Motivated by statistical inference problems in high-dimensional time series data analysis, we first derive non- asymptotic error bounds for Gaussian approximations of sums of high-dimensional dependent random vectors on hyper-rectangles, simple convex sets and sparsely convex sets. We investigate the quantitative effect of temporal dependence on the rates of convergence to a Gaussian random vector over three different dependency frameworks (α-mixing, m-dependent, and physical dependence measure). In particular, we establish new error bounds under the α-mixing framework and derive faster rate over existing results under the physical dependence measure. To implement the proposed results in practical statistical inference problems, we also derive a data-driven parametric bootstrap procedure based on a kernel-type estimator for the long-run covariance matrices. The unified Gaussian and parametric bootstrap approximation results can be used to test mean vectors with combined l2 and l∞ type statistics, do change point detection, and construct confidence regions for covariance and precision matrices, all for time series data.

Gaussian processes (GPs), known for their flexibility as non-parametric models, have been widely used inpractice involving sensitive data (e.g., healthcare, finance) from multiple sources. With the challenge of dataisolation and the need for high-performance models, how to jointly develop privacy-preserving GP for multipleparties has emerged as a crucial topic, In this paper, we propose a new privacy-preserving GP algorithm, namelyPP-GP, which employs secret sharing ($$) techniques, Specifically, we introduce a new ss-based exponentiationoperation (PP-Exp) through confusion correction and an SS-based matrix inversion operation (PP-Ml) basedon Cholesky decomposition. However, the advantages of the GP come with a great computational burden andspace cost. To further enhance the efficiency, we propose an efficient split learning framework for privacy.preserving GP, named Split-GP, which demonstrably improves performance on large-scale data. We leave theDrivate data-related and SMPC-hostile computations (i.., random features) on data holders, and delegate therest of SMPC-friendly computations (i.e., low-rank approximation, model construction, and prediction) to semihonest servers. The resulting algorithm significantly reduces computational and communication costs comparedto Pp-GPp, making it well-suited for application to large-scale datasets. We provide a theoretical analysis interms of the correctness and security of the proposed Ss-based operations. Extensive experiments show thatour methods can achieve competitive performance and efficiency under the premise of preserving privacy.

Meta-learning accelerates the learning process onunseen learning tasks by acquiring prior knowledgethrough previous related tasks. The PAC-Bayesiantheory provides a theoretical framework to analyzethe generalization of meta-learning to unseen tasks.However, previous works still encounter two notablelimitations:(l)they merely focus on the data-freepriors, which often result in inappropriate regular-ization and loose generalization bounds, (2)moreimportantly, their optimization process usually in.volves nested optimization problems, incurring significant computational costs. To address these issues, we derive new generalization bounds and introduce a novel PAC-Bayesian framework for meta-learning that integrates data-dependent priors. Thisframework enables the extraction of optimal posteriors for each task in closed form, thereby allow-ing us to minimize generalization bounds incorporated data-dependent priors with only a simple localentropy. The resulting algorithm, which employsSGLD for sampling from the optimal posteriors, isstable, efficient, and computationally lightweight,eliminating the need for nested optimization. Extensive experimental results demonstrate that ourproposed method outperforms the other baselines.

In this paper, we consider testing the martingale difference hypothesis for high- dimensional time series. Our test is built on the sum of squares of the element-wise max-norm of the proposed matrix-valued nonlinear dependence measure at different lags.

Models defined by moment conditions are at the center of structural econometric estimation, but economic theory is mostly agnostic about moment selection. While a large pool of valid moments can potentially improve estimation efficiency, in the meantime a few invalid ones may undermine consistency. This article investigates the empirical likelihood estimation of these moment-defined models in high-dimensional settings. We propose a penalized empirical likelihood (PEL) estimation and establish its oracle property with consistent detection of invalid moments. The PEL estimator is asymptotically normally distributed, and a projected PEL procedure further eliminates its asymptotic bias and provides more accurate normal approx- imation to the finite sample behavior. Simulation exercises demonstrate excellent numerical performance of these methods in estimation and inference.

