Consider a service system in which individuals are served by facilities at different locations within a geographical region. For example, the facilities could represent schools, polling places, or commercial fulfillment centers. The geographical partitioning problem (Carlsson & Devulapalli 2013) divides the region into non-overlapping districts, such that all individuals residing in the same district are served by the same facility. The goal is to choose a partition that optimizes some measure of social welfare, most commonly the average travel cost per individual (Carlsson et al. 2016). We formulate and study a novel variant of this problem where the population is heterogeneous, consisting of multiple demographic groups, each with a different spatial distribution throughout the region. Again we optimize the expected cost, but now we also impose a new group fairness condition: each subpopulation can be neither over-nor under-represented at any facility. In other words, the districts are designed in such a way that the proportion of the population belonging to a particular group in any district must match that group's incidence in the entire population. This condition is also known as "demographic parity" in the literature (Dwork et al. 2012).

Aerial manipulation combines the maneuverability of multirotors with the dexterity of robotic arms to perform complex tasks in cluttered spaces. Yet planning safe, dynamically feasible trajectories remains difficult due to whole-body collision avoidance and the conservativeness of common geometric abstractions such as bounding boxes or ellipsoids. We present a whole-body motion planning and safety-critical control framework for aerial manipulators built on superquadrics (SQs). Using an SQ-plus-proxy representation, we model both the vehicle and obstacles with differentiable, geometry-accurate surfaces. Leveraging this representation, we introduce a maximum-clearance planner that fuses Voronoi diagrams with an equilibrium-manifold formulation to generate smooth, collision-aware trajectories. We further design a safety-critical controller that jointly enforces thrust limits and collision avoidance via high-order control barrier functions. In simulation, our approach outperforms sampling-based planners in cluttered environments, producing faster, safer, and smoother trajectories and exceeding ellipsoid-based baselines in geometric fidelity. Actual experiments on a physical aerial-manipulation platform confirm feasibility and robustness, demonstrating consistent performance across simulation and hardware settings. The video can be found at https://youtu.be/hQYKwrWf1Ak.

I present a regression algorithm that provides a continuous, piecewise-smooth function approximating scattered data. It is based on composing and blending linear functions over Voronoi cells, and it scales to high dimensions. The algorithm infers Voronoi cells from seed vertices and constructs a linear function for the input data in and around each cell. As the algorithm does not explicitly compute the Voronoi diagram, it avoids the curse of dimensionality. An accuracy of around 98.2% on the MNIST dataset with 722,200 degrees of freedom (without data augmentation, convolution, or other geometric operators) demonstrates the applicability and scalability of the algorithm.

Coordinating a fully distributed multi-agent system (MAS) can be challenging when the communication channel has very limited capabilities in terms of sending rate and packet payload. When the MAS has to deal with active obstacles in a highly partially observable environment, the communication channel acquires considerable relevance. In this paper, we present an approach to deal with task assignments in extremely active scenarios, where tasks need to be frequently reallocated among the agents participating in the coordination process. Inspired by market-based task assignments, we introduce a novel distributed coordination method to orchestrate autonomous agents' actions efficiently in low communication scenarios. In particular, our algorithm takes into account asymmetric obstacles. While in the real world, the majority of obstacles are asymmetric, they are usually treated as symmetric ones, thus limiting the applicability of existing methods. To summarize, the presented architecture is designed to tackle scenarios where the obstacles are active and asymmetric, the communication channel is poor and the environment is partially observable. Our approach has been validated in simulation and in the real world, using a team of NAO robots during official RoboCup competitions. Experimental results show a notable reduction in task overlaps in limited communication settings, with a decrease of 52% in the most frequent reallocated task.

Due to their inherent ability to process data in a graph-based format, GCNs are especially well-suited for applications where relational context and structural information play a crucial role. This introduction explores recent advancements in image classification using GCNs, with a particular focus on the integration of superpixels and wavelet techniques. Graph Convolutional Networks (GCNs) have become a powerful framework for image classification, largely due to their ability to capture and process the non-Euclidean structure of data. Zhou et al. provided a comprehensive review of GCN applications across various domains, highlighting their effectiveness in image classification tasks [1]. Their study emphasized that, unlike traditional Convolutional Neural Networks (CNNs), GCNs excel at capturing long-range dependencies and complex relational patterns within images. Meanwhile, wavelet techniques [2] have transformed image processing by introducing a multi-resolution analysis framework that is essential for applications such as image compression, feature extraction, and noise reduction. In this context, Wavelet transforms have been utilized in conjunction with GCNs for feature extraction in image classification, denoising, compression and other tasks. Wavelets provide a multi-resolution image analysis, capturing spatial and frequency domain information [3]. This paper presents an innovative framework that integrates Graph Convolutional Networks (GCNs) with Voronoi diagrams for image classification, leveraging their exceptional ability to model relational data.

