Pi,jCi,j γH(P) subjectto P Rt t+,PT1t =1t,P1t =1t, (6) where Pi,j is the transport plan andCi,j is the ground metric that measures the distance between point i in the source andj in the target. This will induce some smoothness and wiggle room in the solutionofourobjective. To increase the diversity of the observed trajectories, we inject Gaussian noise (σ = 0.05) into trajectories by perturbing the initial velocities. Since two-body systems are non-chaotic systems, we divide training set and testing set such that for training set[mmin,mmax] = [0.8,1.2], while testing set[mmin,mmax] = [0.9,1.3] to create domain distribution shifting. The initial velocities of all bodies are based on their initial positions by rotating itby 90 andscalingitbyr1.5.

--The integration of digital twinning technologies is driving next-generation networks toward new capabilities, allowing operators to thoroughly understand network conditions, efficiently analyze valuable radio data, and innovate applications through user-friendly, immersive interfaces. Building on this foundation, network digital twins (NDTs) accurately depict the operational processes and attributes of network infrastructures, facilitating predictive management through real-time analysis and measurement. However, constructing precise NDTs poses challenges, such as integrating diverse data sources, mapping necessary attributes from physical networks, and maintaining scalability for various downstream tasks. Unlike previous works that focused on the creation and mapping of NDTs from scratch, we explore intra-and inter-operations among NDTs within an Unified Twin Transformation (UTT) framework, which uncovers a new computing paradigm for efficient transfer, merging, and splitting of NDTs to create task-oriented twins. By leveraging joint multi-modal and distributed mapping mechanisms, UTT optimizes resource utilization and reduces the cost of creating NDTs, while ensuring twin model consistency. A theoretical analysis of the distributed mapping problem is conducted to establish convergence bounds for this multi-modal gated aggregation process. Evaluations on real-world twin-assisted applications, such as trajectory reconstruction, human localization, and sensory data generation, demonstrate the feasibility and effectiveness of interoperability among NDTs for corresponding task development. In the domain of telecommunications, wireless networks are experiencing a paradigmatic evolution, driven by the integration of advanced technologies such as edge computing [1], millimeter-wave communication [2], and machine learning [3]. These technologies are instrumental in laying groundwork for an array of novel applications and services in mixed physical and digital contexts, boosting capabilities of mobile broadband and enabling thorough integration of cyber-physical interactive systems.

Relocalization, the process of re-establishing a robot's position within an environment, is crucial for ensuring accurate navigation and task execution when external positioning information, such as GPS, is unavailable or has been lost. Subterranean environments present significant challenges for relocalization due to limited external positioning information, poor lighting that affects camera localization, irregular and often non-distinct surfaces, and dust, which can introduce noise and occlusion in sensor data. In this work, we propose a robust, computationally friendly framework for relocalization through point cloud registration utilizing a prior point cloud map. The framework employs Intrinsic Shape Signatures (ISS) to select feature points in both the target and prior point clouds. The Fast Point Feature Histogram (FPFH) algorithm is utilized to create descriptors for these feature points, and matching these descriptors yields correspondences between the point clouds. A 3D transformation is estimated using the matched points, which initializes a Normal Distribution Transform (NDT) registration. The transformation result from NDT is further refined using the Iterative Closest Point (ICP) registration algorithm. This framework enhances registration accuracy even in challenging conditions, such as dust interference and significant initial transformations between the target and source, making it suitable for autonomous robots operating in underground mines and tunnels. This framework was validated with experiments in simulated and real-world mine datasets, demonstrating its potential for improving relocalization.

Neurosymbolic (NeSy) AI studies the integration of neural networks (NNs) and symbolic reasoning based on logic. Usually, NeSy techniques focus on learning the neural, probabilistic and/or fuzzy parameters of NeSy models. Learning the symbolic or logical structure of such models has, so far, received less attention. We introduce neurosymbolic decision trees (NDTs), as an extension of decision trees together with a novel NeSy structure learning algorithm, which we dub NeuID3. NeuID3 adapts the standard top-down induction of decision tree algorithms and combines it with a neural probabilistic logic representation, inherited from the DeepProbLog family of models. The key advantage of learning NDTs with NeuID3 is the support of both symbolic and subsymbolic data (such as images), and that they can exploit background knowledge during the induction of the tree structure, In our experimental evaluation we demonstrate the benefits of NeSys structure learning over more traditonal approaches such as purely data-driven learning with neural networks.

In autonomous driving, predicting the behavior (turning left, stopping, etc.) of target vehicles is crucial for the self-driving vehicle to make safe decisions and avoid accidents. Existing deep learning-based methods have shown excellent and accurate performance, but the black-box nature makes it untrustworthy to apply them in practical use. In this work, we explore the interpretability of behavior prediction of target vehicles by an Episodic Memory implanted Neural Decision Tree (abbrev. eMem-NDT). The structure of eMem-NDT is constructed by hierarchically clustering the text embedding of vehicle behavior descriptions. eMem-NDT is a neural-backed part of a pre-trained deep learning model by changing the soft-max layer of the deep model to eMem-NDT, for grouping and aligning the memory prototypes of the historical vehicle behavior features in training data on a neural decision tree. Each leaf node of eMem-NDT is modeled by a neural network for aligning the behavior memory prototypes. By eMem-NDT, we infer each instance in behavior prediction of vehicles by bottom-up Memory Prototype Matching (MPM) (searching the appropriate leaf node and the links to the root node) and top-down Leaf Link Aggregation (LLA) (obtaining the probability of future behaviors of vehicles for certain instances). We validate eMem-NDT on BLVD and LOKI datasets, and the results show that our model can obtain a superior performance to other methods with clear explainability. The code is available at https://github.com/JWFangit/eMem-NDT.

