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.
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.
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.
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.
Kernel method outperforms deep neural networks in speech enhancement.
problem Improving single-channel speech enhancement performance.
method Kernel regression with an exponential power kernel and EigenPro iterative method.
result Kernel method consistently outperforms deep neural networks in speech enhancement.
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.
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. 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.
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…
Study asymptotics of extension and orthogonal Bergman kernels for high tensor powers of positive line bundles.
problem Asymptotic behavior of Bergman kernels for high tensor powers of positive line bundles.
method Analyzing the Schwartz kernel of the Ohsawa-Takegoshi extension operator and orthogonal Bergman projector, proving exponential estimates and asymptotic expansions.
result Explicit asymptotic expansions for the Ohsawa-Takegoshi extension operator and orthogonal Bergman projector.
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…
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…
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.
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 this paper we study the variational problem associated to support vector regression in Banach function spaces. Using the Fenchel-Rockafellar duality theory, we give explicit formulation of the dual problem as well as of the related optimality conditions. Moreover, we provide a new computational framework for solving…
Topological Data Analysis (TDA) is a recent and growing branch of statistics devoted to the study of the shape of the data. In this work we investigate the predictive power of TDA in the context of supervised learning. Since topological summaries, most noticeably the Persistence Diagram, are typically defined in comple…
We propose a representation of graph as a functional object derived from the power iteration of the underlying adjacency matrix. The proposed functional representation is a graph invariant, i.e., the functional remains unchanged under any reordering of the vertices. This property eliminates the difficulty of handling e…
Unified score and distance-based GoF tests for model adequacy.
problem Difficulty in extending score-based GoF tests to nonparametric alternatives.
method Introducing semiparametric kernelized Stein discrepancy (SKSD) test.
result SKSD test is computationally efficient and universally consistent.
The paper explores a new type of kernel using Wasserstein distance for better classification of shapes.
problem Improving kernel methods for shape classification.
method Defined and studied exponential kernels based on regularized Wasserstein distance.
result Wasserstein squared exponential kernels perform better on small shape datasets.
Gaussian processes are powerful, yet analytically tractable models for supervised learning. A Gaussian process is characterized by a mean function and a covariance function (kernel), which are determined by a model selection criterion. The functions to be compared do not just differ in their parametrization but in thei…
Two new models for forward power prices capture clustering jumps.
problem Describing forward power prices with clustering jumps.
method Continuous branching processes with immigration and Hawkes processes with exponential kernel.
result Models adequately describe forward prices evolution in French power market.
New research sets the minimax lower bound for KSD estimation at sqrt(n).
problem Estimating goodness-of-fit using Kernel Stein Discrepancy (KSD) on high-dimensional spaces.
method Two complementary results proving the minimax lower bound of KSD estimation.
result The minimax lower bound of KSD estimation is n^(-1/2), indicating exponential difficulty with dimensionality.
Study the Bochner-Schrödinger operator on symplectic manifolds, proving gap existence and asymptotic kernel behavior.
problem Analyzing the spectrum and asymptotic behavior of the Bochner-Schrödinger operator on symplectic manifolds.
method Rough asymptotic description, existence proof, off-diagonal exponential estimate, complete asymptotic expansion.
result Existence of gaps in the spectrum and asymptotic kernel behavior.
Estimates exponential family distributions using a novel doubly dual embedding technique.
problem Estimating exponential family distributions with smoothness and efficiency.
method Doubly dual embedding for avoiding partition function computation and flexible sampling.
result Improves memory and time efficiency while offering stronger statistical properties.
A quantum-inspired classical algorithm speeds up LS-SVM classification.
problem Big data challenge in SVM classification.
method Improved indirect sampling technique for LS-SVM.
result Algorithm achieves logarithmic runtime for low rank data matrices.
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.
A new kernel improves tensor classification accuracy and reduces computation time.
problem Challenges in classifying high-dimensional tensor data.
method Proposes a weighted subspace exponential kernel based on Tucker decomposition.
result The new kernel outperforms existing methods in accuracy and computational efficiency.
