided in certain anisotropic Besov spaces, we establish the almost optimal learning rates, more precisely, optimal up to some logarithmic factor, presented …