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Recent approaches to VO have significantly improved performance by using deep networks to predict optical flow between video frames. However, existing methods still suffer from noisy and inconsistent flow matching, making it difficult to handle challenging scenarios and long-sequence estimation. To overcome these challenges, we introduce Spatio-Temporal Visual Odometry (STVO), a novel deep network architecture that effectively leverages inherent spatio-temporal cues to enhance the accuracy and consistency of multi-frame flow matching. With more accurate and consistent flow matching, STVO can achieve better pose estimation through the bundle adjustment (BA). Specifically, STVO introduces two innovative components: 1) the Temporal Propagation Module that utilizes multi-frame information to extract and propagate temporal cues across adjacent frames, maintaining temporal consistency; 2) the Spatial Activation Module that utilizes geometric priors from the depth maps to enhance spatial consistency while filtering out excessive noise and incorrect matches. Our STVO achieves state-of-the-art performance on TUM-RGBD, EuRoc MAV, ETH3D and KITTI Odometry benchmarks. Notably, it improves accuracy by 77.8% on ETH3D benchmark and 38.9% on KITTI Odometry benchmark over the previous best methods.

Visual odometry (VO) aims to estimate camera poses from visual inputs -- a fundamental building block for many applications such as VR/AR and robotics. This work focuses on monocular RGB VO where the input is a monocular RGB video without IMU or 3D sensors. Existing approaches lack robustness under this challenging scenario and fail to generalize to unseen data (especially outdoors); they also cannot recover metric-scale poses. We propose Robust Metric Visual Odometry (RoMeO), a novel method that resolves these issues leveraging priors from pre-trained depth models. RoMeO incorporates both monocular metric depth and multi-view stereo (MVS) models to recover metric-scale, simplify correspondence search, provide better initialization and regularize optimization. Effective strategies are proposed to inject noise during training and adaptively filter noisy depth priors, which ensure the robustness of RoMeO on in-the-wild data. As shown in Fig.1, RoMeO advances the state-of-the-art (SOTA) by a large margin across 6 diverse datasets covering both indoor and outdoor scenes. Compared to the current SOTA DPVO, RoMeO reduces the relative (align the trajectory scale with GT) and absolute trajectory errors both by >50%. The performance gain also transfers to the full SLAM pipeline (with global BA & loop closure). Code will be released upon acceptance.

This article considers a novel and widely applicable approach to modeling high-dimensional dependent data when a large number of explanatory variables are available and the signal-to-noise ratio is low. We postulate that a $p$-dimensional response series is the sum of a linear regression with many observable explanatory variables and an error term driven by some latent common factors and an idiosyncratic noise. The common factors have dynamic dependence whereas the covariance matrix of the idiosyncratic noise can have diverging eigenvalues to handle the situation of low signal-to-noise ratio commonly encountered in applications. The regression coefficient matrix is estimated using penalized methods when the dimensions involved are high. We apply factor modeling to the regression residuals, employ a high-dimensional white noise testing procedure to determine the number of common factors, and adopt a projected Principal Component Analysis when the signal-to-noise ratio is low. We establish asymptotic properties of the proposed method, both for fixed and diverging numbers of regressors, as $p$ and the sample size $T$ approach infinity. Finally, we use simulations and empirical applications to demonstrate the efficacy of the proposed approach in finite samples., Comment: 37 pages

Layout Generation aims to synthesize plausible arrangements from given elements. Currently, the predominant methods in layout generation are Generative Adversarial Networks (GANs) and diffusion models, each presenting its own set of challenges. GANs typically struggle with handling discrete data due to their requirement for differentiable generated samples and have historically circumvented the direct generation of discrete labels by treating them as fixed conditions. Conversely, diffusion-based models, despite achieving state-of-the-art performance across several metrics, require extensive sampling steps which lead to significant time costs. To address these limitations, we propose \textbf{DogLayout} (\textbf{D}en\textbf{o}ising Diffusion \textbf{G}AN \textbf{Layout} model), which integrates a diffusion process into GANs to enable the generation of discrete label data and significantly reduce diffusion's sampling time. Experiments demonstrate that DogLayout considerably reduces sampling costs by up to 175 times and cuts overlap from 16.43 to 9.59 compared to existing diffusion models, while also surpassing GAN based and other layout methods. Code is available at https://github.com/deadsmither5/DogLayout., Comment: Code is available at https://github.com/deadsmither5/DogLayout

