One of the main limitations of the commonly used Absolute Trajectory Error (ATE) is that it is highly sensitive to outliers. As a result, in the presence of just a few outliers, it often fails to reflect the varying accuracy as the inlier trajectory error or the number of outliers varies. In this work, we propose an alternative error metric for evaluating the accuracy of the reconstructed camera trajectory. Our metric, named Discernible Trajectory Error (DTE), is computed in four steps: (1) Shift the ground-truth and estimated trajectories such that both of their geometric medians are located at the origin. (2) Rotate the estimated trajectory such that it minimizes the sum of geodesic distances between the corresponding camera orientations. (3) Scale the estimated trajectory such that the median distance of the cameras to their geometric median is the same as that of the ground truth. (4) Compute the distances between the corresponding cameras, and obtain the DTE by taking the average of the mean and root-mean-square (RMS) distance. This metric is an attractive alternative to the ATE, in that it is capable of discerning the varying trajectory accuracy as the inlier trajectory error or the number of outliers varies. Using the similar idea, we also propose a novel rotation error metric, named Discernible Rotation Error (DRE), which has similar advantages to the DTE. Furthermore, we propose a simple yet effective method for calibrating the camera-to-marker rotation, which is needed for the computation of our metrics. Our methods are verified through extensive simulations.
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In literature on imprecise probability little attention is paid to the fact that imprecise probabilities are precise on some events. We call these sets system of precision. We show that, under mild assumptions, the system of precision of a lower and upper probability form a so-called (pre-)Dynkin-system. Interestingly, there are several settings, ranging from machine learning on partial data over frequential probability theory to quantum probability theory and decision making under uncertainty, in which a priori the probabilities are only desired to be precise on a specific underlying set system. At the core of all of these settings lies the observation that precise beliefs, probabilities or frequencies on two events do not necessarily imply this precision to hold for the intersection of those events. Here, (pre-)Dynkin-systems have been adopted as systems of precision, too. We show that, under extendability conditions, those pre-Dynkin-systems equipped with probabilities can be embedded into algebras of sets. Surprisingly, the extendability conditions elaborated in a strand of work in quantum physics are equivalent to coherence in the sense of Walley (1991, Statistical reasoning with imprecise probabilities, p. 84). Thus, literature on probabilities on pre-Dynkin-systems gets linked to the literature on imprecise probability. Finally, we spell out a lattice duality which rigorously relates the system of precision to credal sets of probabilities. In particular, we provide a hitherto undescribed, parametrized family of coherent imprecise probabilities.
Road network and trajectory representation learning are essential for traffic systems since the learned representation can be directly used in various downstream tasks (e.g., traffic speed inference, and travel time estimation). However, most existing methods only contrast within the same scale, i.e., treating road network and trajectory separately, which ignores valuable inter-relations. In this paper, we aim to propose a unified framework that jointly learns the road network and trajectory representations end-to-end. We design domain-specific augmentations for road-road contrast and trajectory-trajectory contrast separately, i.e., road segment with its contextual neighbors and trajectory with its detour replaced and dropped alternatives, respectively. On top of that, we further introduce the road-trajectory cross-scale contrast to bridge the two scales by maximizing the total mutual information. Unlike the existing cross-scale contrastive learning methods on graphs that only contrast a graph and its belonging nodes, the contrast between road segment and trajectory is elaborately tailored via novel positive sampling and adaptive weighting strategies. We conduct prudent experiments based on two real-world datasets with four downstream tasks, demonstrating improved performance and effectiveness. The code is available at //github.com/mzy94/JCLRNT.
Real-time monitoring of human behaviours, especially in e-Health applications, has been an active area of research in the past decades. On top of IoT-based sensing environments, anomaly detection algorithms have been proposed for the early detection of abnormalities. Gradual change procedures, commonly referred to as drift anomalies, have received much less attention in the literature because they represent a much more challenging scenario than sudden temporary changes (point anomalies). In this paper, we propose, for the first time, a fully unsupervised real-time drift detection algorithm named DynAmo, which can identify drift periods as they are happening. DynAmo comprises a dynamic clustering component to capture the overall trends of monitored behaviours and a trajectory generation component, which extracts features from the densest cluster centroids. Finally, we apply an ensemble of divergence tests on sliding reference and detection windows to detect drift periods in the behavioural sequence.
Trajectory data has the potential to greatly benefit a wide-range of real-world applications, such as tracking the spread of the disease through people's movement patterns and providing personalized location-based services based on travel preference. However, privay concerns and data protection regulations have limited the extent to which this data is shared and utilized. To overcome this challenge, local differential privacy provides a solution by allowing people to share a perturbed version of their data, ensuring privacy as only the data owners have access to the original information. Despite its potential, existing point-based perturbation mechanisms are not suitable for real-world scenarios due to poor utility, dependence on external knowledge, high computational overhead, and vulnerability to attacks. To address these limitations, we introduce LDPTrace, a novel locally differentially private trajectory synthesis framework. Our framework takes into account three crucial patterns inferred from users' trajectories in the local setting, allowing us to synthesize trajectories that closely resemble real ones with minimal computational cost. Additionally, we present a new method for selecting a proper grid granularity without compromising privacy. Our extensive experiments using real-world data, various utility metrics and attacks, demonstrate the efficacy and efficiency of LDPTrace.
