This paper develops a general asymptotic theory of series estimators for spatial data collected at irregularly spaced locations within a sampling region $R_n \subset \mathbb{R}^d$. We employ a stochastic sampling design that can flexibly generate irregularly spaced sampling sites, encompassing both pure increasing and mixed increasing domain frameworks. Specifically, we focus on a spatial trend regression model and a nonparametric regression model with spatially dependent covariates. For these models, we investigate $L^2$-penalized series estimation of the trend and regression functions. We establish uniform and $L^2$ convergence rates and multivariate central limit theorems for general series estimators as main results. Additionally, we show that spline and wavelet series estimators achieve optimal uniform and $L^2$ convergence rates and propose methods for constructing confidence intervals for these estimators. Finally, we demonstrate that our dependence structure conditions on the underlying spatial processes include a broad class of random fields, including L\'evy-driven continuous autoregressive and moving average random fields.
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This paper introduces a new objective measure for assessing treatment response in asthmatic patients using computed tomography (CT) imaging data. For each patient, CT scans were obtained before and after one year of monoclonal antibody treatment. Following image segmentation, the Hounsfield unit (HU) values of the voxels were encoded through quantile functions. It is hypothesized that patients with improved conditions after treatment will exhibit better expiration, reflected in higher HU values and an upward shift in the quantile curve. To objectively measure treatment response, a novel linear regression model on quantile functions is developed, drawing inspiration from Verde and Irpino (2010). Unlike their framework, the proposed model is parametric and incorporates distributional assumptions on the errors, enabling statistical inference. The model allows for the explicit calculation of regression coefficient estimators and confidence intervals, similar to conventional linear regression. The corresponding data and R code are available on GitHub to facilitate the reproducibility of the analyses presented.
We characterise the behaviour of the maximum Diaconis-Ylvisaker prior penalized likelihood estimator in high-dimensional logistic regression, where the number of covariates is a fraction $\kappa \in (0,1)$ of the number of observations $n$, as $n \to \infty$. We derive the estimator's aggregate asymptotic behaviour under this proportional asymptotic regime, when covariates are independent normal random variables with mean zero and the linear predictor has asymptotic variance $\gamma^2$. From this foundation, we devise adjusted $Z$-statistics, penalized likelihood ratio statistics, and aggregate asymptotic results with arbitrary covariate covariance. While the maximum likelihood estimate asymptotically exists only for a narrow range of $(\kappa, \gamma)$ values, the maximum Diaconis-Ylvisaker prior penalized likelihood estimate not only exists always but is also directly computable using maximum likelihood routines. Thus, our asymptotic results also hold for $(\kappa, \gamma)$ values where results for maximum likelihood are not attainable, with no overhead in implementation or computation. We study the estimator's shrinkage properties, compare it to alternative estimation methods that can operate with proportional asymptotics, and present procedures for the estimation of unknown constants that describe the asymptotic behaviour of our estimator. We also provide a conjecture about the behaviour of our estimator when an intercept parameter is present in the model. We present results from extensive numerical studies to demonstrate the theoretical advances and strong evidence to support the conjecture, and illustrate the methodology we put forward through the analysis of a real-world data set on digit recognition.
In this manuscript we present the tensor-train reduced basis method, a novel projection-based reduced-order model for the efficient solution of parameterized partial differential equations. Despite their popularity and considerable computational advantages with respect to their full order counterparts, reduced-order models are typically characterized by a considerable offline computational cost. The proposed approach addresses this issue by efficiently representing high dimensional finite element quantities with the tensor train format. This method entails numerous benefits, namely, the smaller number of operations required to compute the reduced subspaces, the cheaper hyper-reduction strategy employed to reduce the complexity of the PDE residual and Jacobian, and the decreased dimensionality of the projection subspaces for a fixed accuracy. We provide a posteriori estimates that demonstrate the accuracy of the proposed method, we test its computational performance for the heat equation and transient linear elasticity on three-dimensional Cartesian geometries.
We present a novel class of projected gradient (PG) methods for minimizing a smooth but not necessarily convex function over a convex compact set. We first provide a novel analysis of the "vanilla" PG method, achieving the best-known iteration complexity for finding an approximate stationary point of the problem. We then develop an "auto-conditioned" projected gradient (AC-PG) variant that achieves the same iteration complexity without requiring the input of the Lipschitz constant of the gradient or any line search procedure. The key idea is to estimate the Lipschitz constant using first-order information gathered from the previous iterations, and to show that the error caused by underestimating the Lipschitz constant can be properly controlled. We then generalize the PG methods to the stochastic setting, by proposing a stochastic projected gradient (SPG) method and a variance-reduced stochastic gradient (VR-SPG) method, achieving new complexity bounds in different oracle settings. We also present auto-conditioned stepsize policies for both stochastic PG methods and establish comparable convergence guarantees.
Tensor data are multi-dimension arrays. Low-rank decomposition-based regression methods with tensor predictors exploit the structural information in tensor predictors while significantly reducing the number of parameters in tensor regression. We propose a method named NA$_0$CT$^2$ (Noise Augmentation for $\ell_0$ regularization on Core Tensor in Tucker decomposition) to regularize the parameters in tensor regression (TR), coupled with Tucker decomposition. We establish theoretically that NA$_0$CT$^2$ achieves exact $\ell_0$ regularization on the core tensor from the Tucker decomposition in linear TR and generalized linear TR. To our knowledge, NA$_0$CT$^2$ is the first Tucker decomposition-based regularization method in TR to achieve $\ell_0$ in core tensors. NA$_0$CT$^2$ is implemented through an iterative procedure and involves two straightforward steps in each iteration -- generating noisy data based on the core tensor from the Tucker decomposition of the updated parameter estimate and running a regular GLM on noise-augmented data on vectorized predictors. We demonstrate the implementation of NA$_0$CT$^2$ and its $\ell_0$ regularization effect in both simulation studies and real data applications. The results suggest that NA$_0$CT$^2$ can improve predictions compared to other decomposition-based TR approaches, with or without regularization and it identifies important predictors though not designed for that purpose.
