Precision medicine aims to personalize treatment by identifying the optimal individualized treatment rule (ITR) that maximizes expected clinical outcomes. Outcome weighted learning (OWL) is an approach that has caught attention for estimating optimal ITR by leveraging each patient’s unique characteristics within a weighted classification framework. The curse of dimensionality in statistical learning significantly impacts the convergence rates of OWL algorithms, especially when analyzing high-dimensional biomedical data. We establish improved learning rates for OWL algorithms with varying Gaussian kernels, assuming the data lie on a low-dimensional intrinsic structure characterized by the box-counting dimension. Under noise or Besov smoothness assumptions, we derive a novel oracle inequality for OWL and obtain fast learning rates for the hinge loss function and q-norm hinge loss function (1 < q ≤ 2), where the rates explicitly depend on the box-counting dimension of the prognostic variable domain. Additionally, we select the hyperparameters of OWL adaptively based on the data through a training-validation procedure. The same learning rates are obtained without assuming any prior knowledge of the data distribution.
In this paper, we consider a Chemotaxis–Navier–Stokes system in a three-dimensional bounded convex domain, driven by multiplicative noise of transport type, where the noise coefficients depend on the gradients of the solutions. Using a multi-level approximation scheme that combines semi-Galerkin projections with suitable truncations and regularizations, together with a compactness argument based on the Jakubowski–Skorokhod representation theorem for non-metric spaces, we establish the existence of a global martingale solution. The solution is shown to be non-negative almost surely. Our result extends known global existence results for the deterministic 3D and stochastic 2D settings to the stochastic 3D case.
A theoretical framework for general stability is established in this paper for stochastic pantograph delay evolution equations driven by tempered fractional Brownian motion (tfBm) with Hurst index [Formula: see text] and tempered parameter [Formula: see text]. A novel estimate of the stochastic integral with respect to tfBm is the key of our analyses, being valid for both [Formula: see text] and [Formula: see text]. We prove the global existence, mean square stability with general decay rate and global Hölder regularity in time of the mild solution by utilizing the convolution method, the Young convolution inequality, the semigroup theory and the Banach fixed point theorem. These stabilities may be specialized as the exponential, polynomial, and logarithmic one.
Deep neural architectures owe much of their remarkable success to their capacity to approximate high-dimensional data while seemingly bypassing the curse of dimensionality - a phenomenon often linked to the capability to uncover intrinsic lower-dimensional structures within the data. To provide a theoretical foundation for this phenomenon, we investigate the problem of approximating functions defined on a [Formula: see text]-dimensional manifold embedded within a space of potentially much higher ambient dimension [Formula: see text]. We prove that [Formula: see text]-deep ReLU neural networks with fixed width that scales polynomially with [Formula: see text] can approximate [Formula: see text]-times continuously differentiable functions with an error of order [Formula: see text]. The innovation of this result lies in the total removal of the ambient dimension [Formula: see text] from the approximation constant. By ensuring that the constant [Formula: see text] is independent of [Formula: see text], we offer a significant improvement over prior work and provide a definitive theoretical explanation for the network’s resilience to high-dimensional environments.
In this paper, we establish new global regularity estimates for the Hessian of viscosity solutions to non-convex fully nonlinear elliptic equations with oblique boundary conditions in the setting of weighted Lorentz and Morrey–Lorentz spaces. More precisely, our approach relies on an analysis of the ellipticity coefficients of the governing operator and, under suitable small-aperture assumptions, we obtain Calderón–Zygmund-type estimates in weighted Lorentz spaces. As a byproduct of this result, we derive several applications, including Sobolev estimates with variable exponents and the study of the obstacle problem with an oblique boundary condition.