3D Multi-Object Tracking (MOT), a fundamental component of environmental perception, is essential for intelligent systems like autonomous driving and robotic sensing. Although Tracking-by-Detection frameworks have demonstrated excellent performance in recent years, their application in real-world scenarios faces significant challenges. Object movement in complex environments is often highly nonlinear, while existing methods typically rely on linear approximations of motion. Furthermore, system noise is frequently modeled as a Gaussian distribution, which fails to capture the true complexity of the noise dynamics. These oversimplified modeling assumptions can lead to significant reductions in tracking precision. To address this, we propose a GRU-based MOT method, which introduces a learnable Kalman filter into the motion module. This approach is able to learn object motion characteristics through data-driven learning, thereby avoiding the need for manual model design and model error. At the same time, to avoid abnormal supervision caused by the wrong association between annotations and trajectories, we design a semi-supervised learning strategy to accelerate the convergence speed and improve the robustness of the model. Evaluation experiment on the nuScenes and Argoverse2 datasets demonstrates that our system exhibits superior performance and significant potential compared to traditional TBD methods.

Matrix-variate time series data are increasingly popular in economics, statistics, and environmental studies, among other fields. This paper develops regularized estimation methods for analyzing high-dimensional matrix-variate time series using bilinear matrix-variate autoregressive models. The bilinear autoregressive structure is widely used for matrix-variate time series, as it reduces model complexity while capturing interactions between rows and columns. However, when dealing with large dimensions, the commonly used iterated least-squares method results in numerous estimated parameters, making interpretation difficult. To address this, we propose two regularized estimation methods to further reduce model dimensionality. The first assumes banded autoregressive coefficient matrices, where each data point interacts only with nearby points. A two-step estimation method is used: first, traditional iterated least-squares is applied for initial estimates, followed by a banded iterated least-squares approach. A Bayesian Information Criterion (BIC) is introduced to estimate the bandwidth of the coefficient matrices. The second method assumes sparse autoregressive matrices, applying the LASSO technique for regularization. We derive asymptotic properties for both methods as the dimensions diverge and the sample size $T\rightarrow\infty$. Simulations and real data examples demonstrate the effectiveness of our methods, comparing their forecasting performance against common autoregressive models in the literature.

Stereo matching is a core component in many computer vision and robotics systems. Despite significant advances over the last decade, handling matching ambiguities in ill-posed regions and large disparities remains an open challenge. In this paper, we propose a new deep network architecture, called IGEV++, for stereo matching. The proposed IGEV++ builds Multi-range Geometry Encoding Volumes (MGEV) that encode coarse-grained geometry information for ill-posed regions and large disparities and fine-grained geometry information for details and small disparities. To construct MGEV, we introduce an adaptive patch matching module that efficiently and effectively computes matching costs for large disparity ranges and/or ill-posed regions. We further propose a selective geometry feature fusion module to adaptively fuse multi-range and multi-granularity geometry features in MGEV. We then index the fused geometry features and input them to ConvGRUs to iteratively update the disparity map. MGEV allows to efficiently handle large disparities and ill-posed regions, such as occlusions and textureless regions, and enjoys rapid convergence during iterations. Our IGEV++ achieves the best performance on the Scene Flow test set across all disparity ranges, up to 768px. Our IGEV++ also achieves state-of-the-art accuracy on the Middlebury, ETH3D, KITTI 2012, and 2015 benchmarks. Specifically, IGEV++ achieves a 3.23% 2-pixel outlier rate (Bad 2.0) on the large disparity benchmark, Middlebury, representing error reductions of 31.9% and 54.8% compared to RAFT-Stereo and GMStereo, respectively. We also present a real-time version of IGEV++ that achieves the best performance among all published real-time methods on the KITTI benchmarks. The code is publicly available at https://github.com/gangweiX/IGEV-plusplus, Comment: 12 pages, 10 figures

