Composite B-spline regularized delta functions for the immersed boundary method: Divergence-free interpolation and gradient-preserving force spreading.

This paper is devoted to a study on SDEs with a bounded Borel drift $ b $. First, we remark that the original integration by parts formula due to P. Malliavin [14] can be used to deal with derivatives with respect to space variables. Then we obtain a link between the product of heat kernels and iterated divergences in Malliavin calculus. An explicit estimate for the derivative of solutions to SDE is obtained in terms of $ ||b||_\infty $. As a result, we prove that the SDE defines a continuous flow of maps in Sobolev spaces.

In the context of probabilistic programming languages, we put forward an approach to formal semantics and sampling-based inference with guarantees, centered on an action-based language equipped with a small-step operational semantics. We argue that this choice offers benefits in terms of clarity and effective, vectorized implementations. In measure-theoretic terms, a product of Markov kernels is used to formalize the small-step operational semantics. A trace semantics is also introduced based on a probability space of infinite sequences, along with a finite approximation theorem, relating the exact semantics to a truncated-execution semantics. This result directly leads to a sampling algorithm with guarantees, that can be efficiently SIMD-parallelized. Experiments conducted with an implementation based on TensorFlow show that our approach compares very favourably to state-of-the-art tools for probabilistic programming and inference. Keywords : probabilistic programming, operational semantics, measure theory, Monte Carlo simulation, SIMD parallelism.

Gaussian processes (GPs) stand as crucial tools in machine learning and signal processing, with their effectiveness hinging on kernel design and hyperparameter optimization. This article presents a novel GP linear multiple kernel (LMK) and a generic sparsity-aware distributed learning framework to optimize the hyperparameters. The newly proposed grid spectral mixture product (GSMP) kernel is tailored for multidimensional data, effectively reducing the number of hyperparameters while maintaining good approximation capability. We further demonstrate that the associated hyperparameter optimization of this kernel yields sparse solutions. To exploit the inherent sparsity of the solutions, we introduce the sparse linear multiple kernel learning (SLIM-KL) framework. The framework incorporates a quantized alternating direction method of multipliers (ADMMs) scheme for collaborative learning among multiple agents, where the local optimization problem is solved using a distributed successive convex approximation (DSCA) algorithm. SLIM-KL effectively manages large-scale hyperparameter optimization for the proposed kernel, simultaneously ensuring data privacy and minimizing communication costs. The theoretical analysis establishes convergence guarantees for the learning framework, while experiments on diverse datasets demonstrate the superior prediction performance and efficiency of our proposed methods.

Homomorphic encryption (HE) represents an encryption technology that allows for direct computation on encrypted data without requiring decryption. However, the substantial computational complexity and significant latency associated with HE has impeded its broader adoption in practical applications. To address these challenges, we propose a GPU-based acceleration framework, namely HEngine, tailored for homomorphic encryption tasks. Specifically, we first propose a warp shuffle-based optimization method for two key phases, i.e., inverse Chinese Remainder Theorem (ICRT) and number theoretic transformation (NTT), to mitigate synchronization overhead in homomorphic encryption. Secondly, we propose to fuse the NTT kernel with the inner product kernel to address the imbalance between memory access and computation. Thirdly, considering the potential difference in the amount of tasks of users in the real world, we design two different encoding methods for small batch and large batch inference tasks to improve computational efficiency. Finally, experiments demonstrate that our proposed framework achieves a 218 × speedup on homomorphic multiplication tasks compared with the CPU-based SEAL library. In addition, for convolutional neural network inference tasks on shallow network structures, our proposed framework achieves amortized inference performance at the millisecond level and sub-millisecond level on small batch and large batch data, respectively. For convolutional neural network inference tasks on deeper network structures (i.e., ResNet-20), our proposed framework achieves second-level inference.

ABSTRACT A recent advance in modeling nonergodic path effects by Sung et al. (2023) includes a path term for the 3D velocity structure effects. For the spatial correlation of the path effects, the difference between two paths was parameterized by the vector sum of the distance between the two source locations and the distance between two site locations. This metric works well for very different paths with small correlations (0–0.1) and very similar paths with large correlations (0.9–1.0), but it overestimates the correlation for paths with intermediate correlations. We propose a new metric for measuring the difference between two ray paths based on the difference in site locations, azimuths, and rupture distances. This new metric leads to a correlation model that better captures the observed spatial correlation structure of the ground motions than the existing metric. The new metric has a better physical basis for parameterizing the difference in the path effects and improves the accuracy of the extrapolation of the model to scenarios outside of the range constrained by the data. In addition, we present an efficient numerical method (Scalable Kernel Interpolation for Product Kernels [SKIP]) that can be applied to large data sets (more than 100,000 ground motions), as is common in large simulation data sets. Using SKIP, the forward predictions of path effects for a grid of source locations for a given site can be computed with a calculation time of about a minute using traditional laptop computers with standard memory.