Because scanning-LIDAR sensors require finite time to create a point cloud, sensor motion during a scan warps the resulting image, a phenomenon known as motion distortion or rolling shutter. Motion-distortion correction methods exist, but they rely on external measurements or Bayesian filtering over multiple LIDAR scans. In this paper we propose a novel algorithm that performs snapshot processing to obtain a motion-distortion correction. Snapshot processing, which registers a current LIDAR scan to a reference image without using external sensors or Bayesian filtering, is particularly relevant for localization to a high-definition (HD) map. Our approach, which we call Velocity-corrected Iterative Compact Ellipsoidal Transformation (VICET), extends the well-known Normal Distributions Transform (NDT) algorithm to solve jointly for both a 6 Degree-of-Freedom (DOF) rigid transform between two LIDAR scans and a set of 6DOF motion states that describe distortion within the current LIDAR scan. Using experiments, we show that VICET achieves significantly higher accuracy than NDT or Iterative Closest Point (ICP) algorithms when localizing a distorted raw LIDAR scan against an undistorted HD Map. We recommend the reader explore our open-source code and visualizations at https://github.com/mcdermatt/VICET, which supplements this manuscript.

Distribution-to-Distribution (D2D) point cloud registration algorithms are fast, interpretable, and perform well in unstructured environments. Unfortunately, existing strategies for predicting solution error for these methods are overly optimistic, particularly in regions containing large or extended physical objects. In this paper we introduce the Iterative Closest Ellipsoidal Transform (ICET), a novel 3D LIDAR scan-matching algorithm that re-envisions NDT in order to provide robust accuracy prediction from first principles. Like NDT, ICET subdivides a LIDAR scan into voxels in order to analyze complex scenes by considering many smaller local point distributions, however, ICET assesses the voxel distribution to distinguish random noise from deterministic structure. ICET then uses a weighted least-squares formulation to incorporate this noise/structure distinction into computing a localization solution and predicting the solution-error covariance. In order to demonstrate the reasonableness of our accuracy predictions, we verify 3D ICET in three LIDAR tests involving real-world automotive data, high-fidelity simulated trajectories, and simulated corner-case scenes. For each test, ICET consistently performs scan matching with sub-centimeter accuracy. This level of accuracy, combined with the fact that the algorithm is fully interpretable, make it well suited for safety-critical transportation applications. Code is available at https://github.com/mcdermatt/ICET

Existing radar sensors can be classified into automotive and scanning radars. While most radar odometry (RO) methods are only designed for a specific type of radar, our RO method adapts to both scanning and automotive radars. Our RO is simple yet effective, where the pipeline consists of thresholding, probabilistic submap building, and an NDT-based radar scan matching. The proposed RO has been tested on two public radar datasets: the Oxford Radar RobotCar dataset and the nuScenes dataset, which provide scanning and automotive radar data respectively. The results show that our approach surpasses state-of-the-art RO using either automotive or scanning radar by reducing translational error by 51% and 30%, respectively, and rotational error by 17% and 29%, respectively. Besides, we show that our RO achieves centimeter-level accuracy as lidar odometry, and automotive and scanning RO have similar accuracy.

In this paper, we propose a novel probabilistic variant of iterative closest point (ICP) dubbed as CoBigICP. The method leverages both local geometrical information and global noise characteristics. Locally, the 3D structure of both target and source clouds are incorporated into the objective function through bidirectional correspondence. Globally, error metric of correntropy is introduced as noise model to resist outliers. Importantly, the close resemblance between normal-distributions transform (NDT) and correntropy is revealed. To ease the minimization step, an on-manifold parameterization of the special Euclidean group is proposed. Extensive experiments validate that CoBigICP outperforms several well-known and state-of-the-art methods.

When analyzing a dataset, it can be useful to assess how smooth the decision boundaries need to be for a model to better fit the data. This paper addresses this question by proposing the quantification of how much should the 'rigid' decision boundaries, produced by an algorithm that naturally finds such solutions, be relaxed to obtain a performance improvement. The approach we propose starts with the rigid decision boundaries of a seed Decision Tree (seed DT), which is used to initialize a Neural DT (NDT). The initial boundaries are challenged by relaxing them progressively through training the NDT. During this process, we measure the NDT's performance and decision agreement to its seed DT. We show how these two measures can help the user in figuring out how expressive his model should be, before exploring it further via model selection. The validity of our approach is demonstrated with experiments on simulated and benchmark datasets.