Leverage scores have become essential in statistics and machine learning, aiding regression analysis, randomized matrix computations, and various other tasks. This paper delves into the inverse problem, aiming to recover the intrinsic model parameters given the leverage scores gradient. This endeavor not only enriches the theoretical understanding of models trained with leverage score techniques but also has substantial implications for data privacy and adversarial security. We specifically scrutinize the inversion of the leverage score gradient, denoted as $g(x)$. An innovative iterative algorithm is introduced for the approximate resolution of the regularized least squares problem stated as $\min_{x \in \mathbb{R}^d} 0.5 \|g(x) - c\|_2^2 + 0.5\|\mathrm{diag}(w)Ax\|_2^2$. Our algorithm employs subsampled leverage score distributions to compute an approximate Hessian in each iteration, under standard assumptions, considerably mitigating the time complexity. Given that a total of $T = \log(\| x_0 - x^* \|_2/ \epsilon)$ iterations are required, the cost per iteration is optimized to the order of $O( (\mathrm{nnz}(A) + d^{\omega} ) \cdot \mathrm{poly}(\log(n/\delta))$, where $\mathrm{nnz}(A)$ denotes the number of non-zero entries of $A$., Comment: arXiv admin note: text overlap with arXiv:2404.13785

This paper proposes a novel method for sparse latent factor modeling using a new sparse asymptotic Principal Component Analysis (APCA). This approach analyzes the co-movements of large-dimensional panel data systems over time horizons within a general approximate factor model framework. Unlike existing sparse factor modeling approaches based on sparse PCA, which assume sparse loading matrices, our sparse APCA assumes that factor processes are sparse over the time horizon, while the corresponding loading matrices are not necessarily sparse. This development is motivated by the observation that the assumption of sparse loadings may not be appropriate for financial returns, where exposure to market factors is generally universal and non-sparse. We propose a truncated power method to estimate the first sparse factor process and a sequential deflation method for multi-factor cases. Additionally, we develop a data-driven approach to identify the sparsity of risk factors over the time horizon using a novel cross-sectional cross-validation method. Theoretically, we establish that our estimators are consistent under mild conditions. Monte Carlo simulations demonstrate that the proposed method performs well in finite samples. Empirically, we analyze daily stock returns for a balanced panel of S&P 500 stocks from January 2004 to December 2016. Through textual analysis, we examine specific events associated with the identified sparse factors that systematically influence the stock market. Our approach offers a new pathway for economists to study and understand the systematic risks of economic and financial systems over time., Comment: 66 pages, 6 figures

Driven by the algorithmic advancements in reinforcement learning and the increasing number of implementations of human-AI collaboration, Collaborative Reinforcement Learning (CRL) has been receiving growing attention. Despite this recent upsurge, this area is still rarely systematically studied. In this paper, we provide an extensive survey, investigating CRL methods based on both interactive reinforcement learning algorithms and human-AI collaborative frameworks that were proposed in the past decade. We elucidate and discuss via synergistic analysis methods both the growth of the field and the state-of-the-art; we conceptualise the existing frameworks from the perspectives of design patterns, collaborative levels, parties and capabilities, and review interactive methods and algorithmic models. Specifically, we create a new Human-AI CRL Design Trajectory Map, as a systematic modelling tool for the selection of existing CRL frameworks, as well as a method of designing new CRL systems, and finally of improving future CRL designs. Furthermore, we elaborate generic Human-AI CRL challenges, providing the research community with a guide towards novel research directions. The aim of this paper is to empower researchers with a systematic framework for the design of efficient and 'natural' human-AI collaborative methods, making it possible to work on maximised realisation of humans' and AI's potentials.

Sign language has been extensively studied as a means of facilitating effective communication between hearing individuals and the deaf community. With the continuous advancements in virtual reality (VR) and gamification technologies, an increasing number of studies have begun to explore the application of these emerging technologies in sign language learning. This paper describes a user study that compares the impact of 2D and 3D games on the user experience in ASL learning. Empirical evidence gathered through questionnaires supports the positive impact of 3D game environments on user engagement and overall experience, particularly in relation to attractiveness, usability, and efficiency. Moreover, initial findings demonstrate a similar behaviour of 2D and 3D games in terms of enhancing user experience. Finally, the study identifies areas where improvements can be made to enhance the dependability and clarity of 3D game environments. These findings contribute to the understanding of how game-based approaches, and specifically the utilisation of 3D environments, can positively influence the learning experience of ASL.

