The Gaussian process state-space model (GPSSM) has attracted extensive attention for modeling complex nonlinear dynamical systems. However, the existing GPSSM employs separate Gaussian processes (GPs) for each latent state dimension, leading to escalating computational complexity and parameter proliferation, thus posing challenges for modeling dynamical systems with high-dimensional latent states. To surmount this obstacle, we propose to integrate the efficient transformed Gaussian process (ETGP) into the GPSSM, which involves pushing a shared GP through multiple normalizing flows to efficiently model the transition function in high-dimensional latent state space. Additionally, we develop a corresponding variational inference algorithm that surpasses existing methods in terms of parameter count and computational complexity. Experimental results on diverse synthetic and real-world datasets corroborate the efficiency of the proposed method, while also demonstrating its ability to achieve similar inference performance compared to existing methods. Code is available at \url{https://github.com/zhidilin/gpssmProj}.

The Gaussian process state-space model (GPSSM) has garnered considerable attention over the past decade. However, the standard GP with a preliminary kernel, such as the squared exponential kernel or Mat\'{e}rn kernel, that is commonly used in GPSSM studies, limits the model's representation power and substantially restricts its applicability to complex scenarios. To address this issue, we propose a new class of probabilistic state-space models called TGPSSMs, which leverage a parametric normalizing flow to enrich the GP priors in the standard GPSSM, enabling greater flexibility and expressivity. Additionally, we present a scalable variational inference algorithm that offers a flexible and optimal structure for the variational distribution of latent states. The proposed algorithm is interpretable and computationally efficient due to the sparse GP representation and the bijective nature of normalizing flow. Moreover, we incorporate a constrained optimization framework into the algorithm to enhance the state-space representation capabilities and optimize the hyperparameters, leading to superior learning and inference performance. Experimental results on synthetic and real datasets corroborate that the proposed TGPSSM outperforms several state-of-the-art methods. The accompanying source code is available at \url{https://github.com/zhidilin/TGPSSM}.

Gaussian process state-space model (GPSSM) is a fully probabilistic state-space model that has attracted much attention over the past decade. However, the outputs of the transition function in the existing GPSSMs are assumed to be independent, meaning that the GPSSMs cannot exploit the inductive biases between different outputs and lose certain model capacities. To address this issue, this paper proposes an output-dependent and more realistic GPSSM by utilizing the well-known, simple yet practical linear model of coregionalization (LMC) framework to represent the output dependency. To jointly learn the output-dependent GPSSM and infer the latent states, we propose a variational sparse GP-based learning method that only gently increases the computational complexity. Experiments on both synthetic and real datasets demonstrate the superiority of the output-dependent GPSSM in terms of learning and inference performance.

Graph neural networks (GNNs) are popular to use for classifying structured data in the context of machine learning. But surprisingly, they are rarely applied to regression problems. In this work, we adopt GNN for a classic but challenging nonlinear regression problem, namely the network localization. Our main findings are in order. First, GNN is potentially the best solution to large-scale network localization in terms of accuracy, robustness and computational time. Second, thresholding of the communication range is essential to its superior performance. Simulation results corroborate that the proposed GNN based method outperforms all benchmarks by far. Such inspiring results are further justified theoretically in terms of data aggregation, non-line-of-sight (NLOS) noise removal and lowpass filtering effect, all affected by the threshold for neighbor selection. Code is available at https://github.com/Yanzongzi/GNN-For-localization.

This paper considers semi-supervised learning for tabular data. It is widely known that Xgboost based on tree model works well on the heterogeneous features while transductive support vector machine can exploit the low density separation assumption. However, little work has been done to combine them together for the end-to-end semi-supervised learning. In this paper, we find these two methods have complementary properties and larger diversity, which motivates us to propose a new semi-supervised learning method that is able to adaptively combine the strengths of Xgboost and transductive support vector machine. Instead of the majority vote rule, an optimization problem in terms of ensemble weight is established, which helps to obtain more accurate pseudo labels for unlabeled data. The experimental results on the UCI data sets and real commercial data set demonstrate the superior classification performance of our method over the five state-of-the-art algorithms improving test accuracy by about $3\%-4\%$. The partial code can be found at https://github.com/hav-cam-mit/CTO.

Kernel-based machine learning approaches are gaining increasing interest for exploring and modeling large dataset in recent years. Gaussian process (GP) is one example of such kernel-based approaches, which can provide very good performance for nonlinear modeling problems. In this work, we first propose a grey-box modeling approach to analyze the forces in cross country skiing races. To be more precise, a disciplined set of kinetic motion model formulae is combined with data-driven Gaussian process regression model, which accounts for everything unknown in the system. Then, a modeling approach is proposed to analyze the kinetic flow of both individual and clusters of skiers. The proposed approaches can be generally applied to use cases where positioned trajectories and kinetic measurements are available. The proposed approaches are evaluated using data collected from the Falun Nordic World Ski Championships 2015, in particular the Men's cross country $4\times10$ km relay. Forces during the cross country skiing races are analyzed and compared. Velocity models for skiers at different competition stages are also evaluated. Finally, the comparisons between the grey-box and black-box approach are carried out, where the grey-box approach can reduce the predictive uncertainty by $30\%$ to $40\%$.

