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The multivariate adaptive regression spline (MARS) is one of the popular estimation methods for nonparametric multivariate regressions. However, as MARS is based on marginal splines, to incorporate interactions of covariates, products of the marginal splines must be used, which leads to an unmanageable number of basis functions when the order of interaction is high and results in low estimation efficiency. In this paper, we improve the performance of MARS by using linear combinations of the covariates which achieve sufficient dimension reduction. The special basis functions of MARS facilitate calculation of gradients of the regression function, and estimation of the linear combinations is obtained via eigen-analysis of the outer-product of the gradients. Under some technical conditions, the asymptotic theory is established for the proposed estimation method. Numerical studies including both simulation and empirical applications show its effectiveness in dimension reduction and improvement over MARS and other commonly-used nonparametric methods in regression estimation and prediction.

In this paper, a dense Internet of Things (IoT) monitoring system is considered in which a large number of IoT devices contend for channel access so as to transmit timely status updates to the corresponding receivers using a carrier sense multiple access (CSMA) scheme. Under two packet management schemes with and without preemption in service, the closed-form expressions of the average age of information (AoI) and the average peak AoI of each device is characterized. It is shown that the scheme with preemption in service always leads to a smaller average AoI and a smaller average peak AoI, compared to the scheme without preemption in service. Then, a distributed noncooperative medium access control game is formulated in which each device optimizes its waiting rate so as to minimize its average AoI or average peak AoI under an average energy cost constraint on channel sensing and packet transmitting. To overcome the challenges of solving this game for an ultra-dense IoT, a mean-field game (MFG) approach is proposed to study the asymptotic performance of each device for the system in the large population regime. The accuracy of the MFG is analyzed, and the existence, uniqueness, and convergence of the mean-field equilibrium (MFE) are investigated. Simulation results show that the proposed MFG is accurate even for a small number of devices; and the proposed CSMA-type scheme under the MFG analysis outperforms three baseline schemes with fixed and dynamic waiting rates. Moreover, it is observed that the average AoI and the average peak AoI under the MFE do not necessarily decrease with the arrival rate.

A computational framework that leverages data from self-consistent field theory simulations with deep learning to accelerate the exploration of parameter space for block copolymers is presented. This is a substantial two-dimensional extension of the framework introduced in [1]. Several innovations and improvements are proposed. (1) A Sobolev space-trained, convolutional neural network (CNN) is employed to handle the exponential dimension increase of the discretized, local average monomer density fields and to strongly enforce both spatial translation and rotation invariance of the predicted, field-theoretic intensive Hamiltonian. (2) A generative adversarial network (GAN) is introduced to efficiently and accurately predict saddle point, local average monomer density fields without resorting to gradient descent methods that employ the training set. This GAN approach yields important savings of both memory and computational cost. (3) The proposed machine learning framework is successfully applied to 2D cell size optimization as a clear illustration of its broad potential to accelerate the exploration of parameter space for discovering polymer nanostructures. Extensions to three-dimensional phase discovery appear to be feasible.

This paper proposes a unified approach for dynamic modeling and simulations of general tensegrity structures with rigid bars and rigid bodies of arbitrary shapes. The natural coordinates are adopted as a non-minimal description in terms of different combinations of basic points and base vectors to resolve the heterogeneity between rigid bodies and rigid bars in three-dimensional space. This leads to a set of differential-algebraic equations with a constant mass matrix and free from trigonometric functions. Formulations for linearized dynamics are derived to enable modal analysis around static equilibrium. For numerical analysis of nonlinear dynamics, we derive a modified symplectic integration scheme which yields realistic results for long-time simulations, and accommodates non-conservative forces as well as boundary conditions. Numerical examples demonstrate the efficacy of the proposed approach for dynamic simulations of Class-1-to-$k$ general tensegrity structures under complex situations, including dynamic external loads, cable-based deployments, and moving boundaries. The novel tensegrity structures also exemplify new ways to create multi-functional structures.

We consider the problem of heteroscedastic linear regression, where, given $n$ samples $(\mathbf{x}_i, y_i)$ from $y_i = \langle \mathbf{w}^{*}, \mathbf{x}_i \rangle + \epsilon_i \cdot \langle \mathbf{f}^{*}, \mathbf{x}_i \rangle$ with $\mathbf{x}_i \sim N(0,\mathbf{I})$, $\epsilon_i \sim N(0,1)$, we aim to estimate $\mathbf{w}^{*}$. Beyond classical applications of such models in statistics, econometrics, time series analysis etc., it is also particularly relevant in machine learning when data is collected from multiple sources of varying but apriori unknown quality. Our work shows that we can estimate $\mathbf{w}^{*}$ in squared norm up to an error of $\tilde{O}\left(\|\mathbf{f}^{*}\|^2 \cdot \left(\frac{1}{n} + \left(\frac{d}{n}\right)^2\right)\right)$ and prove a matching lower bound (upto log factors). This represents a substantial improvement upon the previous best known upper bound of $\tilde{O}\left(\|\mathbf{f}^{*}\|^2\cdot \frac{d}{n}\right)$. Our algorithm is an alternating minimization procedure with two key subroutines 1. An adaptation of the classical weighted least squares heuristic to estimate $\mathbf{w}^{*}$, for which we provide the first non-asymptotic guarantee. 2. A nonconvex pseudogradient descent procedure for estimating $\mathbf{f}^{*}$ inspired by phase retrieval. As corollaries, we obtain fast non-asymptotic rates for two important problems, linear regression with multiplicative noise and phase retrieval with multiplicative noise, both of which are of independent interest. Beyond this, the proof of our lower bound, which involves a novel adaptation of LeCam's method for handling infinite mutual information quantities (thereby preventing a direct application of standard techniques like Fano's method), could also be of broader interest for establishing lower bounds for other heteroscedastic or heavy-tailed statistical problems.

