Paper develops a new kernel approximation framework.
problem High time and space complexity of kernel methods for large datasets.
method Perturbation-based kernel approximation framework using classical perturbation theory.
result Framework generalizes and improves upon existing methods.
The paper studies heat kernels on modified manifolds and bounds their properties.
problem Bounding heat kernels on modified Riemannian manifolds.
method Derives upper bounds and gradient estimates for the heat kernel of ( M , i l d e g ) (M, ilde{g}) ( M , i l d e g ) . result Establishes upper bounds and gradient estimates for the heat kernel of modified manifolds.
Adding node feature kernels improves GCN robustness to graph perturbations.
problem GCNs' robustness to graph perturbations is a concern.
method Introduced random GCN and added node feature kernels to message passing.
result Perturbations of the graph structure can significantly degrade GCN performance.
Paper studies Riesz transform stability under metric perturbations.
problem Stability of Riesz transform boundedness under metric perturbations.
method Derives conditions for stability of L p L^p L p -boundedness of Riesz transform. result Provides counter-examples for instability of Riesz transform boundedness.
We give an alternate proof of the existence of the asymptotic expansion of the Bergman kernel associated to the k k k -th tensor powers of a positive line bundle L L L in a 1 k \frac{1}{\sqrt{k}} k 1 -neighborhood of the diagonal using elementary methods. We use the observation that after rescaling the Kähler potential k φ k\varphi k φ …
Boosts SVMs by perturbing kernels to improve classification of imbalanced and small disjuncts.
problem Class imbalance and small disjuncts in datasets.
method Kernel perturbation to diversify SVMs for boosting, identifying disjuncts.
result Proposed method outperforms state-of-the-art methods on various datasets.
Improved KSD test for better detection of differences in distributions.
problem Low power of KSD test when distributions have same modes but different mixing proportions.
method Perturb the observed sample using Markov transition kernels to improve KSD test power.
result Perturbed KSD test can lead to substantially higher power than the original KSD test.
Improved capsule networks with kernel methods for robustness.
problem Adversarial robustness of capsule networks.
method Kernelized capsule networks using Gaussian processes.
result Improved robustness to adversarial perturbations.
We propose a novel adversarial training method in feature space that improves model robustness and computational efficiency.
problem Improving model robustness against adversarial input perturbations with computational efficiency.
method Shift from input to feature-space perturbations, reformulating the adversarial training problem in reproducing kernel Hilbert spaces, enabling exact solution of inner maximization and efficient optimization.
result The feature-perturbed formulation is a relaxation of the original problem and provides a regularized estimator that adapts to noise and function smoothness.
New robustness test for kernel goodness-of-fit tests.
problem Lack of robustness in existing kernel goodness-of-fit tests.
method Proposes a new robust kernel goodness-of-fit test using kernel Stein discrepancy (KSD) balls.
result First robust kernel goodness-of-fit test addressing both qualitative and quantitative robustness.
New method quantifies uncertainty in kernel models without distributional assumptions.
problem Uncertainty quantification in kernel methods without strong distributional assumptions.
method Gradient perturbation to extract uncertainty information.
result Exact, non-asymptotic confidence regions for kernel models.
Perturbation theory improves nonparametric instrumental variable estimation accuracy.
problem Improving nonparametric instrumental variable estimation accuracy in high-dimensional settings.
method Perturbative approach based on physics perturbation theory, extending kernel ridge methods with higher-order corrections.
result First-order perturbative corrections reduce prediction error by up to 99% in high-dimensional ill-defined cases.
The study analyzes prediction errors in systems with memory kernels, providing bounds and stability results.
problem Prediction errors in stochastic dynamical systems with memory kernels.
method Analysis of generalized Langevin equations (GLEs) with Volterra equations, integrating synchronized noise coupling and weighted norms.
result Prediction discrepancies decay at a rate determined by the memory kernel's decay, quantitatively bounded by kernel estimation errors.
The paper ensures stability of kernel methods under slight changes in probability measure, regularization, and kernel.
problem Stability of kernel-based methods under perturbations of probability measure, regularization, and kernel.
method Conditions for stability are derived based on convex Lipschitz loss functions and smooth kernels.
result Conditions for stability are given under simultaneous changes in probability measure, regularization, and kernel.
Study of regularized least squares in RKKS with indefinite kernels.
problem Asymptotic properties of regularized least squares with indefinite kernels in RKKS.
method Introducing a bounded hyper-sphere constraint, theoretical demonstration of globally optimal solution, modified error decomposition techniques, matrix perturbation theory.
result Derivation of learning rates in RKKS, same as RKHS under certain conditions.
Quantum Kerr learning shows enhancements in convergence and generalization for kernel-based methods.
problem Improving convergence and generalization in kernel-based methods for quantum computing.
method Combining quantum mechanics with neural tangent kernel theory and first-order perturbation theory.
result Quantum enhancements in terms of convergence time and generalization error.
