One of the main computational bottlenecks when working with kernel based learning is dealing with the large and typically dense kernel matrix. Techniques dealing with fast approximations of the matrix vector product for these kernel matrices typically deteriorate in their performance if the feature vectors reside in higher-dimensional feature spaces. We here present a technique based on the non-equispaced fast Fourier transform (NFFT) with rigorous error analysis. We show that this approach is also well suited to allow the approximation of the matrix that arises when the kernel is differentiated with respect to the kernel hyperparameters; a problem often found in the training phase of methods such as Gaussian processes. We also provide an error analysis for this case. We illustrate the performance of the additive kernel scheme with fast matrix vector products on a number of data sets.

Real-world autonomous driving systems must make safe decisions in the face of rare and diverse traffic scenarios. Current state-of-the-art planners are mostly evaluated on real-world datasets like nuScenes (open-loop) or nuPlan (closed-loop). In particular, nuPlan seems to be an expressive evaluation method since it is based on real-world data and closed-loop, yet it mostly covers basic driving scenarios. This makes it difficult to judge a planner's capabilities to generalize to rarely-seen situations. Therefore, we propose a novel closed-loop benchmark interPlan containing several edge cases and challenging driving scenarios. We assess existing state-of-the-art planners on our benchmark and show that neither rule-based nor learning-based planners can safely navigate the interPlan scenarios. A recently evolving direction is the usage of foundation models like large language models (LLM) to handle generalization. We evaluate an LLM-only planner and introduce a novel hybrid planner that combines an LLM-based behavior planner with a rule-based motion planner that achieves state-of-the-art performance on our benchmark.

We challenge the perceived consensus that the application of deep learning to solve the automated driving planning task necessarily requires huge amounts of real-world data or highly realistic simulation. Focusing on a roundabout scenario, we show that this requirement can be relaxed in favour of targeted, simplistic simulated data. A benefit is that such data can be easily generated for critical scenarios that are typically underrepresented in realistic datasets. By applying vanilla behavioural cloning almost exclusively to lightweight simulated data, we achieve reliable and comfortable driving in a real-world test vehicle. We leverage an incremental development approach that includes regular in-vehicle testing to identify sim-to-real gaps, targeted data augmentation, and training scenario variations. In addition to a detailed description of the methodology, we share our lessons learned, touching upon scenario generation, simulation features, and evaluation metrics.

The training of support vector machines (SVMs) leads to large-scale quadratic programs (QPs) [1]. An efficient way to solve these optimization problems is the sequential minimal optimization (SMO) algorithm introduced in [2]. The main motivation for the design of the SMO algorithm comes from the fact that existing optimization methods, i.e., quadratic programming approaches, cannot handle the large-scale dense kernel matrix efficiently. The SMO algorithm is motivated by the result obtained in [3] that showed the optimization problem can be decomposed into the solution of smaller subproblems, which avoids the large-scale dense matrices. When tackling the training as a QP programming task, the use of interior point methods (IPM) has also been studied in the seminal paper of Fine and Scheinberg in [4]: the authors use a low-rank approximation of the kernel matrix and propose a pivoted low-rank Cholesky decomposition to approximate the kernel matrix. A similar matrix approximation was also proposed in [5].

The field of motion prediction for automated driving has seen tremendous progress recently, bearing ever-more mighty neural network architectures. Leveraging these powerful models bears great potential for the closely related planning task. In this letter we propose a novel goal-conditioning method and show its potential to transform a state-of-the-art prediction model into a goal-directed planner. Our key insight is that conditioning prediction on a navigation goal at the behaviour level outperforms other widely adopted methods, with the additional benefit of increased model interpretability. We train our model on a large open-source dataset and show promising performance in a comprehensive benchmark.

Automated driving has the potential to revolutionize personal, public, and freight mobility. Besides the enormous challenge of perception, i.e. accurately perceiving the environment using available sensor data, automated driving comprises planning a safe, comfortable, and efficient motion trajectory. To promote safety and progress, many works rely on modules that predict the future motion of surrounding traffic. Modular automated driving systems commonly handle prediction and planning as sequential separate tasks. While this accounts for the influence of surrounding traffic on the ego-vehicle, it fails to anticipate the reactions of traffic participants to the ego-vehicle's behavior. Recent works suggest that integrating prediction and planning in an interdependent joint step is necessary to achieve safe, efficient, and comfortable driving. While various models implement such integrated systems, a comprehensive overview and theoretical understanding of different principles are lacking. We systematically review state-of-the-art deep learning-based prediction, planning, and integrated prediction and planning models. Different facets of the integration ranging from model architecture and model design to behavioral aspects are considered and related to each other. Moreover, we discuss the implications, strengths, and limitations of different integration methods. By pointing out research gaps, describing relevant future challenges, and highlighting trends in the research field, we identify promising directions for future research.