Ridge regression is an indispensable tool in big data analysis. Yet its inherent bias poses a significant and longstanding challenge, compromising both statistical efficiency and scalability across various applications. To tackle this critical issue, we introduce an iterative strategy to correct bias effectively when the dimension $p$ is less than the sample size $n$. For $p>n$, our method optimally mitigates the bias such that any remaining bias in the proposed de-biased estimator is unattainable through linear transformations of the response data. To address the remaining bias when $p>n$, we employ a Ridge-Screening (RS) method, producing a reduced model suitable for bias correction. Crucially, under certain conditions, the true model is nested within our selected one, highlighting RS as a novel variable selection approach. Through rigorous analysis, we establish the asymptotic properties and valid inferences of our de-biased ridge estimators for both $pn$, where, both $p$ and $n$ may increase towards infinity, along with the number of iterations. We further validate these results using simulated and real-world data examples. Our method offers a transformative solution to the bias challenge in ridge regression inferences across various disciplines., Comment: 48 pages, 10 figures

The field of Knowledge Tracing aims to understand how students learn and master knowledge over time by analyzing their historical behaviour data. To achieve this goal, many researchers have proposed Knowledge Tracing models that use data from Intelligent Tutoring Systems to predict students' subsequent actions. However, with the development of Intelligent Tutoring Systems, large-scale datasets containing long-sequence data began to emerge. Recent deep learning based Knowledge Tracing models face obstacles such as low efficiency, low accuracy, and low interpretability when dealing with large-scale datasets containing long-sequence data. To address these issues and promote the sustainable development of Intelligent Tutoring Systems, we propose a LSTM BERT-based Knowledge Tracing model for long sequence data processing, namely LBKT, which uses a BERT-based architecture with a Rasch model-based embeddings block to deal with different difficulty levels information and an LSTM block to process the sequential characteristic in students' actions. LBKT achieves the best performance on most benchmark datasets on the metrics of ACC and AUC. Additionally, an ablation study is conducted to analyse the impact of each component of LBKT's overall performance. Moreover, we used t-SNE as the visualisation tool to demonstrate the model's embedding strategy. The results indicate that LBKT is faster, more interpretable, and has a lower memory cost than the traditional deep learning based Knowledge Tracing methods.

A Nehari manifold optimization method (NMOM) is introduced for finding 1-saddles, i.e., saddle points with the Morse index equal to one, of a generic nonlinear functional in Hilbert spaces. Actually, it is based on the variational characterization that 1-saddles of this functional are local minimizers of the same functional restricted on the associated Nehari manifold. The framework contains two important ingredients: one is the retraction mapping to make the iterative points always lie on the Nehari manifold; the other is the tangential search direction to decrease the functional with suitable step-size search rules. Particularly, the global convergence is rigorously established by virtue of some crucial analysis techniques (including a weak convergence method) that overcome difficulties in the infinite-dimensional setting. In practice, combining with an easy-to-implement Nehari retraction and the negative Riemannian gradient direction, the NMOM is successfully applied to compute the unstable ground-state solutions of a class of typical semilinear elliptic PDEs, such as the H\'enon equation and the stationary nonlinear Schr\"odinger equation. In particular, the symmetry-breaking phenomenon of the ground states of the H\'enon equation is explored numerically in 1D and 2D with interesting numerical findings on the critical value of the symmetry-breaking reported., Comment: 28 pages, 8 figures