Researchers compute Ricci curvature on noncommutative 3-tori.
problem Calculating Ricci curvature on noncommutative spaces.
method Used Connes' pseudodifferential calculus and localized spectral zeta functions.
result Explicitly computed Ricci curvature and scalar curvatures.
A robust method for multiple kernel learning against adversarial inputs.
problem Certifiably robust learning against adversarial perturbations.
method Distributionally robust optimization with min-max formulation and debiasing techniques.
result The method achieves theoretical guarantees and generalization bounds.
LGKDE learns graph density using neural networks and perturbations.
problem Graph density estimation challenges in capturing structural patterns and semantic variations.
method LGKDE uses graph neural networks to represent graphs as discrete distributions and learns graph metrics via maximum mean discrepancy.
result LGKDE outperforms state-of-the-art baselines in graph anomaly detection.
The paper explores how kernel eigenalignments affect generalization in KRR.
problem Achieving robust generalization in kernel methods.
method Direct connection between generalization and matrix eigenvectors/eigenvalues, focusing on finite-sample settings.
result Strong generalization requires increasing eigenvector alignment, eigenvalue magnitude, or gaps between eigenvalues.
Functional central limit theorem for kernel gradient flow and infinitesimal gradient boosting
problem Fluctuations of boosting processes around their deterministic limit
method Stochastic perturbation analysis of ODEs in Banach spaces
result Rescaled deviations converge to a Gaussian process
In this paper we study the small time asymptotics for the heat kernel on a sub-Riemannian manifold, using a perturbative approach. We then explicitly compute, in the case of a 3D contact structure, the first two coefficients of the small time asymptotics expansion of the heat kernel on the diagonal, expressing them in …
Wider networks improve natural accuracy but worsen perturbation stability, affecting overall robustness.
problem Understanding the tradeoff between natural accuracy and perturbation stability in wider neural networks for adversarial robustness.
method Careful examination of the relationship between network width, robust regularization parameter λ, and perturbation stability using neural tangent kernels.
result Wider networks can achieve better natural accuracy but worse perturbation stability, leading to potentially worse overall model robustness.
Dynamic programming helps manage fisheries affected by random disturbances.
problem Managing fisheries affected by random disturbances.
method Dynamic programming approach to analyze and optimize management strategies.
result Dynamic programming is crucial for fully characterizing optimal harvest strategies.
For any n-dimensional compact Riemannian manifold (M,g), we construct a canonical t-family of isometric embeddings I_{t}: M->R^{q(t)}, with t>0 sufficiently small and q(t)>>t^{-n/2}. This is done by intrinsically perturbing the heat kernel embedding introduced in [BBG]. As t->0, asymptotic geometry of the embedded imag…
The paper improves SVM and localized SVM stability under triple perturbations.
problem Stability of SVMs and localized SVMs under triple perturbations.
method Generalizes and improves existing results, considering simultaneous variations in probability measure, regularization parameter, and kernel.
result Improved stability of SVMs and localized SVMs under triple perturbations.
In this thesis, we study singular pseudo-differential operators defined by groupoids satisfying the Lauter-Nistor condition, by a method parallel to that of manifolds with boundary and edge differential operators. The example of the Bruhat sphere is studied in detail. In particular, we construct an extension to the cal…
Study on fluctuations in neural network kernels and predictions, focusing on finite width effects.
problem Characterizing fluctuations in finite width neural networks.
method Dynamical mean field theory analysis of wide but finite feature learning neural networks.
result Fluctuations in kernels and predictions are dynamically coupled, leading to reduced variance in feature learning regimes.
Fuzzy hashes learn from data to improve file similarity detection.
problem Measuring similarity between files, especially malware.
method Learned fuzzy hashes using a minimax training framework.
result Learned fuzzy hashes outperform traditional methods for file similarity.
Kernel regression predicts graph signals in noisy environments.
problem Predicting smooth graph signals in the presence of sparse noise.
method Kernel regression with ℓ 1 \ell_1 ℓ 1 -norm and ℓ 2 \ell_2 ℓ 2 -norm optimization using IRLS. result Efficacy demonstrated on real-world temperature data.
FHBI enhances generalization in Bayesian inference with iterative steps in functional spaces.
problem Improving generalization in Bayesian inference models.
method Iterative two-step procedure with adversarial and functional descent steps in a reproducing kernel Hilbert space.
result FHBI consistently outperforms nine baseline methods on the VTAB-1K benchmark.
New method improves solving combinatorial optimization problems with smoothed policies.
problem Solving combinatorial optimization problems repeatedly with varying instances.
method Smoothed policies with controlled random perturbations to linear oracle, leading to differentiable surrogate risk.
result Generalization bound decomposes excess risk into bias, estimation, and optimization components.
