To reduce the computational cost of humanoid motion generation, we introduce a new approach to representing robot kinematic reachability: the differentiable reachability map. This map is a scalar-valued function defined in the task space that takes positive values only in regions reachable by the robot's end-effector. A key feature of this representation is that it is continuous and differentiable with respect to task-space coordinates, enabling its direct use as constraints in continuous optimization for humanoid motion planning. We describe a method to learn such differentiable reachability maps from a set of end-effector poses generated using a robot's kinematic model, using either a neural network or a support vector machine as the learning model. By incorporating the learned reachability map as a constraint, we formulate humanoid motion generation as a continuous optimization problem. We demonstrate that the proposed approach efficiently solves various motion planning problems, including footstep planning, multi-contact motion planning, and loco-manipulation planning for humanoid robots.

Recent research has made some progress towards deep learning theory from the perspective of infinite-width NN.

Classical methods, such as weighted cross-entropy, fail when training deep nets to the terminal phase of training (TPT), that is training beyond zero training error.

Graph Neural Networks (GNNs) are a powerful solution in many settings where the data is naturally graph-structured; they enable us to learn not only from individual features of objects, but also from the connections between them [15, 20]. They make the graph topology a key part of the learning process, enabling more accurate results in a range of scenarios [19, 21]. However, graph learning is computationally expensive, and training GNNs is usually much slower than training for state of the art methods on tabular data [11, 5]. In this paper, we aim to achieve graph learning with simpler, more efficient methods for tabular data, after enriching graph nodes with information about the graph topology. Our stepping stone is the Weisfeiler-Leman algorithm (WL), a well-known technique for approximating graph isomorphism [18]. In addition to its many applications in graph theory and other areas of computer science, WL has been very influential in graph learning. It has been used in graph kernels, enabling algorithms such as support vector machines to operate directly on graphs [16, 11]. Crucially, it has been used to characterize the expressive power of several types of GNNs [8, 20, 1], suggesting WL as a suitable tool to distill the topological information that GNNs can learn from.

Cloud detection is essential for atmospheric retrievals, climate studies, and weather forecasting. We analyze infrared radiances from the Infrared Atmospheric Sounding Interferometer (IASI) onboard Meteorological Operational (MetOp) satellites to classify scenes as clear or cloudy. We apply the Support Vector Machine (SVM) approach, based on kernel methods for non-separable data. In this study, the method is implemented for Cloud Identification (CISVM) to classify the test set using radiances or brightness temperatures, with dimensionality reduction through Principal Component Analysis (PCA) and cloud-sensitive channel selection to focus on the most informative features. Our best configuration achieves 88.30 percent agreement with reference labels and shows strong consistency with cloud masks from the Moderate Resolution Imaging Spectroradiometer (MODIS), with the largest discrepancies in polar regions due to sensor differences. These results demonstrate that CISVM is a robust, flexible, and efficient method for automated cloud classification from infrared radiances, suitable for operational retrievals and future missions such as Far infrared Outgoing Radiation Understanding and Monitoring (FORUM), the ninth European Space Agency Earth Explorer Mission.