Sparse autoencoders (SAEs) have received considerable recent attention as tools for mechanistic interpretability, showing success at extracting interpretable features even from very large LLMs. However, this research has been largely empirical, and there have been recent doubts about the true utility of SAEs. In this work, we seek to enhance the theoretical understanding of SAEs, using the spline theory of deep learning. By situating SAEs in this framework: we discover that SAEs generalise ``$k$-means autoencoders'' to be piecewise affine, but sacrifice accuracy for interpretability vs. the optimal ``$k$-means-esque plus local principal component analysis (PCA)'' piecewise affine autoencoder. We characterise the underlying geometry of (TopK) SAEs using power diagrams. And we develop a novel proximal alternating method SGD (PAM-SGD) algorithm for training SAEs, with both solid theoretical foundations and promising empirical results in MNIST and LLM experiments, particularly in sample efficiency and (in the LLM setting) improved sparsity of codes. All code is available at: https://github.com/splInterp2025/splInterp

TabPFN [Hollmann et al., 2023], a Transformer model pretrained to perform in-context learning on fresh tabular classification problems, was presented at the last ICLR conference. To better understand its behavior, we treat it as a black-box function approximator generator and observe its generated function approximations on a varied selection of training datasets. Exploring its learned inductive biases in this manner, we observe behavior that is at turns either brilliant or baffling. We conclude this post with thoughts on how these results might inform the development, evaluation, and application of prior-data fitted networks (PFNs) in the future.

Pitch ownership models allow many types of analysis in soccer and provide valuable assistance to tactical analysts in understanding the game's dynamics. The novelty they provide over event-based analysis is that tracking data incorporates context that event-based data does not possess, like player positioning. This paper proposes a novel approach to building pitch ownership models in soccer games using the K-Nearest Neighbors (KNN) algorithm. Our approach provides a fast inference mechanism that can model different approaches to pitch control using the same algorithm. Despite its flexibility, it uses only three hyperparameters to tune the model, facilitating the tuning process for different player skill levels. The flexibility of the approach allows for the emulation of different methods available in the literature by adjusting a small number of parameters, including adjusting for different levels of uncertainty. In summary, the proposed model provides a new and more flexible strategy for building pitch ownership models, extending beyond just replicating existing algorithms, and can provide valuable insights for tactical analysts and open up new avenues for future research. We thoroughly visualize several examples demonstrating the presented models' strengths and weaknesses. The code is available at github.com/nvsclub/KNNPitchControl.

Deep learning models often struggle with generalization when deploying on real-world data, due to the common distributional shift to the training data. Test-time adaptation (TTA) is an emerging scheme used at inference time to address this issue. In TTA, models are adapted online at the same time when making predictions to test data. Neighbor-based approaches have gained attention recently, where prototype embeddings provide location information to alleviate the feature shift between training and testing data. However, due to their inherit limitation of simplicity, they often struggle to learn useful patterns and encounter performance degradation. To confront this challenge, we study the TTA problem from a geometric point of view. We first reveal that the underlying structure of neighbor-based methods aligns with the Voronoi Diagram, a classical computational geometry model for space partitioning. Building on this observation, we propose the Test-Time adjustment by Voronoi Diagram guidance (TTVD), a novel framework that leverages the benefits of this geometric property. Specifically, we explore two key structures: 1) Cluster-induced Voronoi Diagram (CIVD): This integrates the joint contribution of self-supervision and entropy-based methods to provide richer information. 2) Power Diagram (PD): A generalized version of the Voronoi Diagram that refines partitions by assigning weights to each Voronoi cell. Our experiments under rigid, peer-reviewed settings on CIFAR-10-C, CIFAR-100-C, ImageNet-C, and ImageNet-R shows that TTVD achieves remarkable improvements compared to state-of-the-art methods. Moreover, extensive experimental results also explore the effects of batch size and class imbalance, which are two scenarios commonly encountered in real-world applications. These analyses further validate the robustness and adaptability of our proposed framework.