A new MKL framework improves graph-based clustering and semi-supervised classification.
problem MKL methods often fail to improve performance over single kernels.
method Proposes a new MKL framework based on consensus kernels and automatic weight assignment.
result The proposed method outperforms existing MKL methods on multiple benchmark datasets.
Study shows how transformers classify symbols without naming them, proving a margin-versus-collision criterion.
problem How transformers classify symbols without naming them.
method Logistic classification analysis of transformer-kernel regime, colored collision graph.
result Decomposes learned predictor into ideal template-level classifier and finite-sample perturbation.
Study on reducing dimensionality in high-dimensional regression with kernel methods and stability analysis.
problem Analyzing errors in high-dimensional regression with dimensionality reduction and kernel regression.
method Derive a stability result for kernel regression with Wasserstein distance and apply it to PCA to deduce convergence rates.
result Two-step procedure yields useful convergence rates in semi-supervised settings.
New insights into model robustness for random features and NTK models.
problem Understanding and distinguishing robustness in machine learning models.
method Analyzing empirical risk minimization in random features and NTK models.
result Random features models are not robust under any degree of over-parameterization, even when satisfying the universal law of robustness.
In this paper we establish stability results for symmetric spaces of noncompact type under Ricci flow, i.e. we will show that any small perturbation of the symmetric metric is flown back to the original metric under an appropriately rescaled Ricci flow. It will be important for us which smallness assumptions we have to…
Adversarial training converges to robust neural networks.
problem Vulnerability of neural networks to adversarial examples.
method Adversarial training alternates minimization and maximization steps.
result Adversarial training finds a robust classifier close to optimal robust loss.
Study delocalized eta invariants for signature operators on proper manifolds.
problem Define and analyze delocalized eta invariants for signature operators on proper manifolds.
method Develop detailed heat-kernel analysis and apply to proper manifolds with boundary.
result Prove index formulas relating delocalized eta invariants to Atiyah-Patodi-Singer indices.
RKHS-SHAP uses Shapley values for kernel methods to provide feature attributions.
problem Feature attribution for kernel methods is often heuristic and not individualised.
method RKHS-SHAP uses Shapley values from coalition game theory to compute feature attributions for kernel machines efficiently.
result RKHS-SHAP can compute both Interventional and Observational Shapley values.
Develops robust persistence diagrams using kernel methods.
problem Persistence diagrams are sensitive to data perturbations.
method Constructs robust persistence diagrams from superlevel filtrations of robust density estimators using reproducing kernels.
result Robust persistence diagrams are consistent estimators in bottleneck distance.
New bounds on AE success probability in GP models.
problem Limiting the success of adversarial examples in probabilistic models.
method Investigated upper bounds on AE success probability using Gaussian Processes.
result Proved a new upper bound of AE success probability dependent on perturbation norm, kernel function, and training dataset distance.
Kernel matrix concentration leads to KSC consistency.
problem High-dimensional clustering with noisy data.
method Nonasymptotic concentration inequalities for Lipschitz kernels.
result KSC algorithm consistency for noisy nested manifolds.
Kernel PCA helps analyze multivariate extremes and clusters them effectively.
problem Analyzing the dependence structure of multivariate extremes.
method Kernel PCA as a method for clustering and dimension reduction.
result Kernel PCA preimages effectively identify clusters in multivariate extremes.
New method identifies key features from unlabeled data, improving classification.
problem Identifying important features in unlabeled datasets.
method Adapts kernel PCA to automatically learn a kernel function for data.
result Learned kernel features significantly improve classification performance.
We consider a self-adjoint non-negative operator H H H in a Hilbert space L 2 ( X , d μ ) \mathsf{L}^2(X,{\rm d}μ) L 2 ( X , d μ ) . We assume that the semigroup ( e − t H ) t > 0 (\mathrm{e}^{-t H})_{t>0} ( e − t H ) t > 0 is defined by an integral kernel, p p p , which allows an estimate of the form p ( t , x , x ) ≤ F 1 ( x ) F 2 ( t ) p(t,x,x)\le F_1(x)F_2(t) p ( t , x , x ) ≤ F 1 ( x ) F 2 ( t ) for all ( x , t ) ∈ X × R + (x,t)\in X\times\mathbb{R_+} ( x , t ) ∈ X × R + ; we refer to F 1 F_1 F 1 as…
Unified theory for kernel regression generalizes well under realistic assumptions.
problem Analyzing kernel regression under realistic conditions.
method Unified theory providing rigorous bounds for various settings.
result Self-regularization phenomenon in kernel matrices enables good generalization.
There has been great interest recently in applying nonparametric kernel mixtures in a hierarchical manner to model multiple related data samples jointly. In such settings several data features are commonly present: (i) the related samples often share some, if not all, of the mixture components but with differing weight…