Predicting the future motion of observed vehicles is a crucial enabler for safe autonomous driving. The field of motion prediction has seen large progress recently with State-of-the-Art (SotA) models achieving impressive results on large-scale public benchmarks. However, recent work revealed that learning-based methods are prone to predict off-road trajectories in challenging scenarios. These can be created by perturbing existing scenarios with additional turns in front of the target vehicle while the motion history is left unchanged. We argue that this indicates that SotA models do not consider the map information sufficiently and demonstrate how this can be solved, by representing model inputs and outputs in a Frenet frame defined by lane centreline sequences. To this end, we present a general wrapper that leverages a Frenet representation of the scene and that can be applied to SotA models without changing their architecture. We demonstrate the effectiveness of this approach in a comprehensive benchmark using two SotA motion prediction models. Our experiments show that this reduces the off-road rate on challenging scenarios by more than 90\%, without sacrificing average performance.

High-dimensional data in the form of tensors are challenging for kernel classification methods. To both reduce the computational complexity and extract informative features, kernels based on low-rank tensor decompositions have been proposed. However, what decisive features of the tensors are exploited by these kernels is often unclear. In this paper we propose a novel kernel that is based on the Tucker decomposition. For this kernel the Tucker factors are computed based on re-weighting of the Tucker matrices with tuneable powers of singular values from the HOSVD decomposition. This provides a mechanism to balance the contribution of the Tucker core and factors of the data. We benchmark support tensor machines with this new kernel on several datasets. First we generate synthetic data where two classes differ in either Tucker factors or core, and compare our novel and previously existing kernels. We show robustness of the new kernel with respect to both classification scenarios. We further test the new method on real-world datasets. The proposed kernel has demonstrated a higher test accuracy than the state-of-the-art tensor train multi-way multi-level kernel, and a significantly lower computational time.

The accurate prediction of physicochemical properties of chemical compounds in mixtures (such as the activity coefficient at infinite dilution $\gamma_{ij}^\infty$) is essential for developing novel and more sustainable chemical processes. In this work, we analyze the performance of previously-proposed GNN-based models for the prediction of $\gamma_{ij}^\infty$, and compare them with several mechanistic models in a series of 9 isothermal studies. Moreover, we develop the Gibbs-Helmholtz Graph Neural Network (GH-GNN) model for predicting $\ln \gamma_{ij}^\infty$ of molecular systems at different temperatures. Our method combines the simplicity of a Gibbs-Helmholtz-derived expression with a series of graph neural networks that incorporate explicit molecular and intermolecular descriptors for capturing dispersion and hydrogen bonding effects. We have trained this model using experimentally determined $\ln \gamma_{ij}^\infty$ data of 40,219 binary-systems involving 1032 solutes and 866 solvents, overall showing superior performance compared to the popular UNIFAC-Dortmund model. We analyze the performance of GH-GNN for continuous and discrete inter/extrapolation and give indications for the model's applicability domain and expected accuracy. In general, GH-GNN is able to produce accurate predictions for extrapolated binary-systems if at least 25 systems with the same combination of solute-solvent chemical classes are contained in the training set and a similarity indicator above 0.35 is also present. This model and its applicability domain recommendations have been made open-source at https://github.com/edgarsmdn/GH-GNN.

Kernel matrices are crucial in many learning tasks such as support vector machines or kernel ridge regression. The kernel matrix is typically dense and large-scale. Depending on the dimension of the feature space even the computation of all of its entries in reasonable time becomes a challenging task. For such dense matrices the cost of a matrix-vector product scales quadratically with the dimensionality N , if no customized methods are applied. We propose the use of an ANOVA kernel, where we construct several kernels based on lower-dimensional feature spaces for which we provide fast algorithms realizing the matrix-vector products. We employ the non-equispaced fast Fourier transform (NFFT), which is of linear complexity for fixed accuracy. Based on a feature grouping approach, we then show how the fast matrix-vector products can be embedded into a learning method choosing kernel ridge regression and the conjugate gradient solver. We illustrate the performance of our approach on several data sets.