For many problems, quantum algorithms promise speedups over their classical counterparts. However, these results predominantly rely on asymptotic worst-case analysis, which overlooks significant overheads due to error correction and the fact that real-world instances often contain exploitable structure. In this work, we employ the hybrid benchmarking method to evaluate the potential of quantum Backtracking and Grover's algorithm against the 2023 SAT competition main track winner in solving random $k$-SAT instances with tunable structure, designed to represent industry-like scenarios, using both $T$-depth and $T$-count as cost metrics to estimate quantum run times. Our findings reproduce the results of Campbell, Khurana, and Montanaro (Quantum '19) in the unstructured case using hybrid benchmarking. However, we offer a more sobering perspective in practically relevant regimes: almost all quantum speedups vanish, even asymptotically, when minimal structure is introduced or when $T$-count is considered instead of $T$-depth. Moreover, when the requirement is for the algorithm to find a solution within a single day, we find that only Grover's algorithm has the potential to outperform classical algorithms, but only in a very limited regime and only when using $T$-depth. We also discuss how more sophisticated heuristics could restore the asymptotic scaling advantage for quantum backtracking, but our findings suggest that the potential for practical quantum speedups in more structured $k$-SAT solving will remain limited.
Arbor is a software library designed for efficient simulation of large-scale networks of biological neurons with detailed morphological structures. It combines customizable neuronal and synaptic mechanisms with high-performance computing, supporting multi-core CPU and GPU systems. In humans and other animals, synaptic plasticity processes play a vital role in cognitive functions, including learning and memory. Recent studies have shown that intracellular molecular processes in dendrites significantly influence single-neuron dynamics. However, for understanding how the complex interplay between dendrites and synaptic processes influences network dynamics, computational modeling is required. To enable the modeling of large-scale networks of morphologically detailed neurons with diverse plasticity processes, we have extended the Arbor library to the Plastic Arbor framework, supporting simulations of a large variety of spike-driven plasticity paradigms. To showcase the features of the new framework, we present examples of computational models, beginning with single-synapse dynamics, progressing to multi-synapse rules, and finally scaling up to large recurrent networks. While cross-validating our implementations by comparison with other simulators, we show that Arbor allows simulating plastic networks of multi-compartment neurons at nearly no additional cost in runtime compared to point-neuron simulations. Using the new framework, we have already been able to investigate the impact of dendritic structures on network dynamics across a timescale of several hours, showing a relation between the length of dendritic trees and the ability of the network to efficiently store information. By our extension of Arbor, we aim to provide a valuable tool that will support future studies on the impact of synaptic plasticity, especially, in conjunction with neuronal morphology, in large networks.
This manuscript studies the numerical solution of the time-fractional Burgers-Huxley equation in a reproducing kernel Hilbert space. The analytical solution of the equation is obtained in terms of a convergent series with easily computable components. It is observed that the approximate solution uniformly converges to the exact solution for the aforementioned equation. Also, the convergence of the proposed method is investigated. Numerical examples are given to demonstrate the validity and applicability of the presented method. The numerical results indicate that the proposed method is powerful and effective with a small computational overhead.
This paper presents an analysis of properties of two hybrid discretization methods for Gaussian derivatives, based on convolutions with either the normalized sampled Gaussian kernel or the integrated Gaussian kernel followed by central differences. The motivation for studying these discretization methods is that in situations when multiple spatial derivatives of different order are needed at the same scale level, they can be computed significantly more efficiently compared to more direct derivative approximations based on explicit convolutions with either sampled Gaussian kernels or integrated Gaussian kernels. While these computational benefits do also hold for the genuinely discrete approach for computing discrete analogues of Gaussian derivatives, based on convolution with the discrete analogue of the Gaussian kernel followed by central differences, the underlying mathematical primitives for the discrete analogue of the Gaussian kernel, in terms of modified Bessel functions of integer order, may not be available in certain frameworks for image processing, such as when performing deep learning based on scale-parameterized filters in terms of Gaussian derivatives, with learning of the scale levels. In this paper, we present a characterization of the properties of these hybrid discretization methods, in terms of quantitative performance measures concerning the amount of spatial smoothing that they imply, as well as the relative consistency of scale estimates obtained from scale-invariant feature detectors with automatic scale selection, with an emphasis on the behaviour for very small values of the scale parameter, which may differ significantly from corresponding results obtained from the fully continuous scale-space theory, as well as between different types of discretization methods.
In general $n$-dimensional simplicial meshes, we propose a family of interior penalty nonconforming finite element methods for $2m$-th order partial differential equations, where $m \geq 0$ and $n \geq 1$. For this family of nonconforming finite elements, the shape function space consists of polynomials with a degree not greater than $m$, making it minimal. This family of finite element spaces exhibits natural inclusion properties, analogous to those in the corresponding Sobolev spaces in the continuous case. By applying interior penalty to the bilinear form, we establish quasi-optimal error estimates in the energy norm. Due to the weak continuity of the nonconforming finite element spaces, the interior penalty terms in the bilinear form take a simple form, and an interesting property is that the penalty parameter needs only to be a positive constant of $\mathcal{O}(1)$. These theoretical results are further validated by numerical tests.
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