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…
Efficiently accelerates attention calculation for Transformers with relative positional encoding.
problem Quadratic complexity of attention in long sequences.
method Kernelized attention with Fast Fourier Transform (FFT) for RPE.
result Achieves O(n log n) time complexity, mitigates training instability, and outperforms other models.
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.
The role of kernels is central to machine learning. Motivated by the importance of power-law distributions in statistical modeling, in this paper, we propose the notion of power-law kernels to investigate power-laws in learning problem. We propose two power-law kernels by generalizing Gaussian and Laplacian kernels. Th…
We prove a conjecture about approximating Gaussian Processes on one dimension.
problem Computational scaling issues with Gaussian Processes on one dimension.
method Developed a new family of state-space models (LEG) to approximate any stationary GP on one dimension.
result Proved that any stationary GP on one dimension can be approximated using the LEG family.
Study on expressive power of Euclidean kernels and efficient kernel learning.
problem Limiting the expressive power of kernel methods and improving kernel learning efficiency.
method Define Euclidean kernels, analyze their geometric and spectral properties, and develop efficient algorithms for kernel learning.
result Prove limitations on the expressive power of Euclidean kernels and derive efficient algorithms for kernel learning.
Study on KRR with power-law data, showing better sample complexity.
problem High-dimensional kernel ridge regression with anisotropic power-law covariance.
method Explicit characterization of kernel spectrum and asymptotic analysis of excess risk.
result Sample complexity is governed by effective dimension, not ambient dimension.
A deep learning approach for fitting complex distributions.
problem Limited applicability of simple kernels in fitting complex distributions.
method Learning a deep network to parameterize the kernel of the exponential family.
result The method can fit complex structures on moderate-dimensional problems.
We discover scaling laws for kernel regression loss under various learning rate schedules.
problem Understanding loss dynamics and learning rate schedules in kernel regression.
method Theoretical analysis of stochastic gradient descent on a power-law kernel regression model.
result Established a Functional Scaling Law (FSL) capturing the full loss trajectory under arbitrary learning rate schedules.
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 …
New spectral mixture representation for isotropic kernels simplifies random Fourier features.
problem Applying Random Fourier Features to complex kernels.
method Decompose isotropic kernels into scale mixtures of α-stable random vectors.
result Constructive spectral sampling formula for various kernels.
Study shows Bergman kernel quotient approaches one for punctured surfaces.
problem Analyzing Bergman kernels on punctured Riemann surfaces.
method Examined a punctured Riemann surface with a specific metric and line bundle, calculating quotient of Bergman kernels.
result The quotient of Bergman kernels tends to one as tensor power increases.
We propose an explicit recursive method to approximate a power-law with a finite sum of weighted exponentials. Applications to moving averages with long memory are discussed in relationship with stochastic volatility models.
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.
Develops European power option pricing under correlated interest rate and asset processes.
problem Pricing European power options under correlated interest rate and asset processes.
method Martingale method and Girsannov transform.
result Derives European power option pricing formulae under two market assumptions.
A new method for kernel tests without data splitting increases power.
problem Lack of power in kernel-based tests due to data splitting.
method Selective inference framework to learn hyperparameters and test on full sample.
result Empirically larger test power without data splitting, regardless of split proportion.
CTT compresses samples to test distributions near-linearly, outperforming existing methods.
problem Efficiently testing distributions with high power and near-linear runtime.
method Sample compression followed by permutation testing.
result CTT achieves near-linear runtime while maintaining high statistical power.
Boosts change-point detection power with optimal sub-sampling.
problem Power loss in sequential change-point detection from large history data.
method Optimal sub-sampling of history data before kernel-based detection procedures.
result Improved detection performance in extensive experiments.
Appropriately designing the proposal kernel of particle filters is an issue of significant importance, since a bad choice may lead to deterioration of the particle sample and, consequently, waste of computational power. In this paper we introduce a novel algorithm adaptively approximating the so-called optimal proposal…
New kernels on symmetric groups enable efficient Gaussian process sampling.
problem Efficiently modeling and sampling on symmetric groups.
method Introduced power sum kernels and methods for efficient calculation and sampling.
result Polynomial computational complexity for sampling Gaussian processes.
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…