This paper proposes a new multi-linear projection method for denoising and estimation of high-dimensional matrix-variate factor time series. It assumes that a $p_1\times p_2$ matrix-variate time series consists of a dynamically dependent, lower-dimensional matrix-variate factor process and a $p_1\times p_2$ matrix idiosyncratic series. In addition, the latter series assumes a matrix-variate factor structure such that its row and column covariances may have diverging/spiked eigenvalues to accommodate the case of low signal-to-noise ratio often encountered in applications. We use an iterative projection procedure to reduce the dimensions and noise effects in estimating front and back loading matrices and to obtain faster convergence rates than those of the traditional methods available in the literature. We further introduce a two-way projected Principal Component Analysis to mitigate the diverging noise effects, and implement a high-dimensional white-noise testing procedure to estimate the dimension of the matrix factor process. Asymptotic properties of the proposed method are established if the dimensions and sample size go to infinity. We also use simulations and real examples to assess the performance of the proposed method in finite samples and to compare its forecasting ability with some existing ones in the literature. The proposed method fares well in out-of-sample forecasting. In a supplement, we demonstrate the efficacy of the proposed approach even when the idiosyncratic terms exhibit serial correlations with or without a diverging white noise effect., Comment: 66 Pages

This paper proposes a novel dynamic forecasting method using a new supervised Principal Component Analysis (PCA) when a large number of predictors are available. The new supervised PCA provides an effective way to bridge the gap between predictors and the target variable of interest by scaling and combining the predictors and their lagged values, resulting in an effective dynamic forecasting. Unlike the traditional diffusion-index approach, which does not learn the relationships between the predictors and the target variable before conducting PCA, we first re-scale each predictor according to their significance in forecasting the targeted variable in a dynamic fashion, and a PCA is then applied to a re-scaled and additive panel, which establishes a connection between the predictability of the PCA factors and the target variable. Furthermore, we also propose to use penalized methods such as the LASSO approach to select the significant factors that have superior predictive power over the others. Theoretically, we show that our estimators are consistent and outperform the traditional methods in prediction under some mild conditions. We conduct extensive simulations to verify that the proposed method produces satisfactory forecasting results and outperforms most of the existing methods using the traditional PCA. A real example of predicting U.S. macroeconomic variables using a large number of predictors showcases that our method fares better than most of the existing ones in applications. The proposed method thus provides a comprehensive and effective approach for dynamic forecasting in high-dimensional data analysis., Comment: 58 pages, 7 figures

This paper proposes a new approach to identifying the effective cointegration rank in high-dimensional unit-root (HDUR) time series from a prediction perspective using reduced-rank regression. For a HDUR process $\mathbf{x}_t\in \mathbb{R}^N$ and a stationary series $\mathbf{y}_t\in \mathbb{R}^p$ of interest, our goal is to predict future values of $\mathbf{y}_t$ using $\mathbf{x}_t$ and lagged values of $\mathbf{y}_t$. The proposed framework consists of a two-step estimation procedure. First, the Principal Component Analysis is used to identify all cointegrating vectors of $\mathbf{x}_t$. Second, the co-integrated stationary series are used as regressors, together with some lagged variables of $\mathbf{y}_t$, to predict $\mathbf{y}_t$. The estimated reduced rank is then defined as the effective cointegration rank of $\mathbf{x}_t$. Under the scenario that the autoregressive coefficient matrices are sparse (or of low-rank), we apply the Least Absolute Shrinkage and Selection Operator (or the reduced-rank techniques) to estimate the autoregressive coefficients when the dimension involved is high. Theoretical properties of the estimators are established under the assumptions that the dimensions $p$ and $N$ and the sample size $T \to \infty$. Both simulated and real examples are used to illustrate the proposed framework, and the empirical application suggests that the proposed procedure fares well in predicting stock returns.

The mirror of a toric orbifold is an affine curve called the mirror curve. In this paper, firstly, we recall the basic tools in tropical geometry and give a definition of the mirror curve. Then we calculate the tropical spine of the mirror curve for a smooth toric Calabi-Yau 3-fold near the large radius limit. Finally, we recall a special kind of real algebraic curves called the cyclic-M curve and show that under some special choices of parameters near the large radius limit, the mirror curve is a cyclic-M curve. Applying Mikhalkin's result of cyclic-M curves, we show that the mirror curve is glued from tubes and pairs of pants.

