Paper introduces kernel deformed exponential families for sparse continuous attention.
problem Creating efficient attention mechanisms for sparse data.
method Developed kernel deformed exponential families, theoretically and experimentally.
result Kernel deformed exponential families can attend to multiple compact regions of data.
In the context of kernel methods, the similarity between data points is encoded by the kernel function which is often defined thanks to the Euclidean distance, a common example being the squared exponential kernel. Recently, other distances relying on optimal transport theory - such as the Wasserstein distance between …
Study compares exponential and power-law kernels in modeling high-frequency trading data.
problem Modeling high-frequency trading data with specific kernel types.
method Proposes and analyzes two bivariate Hawkes processes with exponential and power-law kernels.
result Identifies strengths and limitations of exponential and power-law kernels for high-frequency trading data.
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.
We prove an exponential estimate for the asymptotics of Bergman kernels of a positive line bundle under hypotheses of bounded geometry. We give further Bergman kernel proofs of complex geometry results, such as separation of points, existence of local coordinates and holomorphic convexity by sections of positive line b…
Kernel estimator optimally recovers function from noisy exponential Radon transform.
problem Inverting noisy exponential Radon transform of a function.
method Proposed a kernel estimator to estimate the true function.
result The estimator converges to the true function at minimax optimal rate.
We investigate penalized maximum log-likelihood estimation for exponential family distributions whose natural parameter resides in a reproducing kernel Hilbert space. Key to our approach is a novel technique, doubly dual embedding, that avoids computation of the partition function. This technique also allows the develo…
New random feature maps for Laplacian and related kernels.
problem Challenges in approximating the Laplacian kernel and its generalizations.
method Developed random feature maps for Laplacian and related kernels, providing efficient sampling schemes.
result Demonstrated the efficacy of these random feature maps on real datasets.
We present a generalization of the adversarial linear bandits framework, where the underlying losses are kernel functions (with an associated reproducing kernel Hilbert space) rather than linear functions. We study a version of the exponential weights algorithm and bound its regret in this setting. Under conditions on …
We study the asymptotic behavior of the generalized Bergman kernel of the renormalized Bochner-Laplacian on high tensor powers of a positive line bundle on a symplectic manifold of bounded geometry. First, we establish the off-diagonal exponential estimate for the generalized Bergman kernel. As an application, we obtai…
New method for spectral and Bergman kernels under local spectral gap condition.
problem Analyzing spectral and Bergman kernels for complex manifolds.
method Developed a new scaling method to study spectral and Bergman kernels.
result Established pointwise asymptotics of spectral and Bergman kernels.
Orthogonal random features approximate a Bessel kernel, offering sharper bounds than random Fourier features.
problem Approximating Gaussian kernel efficiently for large datasets.
method Use of Haar orthogonal matrices to construct orthogonal random features and analyze their bias and variance.
result Orthogonal random features approximate a Bessel kernel, not the Gaussian kernel, with sharper bounds.
Given two sets of independent samples from unknown distributions P and Q, a two-sample test decides whether to reject the null hypothesis that P=Q. Recent attention has focused on kernel two-sample tests as the test statistics are easy to compute, converge fast, and have low bias with their finite sample estimate…
Kernel thinning compresses distributions more effectively than i.i.d. sampling or standard thinning.
problem Efficiently compressing distributions for better sampling and integration accuracy.
method Introduces kernel thinning, a procedure that compresses an n-point approximation of a distribution into a sqrt(n)-point approximation with comparable integration error.
result Kernel thinning achieves a maximum discrepancy in integration error of O_d(n^(-1/2) sqrt(log n)) in probability for compactly supported distributions and O_d(n^(-1/2) (log n)^(d+1/2) sqrt(log log n)) for sub-exponential distributions.
New algorithm reduces feature count and accelerates error convergence.
problem Exponential error convergence in data classification with optimized random features.
method Optimized random features accelerated by quantum machine learning.
result Achieves exponential error convergence under low-noise condition.
Study on U-statistics with heavy-tailed samples, providing tail bounds and LDP.
problem Deviation of U-statistics with heavy-tailed samples.
method Exponential tail bounds and Large Deviation Principle (LDP) for U-statistics.
result Obtained an exponential upper bound for U-statistics tail decay, showing two regions of decay.
Paper studies kernel hyperparameters for clustering, proposing an efficient search method.
problem Challenges in tuning kernel parameters for clustering, especially for RBF kernels.
method Derives a lower bound for RBF kernel parameters, proposes an efficient hyperparameter search algorithm.
result Proposes an efficient algorithm for hyperparameter search in kernel clustering, improving upon grid search.