We consider to model matrix time series based on a tensor canonical polyadic (CP)-decomposition. Instead of using an iterative algorithm which is the standard practice for estimating CP-decompositions, we propose a new and one-pass estimation procedure based on a generalized eigenanalysis constructed from the serial dependence structure of the underlying process. To overcome the intricacy of solving a rank-reduced generalized eigenequation, we propose a further refined approach which projects it into a lower-dimensional full-ranked eigenequation.

Modeling Hawkes process using deep learning is superior to traditional statistical methods in the goodness of fit. However, methods based on RNN or self-attention are deficient in long-time dependence and recursive induction, respectively. Universal Transformer (UT) is an advanced framework to integrate these two requirements simultaneously due to its continuous transformation of self-attention in the depth of the position. In addition, migration of the UT framework involves the problem of effectively matching Hawkes process modeling. Thus, in this paper, an iterative convolutional enhancing self-attention Hawkes process with time relative position encoding (ICAHP-TR) is proposed, which is based on improved UT. First, the embedding maps from dense layers are carried out on sequences of arrival time points and markers to enrich event representation. Second, the deep network composed of UT extracts hidden historical information from event expression with the characteristics of recursion and the global receptive field. Third, two designed mechanics, including the relative positional encoding on the time step and the convolution enhancing perceptual attention are adopted to avoid losing dependencies between relative and adjacent positions in the Hawkes process. Finally, the hidden historical information is mapped by Dense layers as parameters in Hawkes process intensity function, thereby obtaining the likelihood function as the network loss. The experimental results show that the proposed methods demonstrate the effectiveness of synthetic datasets and realworld datasets from the perspective of both the goodness of fit and predictive ability compared with other baseline methods.

Graph convolutional networks (GCN) are viewed asone of the most popular representations among thevariants of graph neural networks over graph dataand have shown powerful performance in empiricalexperiments. That l2-based graph smoothing enforces the global smoothness of GCN, while (soft)l1-based sparse graph learning tends to promotesignal sparsity to trade for discontinuity. This paper aims to quantify the trade-off of GCN betweensmoothness and sparsity, with the help of a generall, ℓp-regularized (1 < p≤ 2) stochastic learning pro.posed within. While stability-based generalizationanalyses have been given in prior work for a secondderivative objectiveness function, our ℓp-regularized learning scheme does not satisfy such a smooth con.dition. To tackle this issue, we propose a novel SGD proximal algorithm for GCNs with an inexactoperator. For a single-layer GCN, we establish anexplicit theoretical understanding of GCN with the ℓp-regularized stochastic learning by analyzing thestability of our SGD proximal algorithm. We conduct multiple empirical experiments to validate ourtheoretical findings.

We propose a new unit-root test for a stationary null hypothesis H0 against a unit-root alternative H1. Our approach is nonparametric as H0 assumes only that the process concerned is I (0), without specifying any parametric forms. The new test is based on the fact that the sample autocovariance function converges to the finite population autocovariance function for an I(0) process, but diverges to infinity for a process with unit roots.

While it is common practice in applied network analysis to report various standard network summary statistics, these numbers are rarely accompanied by uncertainty quantification. Yet any error inherent in the measurements underlying the construction of the network, or in the network construction procedure itself, necessarily must propagate to any summary statistics reported. Here we study the problem of estimating the density of an arbitrary subgraph, given a noisy version of some underlying network as data. Under a simple model of network error, we show that consistent estimation of such densities is impossible when the rates of error are unknown and only a single network is observed. Accordingly, we develop method-of- moment estimators of network subgraph densities and error rates for the case where a minimal number of network replicates are available. These estimators are shown to be asymptotically normal as the number of vertices increases to infinity. We also provide confidence intervals for quantifying the uncertainty in these estimates based on the asymptotic normality. To construct the confidence intervals, a new and nonstandard bootstrap method is proposed to compute asymptotic variances, which is infeasible otherwise. We illustrate the proposed methods in the context of gene coexpression networks. Supplementary materials for this article are available online.