Motivation: The abundance of gene flow in the Tree of Life challenges the notion that evolution can be represented with a fully bifurcating process, as this process cannot capture important biological realities like hybridization, introgression, or horizontal gene transfer. Coalescent-based network methods are increasingly popular, yet not scalable for big data, because they need to perform a heuristic search in the space of networks as well as numerical optimization that can be NP-hard. Results: Here, we introduce a novel method to reconstruct phylogenetic networks based on algebraic invariants. While there is a long tradition of using algebraic invariants in phylogenetics, our work is the first to define phylogenetic invariants on concordance factors (frequencies of 4-taxon splits in the input gene trees) to identify level-1 phylogenetic networks under the multispecies coalescent model. Our novel inference methodology is optimization-free as it only requires the evaluation of polynomial equations, and as such, it bypasses the traversal of network space, yielding a computational speed at least 10 times faster than the fastest-to-date network methods. We illustrate the accuracy and speed of our new method on a variety of simulated scenarios as well as in the estimation of a phylogenetic network for the genus Canis. Availability and Implementation: We implement our novel theory on an open-source publicly available Julia package PhyloDiamond.jl available at https://github.com/solislemuslab/PhyloDiamond.jl with broad applicability within the evolutionary biology community. Contact: solislemus@wisc.edu

Recently, improving the robustness of policies across different environments attracts increasing attention in the reinforcement learning (RL) community. Existing robust RL methods mostly aim to achieve the max-min robustness by optimizing the policy's performance in the worst-case environment. However, in practice, a user that uses an RL policy may have different preferences over its performance across environments. Clearly, the aforementioned max-min robustness is oftentimes too conservative to satisfy user preference. Therefore, in this paper, we integrate user preference into policy learning in robust RL, and propose a novel User-Oriented Robust RL (UOR-RL) framework. Specifically, we define a new User-Oriented Robustness (UOR) metric for RL, which allocates different weights to the environments according to user preference and generalizes the max-min robustness metric. To optimize the UOR metric, we develop two different UOR-RL training algorithms for the scenarios with or without a priori known environment distribution, respectively. Theoretically, we prove that our UOR-RL training algorithms converge to near-optimal policies even with inaccurate or completely no knowledge about the environment distribution. Furthermore, we carry out extensive experimental evaluations in 4 MuJoCo tasks. The experimental results demonstrate that UOR-RL is comparable to the state-of-the-art baselines under the average and worst-case performance metrics, and more importantly establishes new state-of-the-art performance under the UOR metric.

In recent years, multi-agent reinforcement learning (MARL) has presented impressive performance in various applications. However, physical limitations, budget restrictions, and many other factors usually impose \textit{constraints} on a multi-agent system (MAS), which cannot be handled by traditional MARL frameworks. Specifically, this paper focuses on constrained MASes where agents work \textit{cooperatively} to maximize the expected team-average return under various constraints on expected team-average costs, and develops a \textit{constrained cooperative MARL} framework, named DeCOM, for such MASes. In particular, DeCOM decomposes the policy of each agent into two modules, which empowers information sharing among agents to achieve better cooperation. In addition, with such modularization, the training algorithm of DeCOM separates the original constrained optimization into an unconstrained optimization on reward and a constraints satisfaction problem on costs. DeCOM then iteratively solves these problems in a computationally efficient manner, which makes DeCOM highly scalable. We also provide theoretical guarantees on the convergence of DeCOM's policy update algorithm. Finally, we validate the effectiveness of DeCOM with various types of costs in both toy and large-scale (with 500 agents) environments., Comment: 25 pages

Force control is essential for medical robots when touching and contacting the patient's body. To increase the stability and efficiency in force control, an Adaption Module could be used to adjust the parameters for different contact situations. We propose an adaptive controller with an Adaption Module which can produce control parameters based on force feedback and real-time stiffness detection. We develop methods for learning the optimal policies by value iteration and using the data generated from those policies to train the Adaptive Module. We test this controller on different zones of a person's arm. All the parameters used in practice are learned from data. The experiments show that the proposed adaptive controller can exert various target forces on different zones of the arm with fast convergence and good stability.