Develops kernels for matchings, overcoming computational challenges.
problem Challenges in applying kernel methods to matchings due to their discrete, non-Euclidean nature.
method Characterizes stationary kernels, introduces heat and Matérn kernel families, and develops a sub-exponential algorithm for efficient evaluation.
result Establishes novel negative results and identifies an open problem in transferring the framework to trees.
Paper explores duality in DPPs using embedding structure analysis.
problem Understanding the geometric structure of determinantal point processes.
method Analyzes the exponential family embedding of DPPs and uses the e-embedding curvature tensor.
result Discovers the duality between marginal and L-ensemble kernels.
Price changes are induced by aggressive market orders in stock market. We introduce a bivariate marked Hawkes process to model aggressive market order arrivals at the microstructural level. The order arrival intensity is marked by an exogenous part and two endogenous processes reflecting the self-excitation and cross-e…
Paper explores robust estimators for kernel exponential families using smoothed total variation distances.
problem Outliers can severely impact classical estimators in statistical inference.
method Proposes smoothed total variation (STV) distance as a class of IPMs for robust estimation of kernel exponential families.
result STV-based estimators are robust against distribution contamination for kernel exponential families.
The paper improves error bounds for Bayesian quadrature in noisy settings.
problem Improving error bounds for Bayesian quadrature in noisy settings.
method Develops a two-step meta-algorithm to relate average-case quadrature error to L2-function approximation error. result Provides new average-case results for various kernels and noise settings.
This paper studies the optimality of kernel methods in high-dimensional data clustering. Recent works have studied the large sample performance of kernel clustering in the high-dimensional regime, where Euclidean distance becomes less informative. However, it is unknown whether popular methods, such as kernel k-means, …
A new MMD-based test combines kernels for two-sample testing without splitting data.
problem Efficiently testing if two datasets come from the same distribution without splitting data.
method Proposes a novel statistic based on Maximum Mean Discrepancy (MMD) that combines kernels, proving concentration bounds and showing data-dependent kernel selection.
result Exponential concentration bounds and improved test power compared to existing methods.
Enhanced kernel framework for advanced data forecasting.
problem Complex mechanical behaviors and timefrequency dynamics in aircraft systems.
method Frequency-aware surrogate modeling with SMT kernels, extending kernel types and incorporating derivatives.
result Improved accuracy in forecasting CO2 concentrations and airline passenger traffic.
New bounds for KRR condition number reveal overfitting phenomena.
problem Characterizing overfitting in KRR with varying kernel spectral decay.
method Derived new bounds for kernel matrices, enhanced test error bounds, and identified feature independence role.
result Identified tempered and catastrophic overfitting phenomena.
Study analyzes learnability of RKHS under L∞ norm for kernel methods.
problem Understand performance of kernel methods and random feature models.
method Relate L∞ learnability to kernel spectrum decay and establish sample complexity bounds.
result Conditions for efficient L∞ learning of RKHS identified.
Study evaluates RKHS choices for assessing graph models using KSD tests.
problem Effect of RKHS choice on KSD tests for graph model assessment.
method Investigated power performance and computational runtime of KSD tests for ERGMs and synthetic graph generators.
result Different RKHS choices affect KSD test performance and computational runtime.
The kernel exponential family is a rich class of distributions, which can be fit efficiently and with statistical guarantees by score matching. Being required to choose a priori a simple kernel such as the Gaussian, however, limits its practical applicability. We provide a scheme for learning a kernel parameterized by …
Parameterized state space models in the form of recurrent networks are often used in machine learning to learn from data streams exhibiting temporal dependencies. To break the black box nature of such models it is important to understand the dynamical features of the input driving time series that are formed in the sta…
Deep neural kernels and Laplace kernel have equivalent RKHS on spheres.
problem Comparing RKHS of deep neural tangent and Laplace kernels.
method Proof of RKHS equivalence using sphere restrictions and kernel properties.
result RKHS of deep neural tangent kernel and Laplace kernel are the same on Sd−1. Quantum kernels offer potential speed-ups but require encoding problem-specific knowledge.
problem Generalization difficulty in high-dimensional feature spaces.
method Analysis of spectral properties of quantum kernels and their RKHS.
result Quantum advantage is expected if RKHS is low-dimensional and contains hard-to-compute functions.