Pairwise learning is a specialized form of supervised learning that focuses on predicting outcomes for pairs of objects. In this paper, we formulate the pairwise learning problem as a difference of convex (DC) optimization problem using the Kronecker product kernel, ℓ1- and ℓ0-regularizations, and various, possibly nonsmooth, loss functions. Our aim is to develop an efficient learning algorithm, SparsePKL, that produces accurate predictions with the desired sparsity level. In addition, we propose a novel limited memory bundle DC algorithm (LMB-DCA) for large-scale nonsmooth DC optimization and apply it as an underlying solver in the SparsePKL. The performance of the SparsePKL-algorithm is studied in seven real-world drug-target interaction data and the results are compared with those of the state-of-art methods in pairwise learning.

This work concerns the construction and characterization of product kernels for multivariate approximation from a finite set of discrete samples. To this end, we consider composing different component kernels, each acting on a low-dimensional Euclidean space. Due to Aronszajn (Trans. Am. Math. Soc. 68, 337–404 1950), the product of positive semi-definite kernel functions is again positive semi-definite, where, moreover, the corresponding native space is a particular instance of a tensor product, referred to as Hilbert tensor product. We first analyze the general problem of multivariate interpolation by product kernels. Then, we further investigate the tensor product structure, in particular for grid-like samples. We use this case to show that the product of positive definite kernel functions is again positive definite. Moreover, we develop an efficient computation scheme for the well-known Newton basis. Supporting numerical examples show the good performance of product kernels, especially for their flexibility.

This study addresses the motion and inertia parameter estimation problem of a torque-free, tumbling, non-cooperative space object (target) under long-term occlusions. To solve this problem, we employ a data-driven Gaussian process (GP) to simulate sensor measurements. In particular, we implement the multi-output GP to predict the projection measurements of a stereo-camera system onboard a chaser spacecraft. A product kernel, consisting of two periodic kernels, is used in the GP models to capture the periodic trends from non-periodic projection data. The initial guesses for the periodicity hyper-parameters of the GP models are intelligently derived from fast Fourier transform (FFT) analysis of the projection data. Additionally, we propose an unscented Kalman filter-Gaussian process (UKF-GP) fusion algorithm for target motion and inertia parameter estimation. The predicted projections from the GP models and their derivatives are used as the pseudo-measurements for UKF-GP during long-term occlusion. Results from Monte Carlo (MC) simulations demonstrate that, for varying tumbling frequencies, the UKF-GP can accurately estimate the target's motion variables over hundreds of seconds, a capability the conventional UKF algorithm lacks.

ABSTRACTBy introducing multiple parameters, employing the weight function method based on the “Hardy interpolation problem” and some real analysis techniques, a general Hilbert‐type integral inequality with the kernel as is established. The necessary and sufficient condition for the constant factor of the general inequality to be the best possible is identified; the equivalent inequality with the best possible constant factor is obtained. A Hilbert‐type singular integral operator is defined and utilized to characterize the obtained Hilbert‐type integral inequality and its equivalent form. As an application, by choosing appropriate parameter values, several specific Hilbert‐type integral inequalities are presented.

Understanding the characteristics of temporal correlations in a time series is crucial for developing accurate models in natural and social sciences. The burst-tree decomposition method was recently introduced to reveal temporal correlations in a time series in the form of an event sequence, in particular, the hierarchical structure of bursty trains of events for the entire range of timescales [Jo et al., Sci. Rep. 10, 12202 (2020)10.1038/s41598-020-68157-1]. Such structure cannot be solely captured by the interevent time distribution but can show higher-order correlations beyond interevent times. It has been found to be simply characterized by the burst-merging kernel governing which bursts are merged together as the timescale for defining bursts increases. In this work, we study the effects of kernels on the higher-order temporal correlations in terms of burst-size distributions, memory coefficients for bursts, and the autocorrelation function. We employ several kernels, including the constant, sum, product, and diagonal kernels as well as those inspired by empirical results. We generically find that kernels with preferential merging lead to heavy-tailed burst-size distributions, while kernels with assortative merging lead to positive correlations between burst sizes. The decaying exponent of the autocorrelation function depends not only on the kernel but also on the power-law exponent of the interevent time distribution. In addition, thanks to the analogy to the coagulation process, analytical solutions of burst-size distributions for some kernels could be obtained. Our findings may shed light on the role of burst-merging kernels as underlying mechanisms of higher-order temporal correlations in a time series.