Continuous trajectory tracking control of quadrotors is complicated when considering noise from the environment. Due to the difficulty in modeling the environmental dynamics, tracking methodologies based on conventional control theory, such as model predictive control, have limitations on tracking accuracy and response time. We propose a Time-attenuating Twin Delayed DDPG, a model-free algorithm that is robust to noise, to better handle the trajectory tracking task. A deep reinforcement learning framework is constructed, where a time decay strategy is designed to avoid trapping into local optima. The experimental results show that the tracking error is significantly small, and the operation time is one-tenth of that of a traditional algorithm. The OpenAI Mujoco tool is used to verify the proposed algorithm, and the simulation results show that, the proposed method can significantly improve the training efficiency and effectively improve the accuracy and convergence stability.
Federated Learning (FL) is a distributed machine learning paradigm where clients collaboratively train a model using their local (human-generated) datasets. While existing studies focus on FL algorithm development to tackle data heterogeneity across clients, the important issue of data quality (e.g., label noise) in FL is overlooked. This paper aims to fill this gap by providing a quantitative study on the impact of label noise on FL. We derive an upper bound for the generalization error that is linear in the clients' label noise level. Then we conduct experiments on MNIST and CIFAR-10 datasets using various FL algorithms. Our empirical results show that the global model accuracy linearly decreases as the noise level increases, which is consistent with our theoretical analysis. We further find that label noise slows down the convergence of FL training, and the global model tends to overfit when the noise level is high.
The field of Automatic Machine Learning (AutoML) has recently attained impressive results, including the discovery of state-of-the-art machine learning solutions, such as neural image classifiers. This is often done by applying an evolutionary search method, which samples multiple candidate solutions from a large space and evaluates the quality of each candidate through a long training process. As a result, the search tends to be slow. In this paper, we show that large efficiency gains can be obtained by employing a fast unified functional hash, especially through the functional equivalence caching technique, which we also present. The central idea is to detect by hashing when the search method produces equivalent candidates, which occurs very frequently, and this way avoid their costly re-evaluation. Our hash is "functional" in that it identifies equivalent candidates even if they were represented or coded differently, and it is "unified" in that the same algorithm can hash arbitrary representations; e.g. compute graphs, imperative code, or lambda functions. As evidence, we show dramatic improvements on multiple AutoML domains, including neural architecture search and algorithm discovery. Finally, we consider the effect of hash collisions, evaluation noise, and search distribution through empirical analysis. Altogether, we hope this paper may serve as a guide to hashing techniques in AutoML.
We propose in this paper efficient first/second-order time-stepping schemes for the evolutional Navier-Stokes-Nernst-Planck-Poisson equations. The proposed schemes are constructed using an auxiliary variable reformulation and sophisticated treatment of the terms coupling different equations. By introducing a dynamic equation for the auxiliary variable and reformulating the original equations into an equivalent system, we construct first- and second-order semi-implicit linearized schemes for the underlying problem. The main advantages of the proposed method are: (1) the schemes are unconditionally stable in the sense that a discrete energy keeps decay during the time stepping; (2) the concentration components of the discrete solution preserve positivity and mass conservation; (3) the delicate implementation shows that the proposed schemes can be very efficiently realized, with computational complexity close to a semi-implicit scheme. Some numerical examples are presented to demonstrate the accuracy and performance of the proposed method. As far as the best we know, this is the first second-order method which satisfies all the above properties for the Navier-Stokes-Nernst-Planck-Poisson equations.
Recent advances in sensor and mobile devices have enabled an unprecedented increase in the availability and collection of urban trajectory data, thus increasing the demand for more efficient ways to manage and analyze the data being produced. In this survey, we comprehensively review recent research trends in trajectory data management, ranging from trajectory pre-processing, storage, common trajectory analytic tools, such as querying spatial-only and spatial-textual trajectory data, and trajectory clustering. We also explore four closely related analytical tasks commonly used with trajectory data in interactive or real-time processing. Deep trajectory learning is also reviewed for the first time. Finally, we outline the essential qualities that a trajectory management system should possess in order to maximize flexibility.
Graph Neural Networks (GNNs), which generalize deep neural networks to graph-structured data, have drawn considerable attention and achieved state-of-the-art performance in numerous graph related tasks. However, existing GNN models mainly focus on designing graph convolution operations. The graph pooling (or downsampling) operations, that play an important role in learning hierarchical representations, are usually overlooked. In this paper, we propose a novel graph pooling operator, called Hierarchical Graph Pooling with Structure Learning (HGP-SL), which can be integrated into various graph neural network architectures. HGP-SL incorporates graph pooling and structure learning into a unified module to generate hierarchical representations of graphs. More specifically, the graph pooling operation adaptively selects a subset of nodes to form an induced subgraph for the subsequent layers. To preserve the integrity of graph's topological information, we further introduce a structure learning mechanism to learn a refined graph structure for the pooled graph at each layer. By combining HGP-SL operator with graph neural networks, we perform graph level representation learning with focus on graph classification task. Experimental results on six widely used benchmarks demonstrate the effectiveness of our proposed model.
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