Hyper-parameter optimization remains as the core issue of Gaussian process (GP) for machine learning nowadays. The benchmark method using maximum likelihood (ML) estimation and gradient descent (GD) is impractical for processing big data due to its $O(n^3)$ complexity. Many sophisticated global or local approximation models, for instance, sparse GP, distributed GP, have been proposed to address such complexity issue. In this paper, we propose two novel and general-purpose GP hyper-parameter training schemes (GPCV-ADMM) by replacing ML with cross-validation (CV) as the fitting criterion and replacing GD with a non-linearly constrained alternating direction method of multipliers (ADMM) as the optimization method. The proposed schemes are of $O(n^2)$ complexity for any covariance matrix without special structure. We conduct various experiments based on both synthetic and real data sets, wherein the proposed schemes show excellent performance in terms of convergence, hyper-parameter estimation accuracy, and computational time in comparison with the traditional ML based routines given in the GPML toolbox.

Gaussian processes (GP) for machine learning have been studied systematically over the past two decades and they are by now widely used in a number of diverse applications. However, GP kernel design and the associated hyper-parameter optimization are still hard and to a large extend open problems. In this paper, we consider the task of GP regression for time series modeling and analysis. The underlying stationary kernel can be approximated arbitrarily close by a new proposed grid spectral mixture (GSM) kernel, which turns out to be a linear combination of low-rank sub-kernels. In the case where a large number of the sub-kernels are used, either the Nystr\"{o}m or the random Fourier feature approximations can be adopted to deal efficiently with the computational demands. The unknown GP hyper-parameters consist of the non-negative weights of all sub-kernels as well as the noise variance; their estimation is performed via the maximum-likelihood (ML) estimation framework. Two efficient numerical optimization methods for solving the unknown hyper-parameters are derived, including a sequential majorization-minimization (MM) method and a non-linearly constrained alternating direction of multiplier method (ADMM). The MM matches perfectly with the proven low-rank property of the proposed GSM sub-kernels and turns out to be a part of efficiency, stable, and efficient solver, while the ADMM has the potential to generate better local minimum in terms of the test MSE. Experimental results, based on various classic time series data sets, corroborate that the proposed GSM kernel-based GP regression model outperforms several salient competitors of similar kind in terms of prediction mean-squared-error and numerical stability.

The cloud radio access network (CRAN) is a promising paradigm to meet the stringent requirements of the fifth generation (5G) wireless systems. Meanwhile, wireless traffic prediction is a key enabler for C-RANs to improve both the spectrum efficiency and energy efficiency through load-aware network managements. This paper proposes a scalable Gaussian process (GP) framework as a promising solution to achieve large-scale wireless traffic prediction in a cost-efficient manner. First, to the best of our knowledge, this paper is the first to empower GP regression with the alternating direction method of multipliers (ADMM) for parallel hyper-parameter optimization in the training phase, where such a scalable training framework well balances the local estimation in baseband units (BBUs) and information consensus among BBUs in a principled way for large-scale executions. Second, in the prediction phase, we fuse local predictions obtained from the BBUs via a cross-validation based optimal strategy, which demonstrates itself to be reliable and robust for general regression tasks. Moreover, such a cross-validation based optimal fusion strategy is built upon a well acknowledged probabilistic model to retain the valuable closed-form GP inference properties. Third, we propose a CRAN based scalable wireless prediction architecture, where the prediction accuracy and the time consumption can be balanced by tuning the number of the BBUs according to the real-time system demands. Experimental results show that our proposed scalable GP model can outperform the state-of-the-art approaches considerably, in terms of wireless traffic prediction performance. I. INTRODUCTION The fifth generation (5G) system is expected to provide approximately 1000 times higher wireless capacity and reduce up to 90 percent of energy consumption compared with the current 4G system [1]. A CRAN is composed of two parts: the distributed remote radio heads (RRHs) with basic radio functionalities to provide coverage over a large area, and the centralized baseband units (BBUs) pool with parallel BBUs to support joint processing and cooperative network management. The BBUs can perform dynamic resource allocation in accordance with realtime networkdemands based on the virtualized resources in cloud computing. One major feature for the C-RANs to enable high energy-efficient services is the fast adaptability to nonuniform traffic variations [1]-[4], e.g., the tidal effects. Consequently, wireless traffic prediction techniques stand out as the key enabler to realize such loadaware managementand proactive control in C-RANs, e.g., the load-aware RRH on/off operation [4].