Operator learning with Gaussian processes

Summary The generalization ability of kernel interpolation in large dimensions, ie, $ n\asymp d^{\gamma} $ for some $ \gamma \gt 0 $, could be one of the most interesting problems in the recent renaissance of kernel regression, since it may help us understand the so-called benign overfitting phenomenon reported in the neural networks literature. Focusing on the inner product kernel on the unit sphere, we fully characterize the exact order of both the variance and the bias of large-dimensional kernel interpolation under various source conditions $ s\geqslant 0 $. Consequently, we obtain the $ (s,\gamma) $ phase diagram of large-dimensional kernel interpolation, ie, we determine the regions in the $ (s,\gamma) $ plane where the kernel interpolation is minimax optimal, suboptimal and inconsistent.

We develop an action formalism to calculate probabilities of rare events in cluster-cluster aggregation for arbitrary collision kernels and establish a pathwise large deviation principle with total mass being the rate. As an application, the rate function for the number of surviving particles as well as the optimal evolution trajectory are calculated exactly for the constant, sum, and product kernels. For the product kernel, we argue that the second derivative of the rate function has a discontinuity. The theoretical results agree with simulations tailored to the calculation of rare events.

Effect of dimensionality on convergence rates of kernel ridge regression estimator

This paper is motivated by Phong and Stein's paper on non-standard singular integrals with mixed homogeneities. Our purpose is to study these new non-standard convolution singular integrals and establish the boundedness of these singular integrals on the Hardy spaces.

This paper examines the tractability of multivariate approximation problems under the normalized error criterion for a zero-mean Gaussian measure in an average-case setting. The Gaussian measure is associated with a covariance kernel, which is represented by the tensor product of one-dimensional kernels corresponding to Euler and Wiener integrated processes with non-negative and nondecreasing smoothness parameters {rd}d∈N. We give matching sufficient and necessary conditions for various concepts of tractability in terms of the asymptotic properties of the regularity parameters, except for (s, 0)-WT.

In this paper, we introduce a novel nonsmooth optimization-based method LMBM-Kron ℓ 0 LS for solving large-scale pairwise interaction affinity prediction problems. The aim of LMBM-Kron ℓ 0 LS is to produce accurate predictions using as sparse a model as possible. We apply the least squares approach with Kronecker product kernels for a loss function and a continuous formulation of ℓ 0 pseudonorm for regularization. Thus, we end up solving a nonsmooth optimization problem. In addition, we apply a specific bi-objective criterion to strike a balance between the prediction accuracy of the learned model and the sparsity of the obtained solution. We compare LMBM-Kron ℓ 0 LS with some state-of-the-art methods using three benchmark and two simulated data sets under four distinct experimental settings, including zero-shot learning. Moreover, both binary and continuous interaction affinity labels are considered with LMBM-Kron ℓ 0 LS. The results show that LMBM-Kron ℓ 0 LS finds sparse solutions without sacrificing too much in the prediction performance.

Nagel, Ricci and Stein proved that product kernels are finite sums of flag kernels in the Euclidean space. We show that the product Triebel-Lizorkin space is the intersection of two flag Triebel-Lizorkin spaces. This extends a main result in [Chang D-C, Han Y, Wu X. Relations between product and flag Hardy spaces. J Geom Anal. 2021;31(7):6601–6623]. As an application, we provide a new proof of the boundedness of product singular integral operators on product Triebel-Lizorkin spaces.

We present an event-triggered boundary control scheme for hyperbolic systems. The trigger condition is based on predictions of the state on determinate sets, and the control input is updated only when the predictions deviate from the reference by a given margin. Nominal closed-loop stability, the absence of Zeno behaviour, and robustness to uncertainty and disturbances, are all established analytically. For the special case of linear systems, the trigger condition can be expressed in closed-form as an <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$L_{2}$</tex-math></inline-formula> -scalar product of kernels with the distributed state. The presented controller can also be combined with existing observers to solve the event-triggered output-feedback control problem. A numerical simulation demonstrates the effectiveness of the proposed approach.