Consistent weighted least square estimators are proposed for a wide class of nonparametric regression models with random regression function, where this real-valued random function of $k$ arguments is assumed to be continuous with probability 1. We obtain explicit upper bounds for the rate of uniform convergence in probability of the new estimators to the unobservable random regression function for both fixed or random designs. In contrast to the predecessors' results, the bounds for the convergence are insensitive to the correlation structure of the $k$-variate design points. As an application, we study the problem of estimating the mean and covariance functions of random fields with additive noise under dense data conditions. The theoretical results of the study are illustrated by simulation examples which show that the new estimators are more accurate in some cases than the Nadaraya--Watson ones. An example of processing real data on earthquakes in Japan in 2012--2021 is included.

It is well known that a single anchor can be used to determine the position and orientation of an agent communicating with it. However, it is not clear what information about the anchor or the agent is necessary to perform this localization, especially when the agent is in the near-field of the anchor. Hence, in this paper, to investigate the limits of localizing an agent with some uncertainty in the anchor location, we consider a wireless link consisting of source and destination nodes. More specifically, we present a Fisher information theoretical investigation of the possibility of estimating different combinations of the source and destination's position and orientation from the signal received at the destination. To present a comprehensive study, we perform this Fisher information theoretic investigation under both the near and far field propagation models. One of the key insights is that while the source or destination's $3$D orientation can be jointly estimated with the source or destination's $3$D position in the near-field propagation regime, only the source or destination's $2$D orientation can be jointly estimated with the source or destination's $2$D position in the far-field propagation regime. Also, a simulation of the FIM indicates that in the near-field, we can estimate the source's $3$D orientation angles with no beamforming, but in the far-field, we can not estimate the source's $2$D orientation angles when no beamforming is employed.

Almost all problems in applied mathematics, including the analysis of dynamical systems, deal with spaces of real-valued functions on Euclidean domains in their formulation and solution. In this paper, we describe the the tool Ariadne, which provides a rigorous calculus for working with Euclidean functions. We first introduce the Ariadne framework, which is based on a clean separation of objects as providing exact, effective, validated and approximate information. We then discuss the function calculus as implemented in \Ariadne, including polynomial function models which are the fundamental class for concrete computations. We then consider solution of some core problems of functional analysis, namely solution of algebraic equations and differential equations, and briefly discuss their use for the analysis of hybrid systems. We will give examples of C++ and Python code for performing the various calculations. Finally, we will discuss progress on extensions, including improvements to the function calculus and extensions to more complicated classes of system.

We propose a dynamical low-rank algorithm for a gyrokinetic model that is used to describe strongly magnetized plasmas. The low-rank approximation is based on a decomposition into variables parallel and perpendicular to the magnetic field, as suggested by the physics of the underlying problem. We show that the resulting scheme exactly recovers the dispersion relation even with rank 1. We then perform a simulation of kinetic shear Alfv\'en waves and show that using the proposed dynamical low-rank algorithm a drastic reduction (multiple orders of magnitude) in both computational time and memory consumption can be achieved. We also compare the performance of robust first and second-order projector splitting, BUG (also called unconventional), and augmented BUG integrators as well as a FFT-based spectral and Lax--Wendroff discretization.

The cyber-threat landscape has evolved tremendously in recent years, with new threat variants emerging daily, and large-scale coordinated campaigns becoming more prevalent. In this study, we propose CELEST (CollaborativE LEarning for Scalable Threat detection), a federated machine learning framework for global threat detection over HTTP, which is one of the most commonly used protocols for malware dissemination and communication. CELEST leverages federated learning in order to collaboratively train a global model across multiple clients who keep their data locally, thus providing increased privacy and confidentiality assurances. Through a novel active learning component integrated with the federated learning technique, our system continuously discovers and learns the behavior of new, evolving, and globally-coordinated cyber threats. We show that CELEST is able to expose attacks that are largely invisible to individual organizations. For instance, in one challenging attack scenario with data exfiltration malware, the global model achieves a three-fold increase in Precision-Recall AUC compared to the local model. We deploy CELEST on two university networks and show that it is able to detect the malicious HTTP communication with high precision and low false positive rates. Furthermore, during its deployment, CELEST detected a set of previously unknown 42 malicious URLs and 20 malicious domains in one day, which were confirmed to be malicious by VirusTotal.