We introduce a Gaussian process model of functions which are additive. An additive function is one which decomposes into a sum of low-dimensional functions, each depending on only a subset of the input variables. Additive GPs generalize both Generalized Additive Models, and the standard GP models which use squared-expo…
We propose Kernel Hamiltonian Monte Carlo (KMC), a gradient-free adaptive MCMC algorithm based on Hamiltonian Monte Carlo (HMC). On target densities where classical HMC is not an option due to intractable gradients, KMC adaptively learns the target's gradient structure by fitting an exponential family model in a Reprod…
The recently proposed "generalized min-max" (GMM) kernel can be efficiently linearized, with direct applications in large-scale statistical learning and fast near neighbor search. The linearized GMM kernel was extensively compared in with linearized radial basis function (RBF) kernel. On a large number of classificatio…
Explicit formula for Bergman kernel of abelian varieties proved.
problem Explicit formula for Bergman kernel of polarized abelian varieties.
method Explicit formula for Bergman kernel of polarized abelian varieties.
result Explicit formula for Bergman kernel of polarized abelian varieties.
We prove a variant of the Davies-Gaffney-Grigor'yan Lemma for the continuous time heat kernel on graphs. We use it together with the Li-Yau inequality to obtain strong heat kernel estimates for graphs satisfying the exponential curvature dimension inequality.
Unified analysis of kernel-based and locally adaptive bandit optimization methods.
problem Performance of bandit optimization algorithms in RKHS functions.
method Investigates the relationship between kernel regularity and algorithmic performance, characterizing spectral properties of various kernels.
result Unified framework for analyzing kernel-based and locally adaptive bandit algorithms, deriving explicit regret bounds.
Hawkes processes have seen a number of applications in finance, due to their ability to capture event clustering behaviour typically observed in financial systems. Given a calibrated Hawkes process, of concern is the statistical fit to empirical data, particularly for the accurate quantification of self- and mutual-exc…
We show that under very general assumptions the partial Bergman kernel function of sections vanishing along an analytic hypersurface has exponential decay in a neighborhood of the vanishing locus. Considering an ample line bundle, we obtain a uniform estimate of the Bergman kernel function associated to a singular metr…
We give a short proof of a strong version of the short time asymptotic expansion of heat kernels associated to Laplace type operators acting on sections of vector bundles over compact Riemannian manifolds, including exponential decay of the difference of the approximate heat kernel and the true heat kernel. We use this…
Adapts to estimate functions from noisy ERT data.
problem Estimating functions from noisy Exponential Radon Transform data.
method Locally adaptive kernel type estimator for functions of varying smoothness.
result Achieves minimax optimal rate up to a log(n) factor for Sobolev functions.
Study on biharmonic heat equation on manifolds with curvature constraints.
problem Analyzing entire solutions of biharmonic heat equation on manifolds.
method Exponential decay estimates for biharmonic heat kernel under Ricci curvature and noncollapsing conditions. Proving uniqueness criteria for Cauchy problem.
result Conservation law for biharmonic heat kernel and uniform L-infinity estimate for entire solutions.
Efficiently models event-based data with general parametric kernels.
problem Inference for Hawkes processes with general parametric kernels requires large datasets.
method Developed a fast ℓ2 gradient-based solver using a discretized version of events. result Improved estimation of pattern latency in brain signals.
Bayesian tensor train kernel machine uses Laplace approximation for scalable GP regression.
problem Scalability limitations of Gaussian process regression.
method Bayesian tensor train kernel machine with Laplace approximation and variational inference.
result VI replaces cross-validation and offers up to 65x faster training.
Quantum algorithm speeds up learning from big data exponentially.
problem Scalable learning from big data with optimized random features.
method Quantum algorithm for sampling optimized random features.
result Exponential speedup in runtime compared to classical algorithms.
Study proves existence, uniqueness, and positivity of solutions to a complex volatility model.
problem Modeling equity index and spot volatility with path-dependent features and general kernels.
method Proved existence and uniqueness of a continuous solution to a Stochastic Volterra Equation (SVE) with non-convolutional, non-bounded kernels and non-Lipschitz coefficients.
result Positivity of the volatility process under certain conditions on the kernels.
A new test assesses how well observed networks fit a specified ERGM model.
problem Testing the goodness of fit for ERGMs with a single network observation.
method Kernel Stein discrepancy combined with a discrete Stein operator for ERGMs, Monte Carlo simulation.
result The test provides theoretical and practical support for assessing ERGM fit.