We give a novel analysis of the standard Decentralized Gradient Descent with Adapt-Then-Combine (DGD-ATC) strategy algorithm for decentralized statistical estimation in high-dimensional sparse regression. Our analysis applies to more general smooth loss functions, while the previous analysis is limited to the least squares regression with a worse dependence on dimension even in this special case. Compared to the general convergence analysis of DGD, the difference for analyzing DGD algorithms in high dimensions is related to the fact that the standard strong convexity almost never holds in this setting, and thus establishing its linear convergence should rely on a relaxed concept of restricted strong convexity/smoothness. This relaxation creates significant technical challenges. Note that using a general theory of DGD, for a convex (but not strongly convex) objective function, one can only demonstrate the sub-linear convergence. Along the way, we also introduce a slightly stronger version of restricted strong smoothness condition which is the key in our analysis to extend the analysis beyond least squares. We present some numerical studies to illustrate the performances of the algorithm.

Let [Formula: see text] be a space of homogeneous type. Assume that [Formula: see text] satisfies the monotone geodesic property, then we prove some weighted estimates for Schrödinger-type operators, where the weights are Muckenhoupt-type weights associated with admissible functions modeled on the critical radius functions of Schrödinger operators. To be precise, we obtain the boundedness of the Schrödinger-type maximal Calderón–Zygmund operators on weighted [Formula: see text] with [Formula: see text] and from weighted [Formula: see text] to weighted [Formula: see text]. The same boundedness results extend to the Schrödinger-type Calderón–Zygmund operators as noted in prior work, though the rigorous proof remains absent; we provide the missing derivation in this study. The geometric properties associated with admissible functions and the dyadic cubes on [Formula: see text] play a crucial role in the proof of the results.

In this paper, we investigate Besov–Morrey spaces [Formula: see text] and Besov-type spaces [Formula: see text] of positive smoothness defined on Lipschitz domains [Formula: see text] as well as on [Formula: see text]. We combine the Hedberg–Netrusov approach to function spaces with distinguished kernel representations due to Triebel, in order to derive novel characterizations of these scales in terms of local oscillations provided that some standard conditions concerning the parameters are fulfilled. In connection with that we also obtain new characterizations of [Formula: see text] and [Formula: see text] via differences of higher order. By the way we recover and extend corresponding results for the scale of classical Besov spaces [Formula: see text].

For high-dimensional linear measurement error models with massive data, we develop a doubly bias-corrected score function based on a distributed subsample for the preconceived low-dimensional parameter in the presence of high-dimensional nuisance parameter. The resulting distributed subsample estimator is consistent and asymptotically normal with an explicitly derived asymptotic covariance matrix. To pursue more efficient distributed subsampling, we propose a unified optimal distributed subsampling scheme, which includes the A- and L-optimality criteria and can be implemented using an initial distributed subsample for constructing the doubly bias-corrected score function. The resulting estimator under optimal distributed subsampling is asymptotically normal with some optimality property. The finite-sample performance of the proposed method is studied by simulation and National Longitudinal Survey of Youth Dataset.

In this paper, we consider nonlinear partial integro-differential equations (PIDEs) with gradient-independent Lipschitz continuous nonlinearities and prove that deep neural networks with ReLU activation function can approximate solutions of such semilinear PIDEs without curse of dimensionality in the sense that the required number of parameters in the deep neural networks increases at most polynomially in both the dimension [Formula: see text] of the corresponding PIDE and the reciprocal of the prescribed accuracy [Formula: see text].

We study the following equation involving higher-order fractional Laplacian: [Formula: see text] where [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text]. Under either one of the two integral constraints [Formula: see text] and [Formula: see text] we establish an integral representation formula for any nonconstant classical solution satisfying certain growth at infinity. From this, we prove that these solutions are radially symmetric about some point in [Formula: see text] and monotone decreasing in the radial direction via method of moving planes in integral forms. We also establish a Moser–Trudinger-type inequality associated to the equation [Formula: see text] in bounded domain for [Formula: see text].

We study the asymptotic relaxation of Kuramoto oscillators in an analytic potential field. The Kuramoto model describes the synchronization of weakly coupled limit-cycle oscillators and it casts as a gradient flow on the Euclidean space with an analytic potential. The Kuramoto model corresponds to our proposed model with a constant force field. As a nontrivial local potential field, we consider two types of local potential fields corresponding to quadratic and linear-plus-oscillatory potentials. For the asymptotic relaxation, we provide sufficient frameworks leading to the uniform boundedness of state for each local potential field. The proposed framework is given in terms of system parameters and initial data. Then, we use the gradient flow formulation and uniform boundedness of state to find the existence of equilibrium and convergence toward them, which implies the existence of asymptotic relaxation.

The harmonic balance (HB) method is efficient in predicting the limit cycle oscillations for many dynamic systems including very high dimensional systems in fluid dynamics. Recently, some variations and extensions of the HB method have been developed and employed in the computation of large systems. This study focuses on a new development of the HB method. The new approach, the hybrid harmonic balance method (HHB), is developed based on the ideas in the conventional HB approach and the high dimensional harmonic balance (HDHB) approach. To demonstrate the applications and advantages, the new method is employed on a prototypical dynamic system in comparison with the results from both the HB and HDHB approaches. It is theoretically proved that when twice the number of harmonics in HB are included in the HHB derivation the results from HHB achieve the accuracy of the HB method for the cubic nonlinearity. Numerical simulations reveal that HHB combines the advantages of both the conventional HB and HDHB methods: Ease of implementation for very high dimensional systems regardless of the complexity of the nonlinearities and generation of meaningful solutions while the spurious solutions (from HDHB) are diminished. This new method is more accurate when compared with the HDHB method and much more computationally efficient when compared with the conventional HB method.

Kernel methods are popular in nonlinear and nonparametric regression due to their solid mathematical foundations and optimal statistical properties. However, scalability remains the primary bottleneck in applying kernel methods to large-scale data regression analysis. This paper aims to improve the scalability of kernel methods. We combine Nyström subsampling and the preconditioned conjugate gradient method to solve regularized kernel regression. Our theoretical analysis indicates that achieving optimal convergence rates requires only [Formula: see text] memory and [Formula: see text] time (up to logarithmic factors). Numerical experiments show that our algorithm outperforms existing methods in time efficiency and prediction accuracy on large-scale datasets. Notably, compared to the FALKON algorithm [A. Rudi, L. Carratino and L. Rosasco, Falkon: An optimal large scale kernel method, in Advances in Neural Information Processing Systems (Curran Associates, 2017), pp. 3891–3901], which is known as the optimal large-scale kernel method, our method is more flexible (applicable to non-positive definite kernel functions) and has a lower algorithmic complexity. Additionally, our established theoretical analysis further relaxes the restrictive conditions on hyperparameters previously imposed in convergence analyses.

We give investigations on a kernel function approximation problem arising from learning theory and show the convergence rate from the view of classical Fourier analysis. First, we provide the general definition for a modulus of smoothness and a [Formula: see text]-functional, and show that they are equivalent. In particular, we give explicit representation for some moduli of smoothness. Second, we establish some Jackson-type inequalities for the approximation error associated with some non-radial kernels. Also we apply these results to some concrete classical kernel function spaces and give Jackson-type inequalities for some concrete RKHS approximation problems. Finally, we apply these discussions to learning theory and describe the learning rates with the moduli of smoothness. The tools we used are Fourier analysis and the semigroup operator. The results show that the Jackson-type inequalities of approximation by some radial kernel functions on compact set with nonempty interiors cannot be expressed with the classical moduli of smoothness.