Face reenactment is a challenging task, as it is difficult to maintain accurate expression, pose and identity simultaneously. Most existing methods directly apply driving facial landmarks to reenact source faces and ignore the intrinsic gap between two identities, resulting in the identity mismatch issue. Besides, they neglect the entanglement of expression and pose features when encoding driving faces, leading to inaccurate expressions and visual artifacts on large-pose reenacted faces. To address these problems, we propose a Large-pose Identity-preserving face reenactment network, LI-Net. Specifically, the Landmark Transformer is adopted to adjust driving landmark images, which aims to narrow the identity gap between driving and source landmark images. Then the Face Rotation Module and the Expression Enhancing Generator decouple the transformed landmark image into pose and expression features, and reenact those attributes separately to generate identity-preserving faces with accurate expressions and poses. Both qualitative and quantitative experimental results demonstrate the superiority of our method., Comment: IEEE International Conference on Multimedia and Expo(ICME) 2021 Oral

This paper proposes a hierarchical approximate-factor approach to analyzing high-dimensional, large-scale heterogeneous time series data using distributed computing. The new method employs a multiple-fold dimension reduction procedure using Principal Component Analysis (PCA) and shows great promises for modeling large-scale data that cannot be stored nor analyzed by a single machine. Each computer at the basic level performs a PCA to extract common factors among the time series assigned to it and transfers those factors to one and only one node of the second level. Each 2nd-level computer collects the common factors from its subordinates and performs another PCA to select the 2nd-level common factors. This process is repeated until the central server is reached, which collects common factors from its direct subordinates and performs a final PCA to select the global common factors. The noise terms of the 2nd-level approximate factor model are the unique common factors of the 1st-level clusters. We focus on the case of 2 levels in our theoretical derivations, but the idea can easily be generalized to any finite number of hierarchies. We discuss some clustering methods when the group memberships are unknown and introduce a new diffusion index approach to forecasting. We further extend the analysis to unit-root nonstationary time series. Asymptotic properties of the proposed method are derived for the diverging dimension of the data in each computing unit and the sample size $T$. We use both simulated data and real examples to assess the performance of the proposed method in finite samples, and compare our method with the commonly used ones in the literature concerning the forecastability of extracted factors., Comment: 48 pages, 10 figures

We present a new algorithm A*+BFHS for solving problems with unit-cost operators where A* and IDA* fail due to memory limitations and/or the existence of many distinct paths between the same pair of nodes. A*+BFHS is based on A* and breadth-first heuristic search (BFHS). A*+BFHS combines advantages from both algorithms, namely A*'s node ordering, BFHS's memory savings, and both algorithms' duplicate detection. On easy problems, A*+BFHS behaves the same as A*. On hard problems, it is slower than A* but saves a large amount of memory. Compared to BFIDA*, A*+BFHS reduces the search time and/or memory requirement by several times on a variety of planning domains., Comment: 8 pages, 5 figures, 1 table

We propose a new framework for modeling high-dimensional matrix-variate time series by a two-way transformation, where the transformed data consist of a matrix-variate factor process, which is dynamically dependent, and three other blocks of white noises. Specifically, for a given $p_1\times p_2$ matrix-variate time series, we seek common nonsingular transformations to project the rows and columns onto another $p_1$ and $p_2$ directions according to the strength of the dynamic dependence of the series on the past values. Consequently, we treat the data as nonsingular linear row and column transformations of dynamically dependent common factors and white noise idiosyncratic components. We propose a common orthonormal projection method to estimate the front and back loading matrices of the matrix-variate factors. Under the setting that the largest eigenvalues of the covariance of the vectorized idiosyncratic term diverge for large $p_1$ and $p_2$, we introduce a two-way projected Principal Component Analysis (PCA) to estimate the associated loading matrices of the idiosyncratic terms to mitigate such diverging noise effects. A diagonal-path white noise testing procedure is proposed to estimate the order of the factor matrix. %under the assumption that the idiosyncratic term is a matrix-variate white noise process. Asymptotic properties of the proposed method are established for both fixed and diverging dimensions as the sample size increases to infinity. We use simulated and real examples to assess the performance of the proposed method. We also compare our method with some existing ones in the literature and find that the proposed approach not only provides interpretable results but also performs well in out-of-sample forecasting., Comment: 49 pages, 6 figures