We introduce a new linear-algebraic tool based on group representation theory, and use it to address three key problems in machine learning.1. Past researchers have proposed fast attention algorithms for LLMs by approximating or replace softmax attention with other functions, such as low-degree polynomials. The key property of these functions is that, when applied entry-wise to the matrix $QK^{\top}$, the result is a low rank matrix when $Q$ and $K$ are $n \times d$ matrices and $n \gg d$. This suggests a natural question: what are all functions $f$ with this property? If other $f$ exist and are quickly computable, they can be used in place of softmax for fast subquadratic attention algorithms.

This suggests that the low-rank fast attention only works for functions approx-imable by polynomials.

Many real-world decision-making problems involve optimizing multiple objectives simultaneously, rendering the selection of the most preferred solution a non-trivial problem: All Pareto optimal solutions are viable candidates, and it is typically up to a decision maker to select one for implementation based on their subjective preferences. To reduce the cognitive load on the decision maker, previous work has introduced the Pareto pruning problem, where the goal is to compute a fixed-size subset of Pareto optimal solutions that best represent the full set, as evaluated by a given quality measure. Reframing Pareto pruning as a multiwinner voting problem, we conduct an axiomatic analysis of existing quality measures, uncovering several unintuitive behaviors. Motivated by these findings, we introduce a new measure, directed coverage. We also analyze the computational complexity of optimizing various quality measures, identifying previously unknown boundaries between tractable and intractable cases depending on the number and structure of the objectives. Finally, we present an experimental evaluation, demonstrating that the choice of quality measure has a decisive impact on the characteristics of the selected set of solutions and that our proposed measure performs competitively or even favorably across a range of settings.

Manhattan Distance Mapping (MDM) is a post-training deep neural network (DNN) weight mapping technique for memristive bit-sliced compute-in-memory (CIM) crossbars that reduces parasitic resistance (PR) nonidealities. PR limits crossbar efficiency by mapping DNN matrices into small crossbar tiles, reducing CIM-based speedup. Each crossbar executes one tile, requiring digital synchronization before the next layer. At this granularity, designers either deploy many small crossbars in parallel or reuse a few sequentially-both increasing analog-to-digital conversions, latency, I/O pressure, and chip area. MDM alleviates PR effects by optimizing active-memristor placement. Exploiting bit-level structured sparsity, it feeds activations from the denser low-order side and reorders rows according to the Manhattan distance, relocating active cells toward regions less affected by PR and thus lowering the nonideality factor (NF). Applied to DNN models on ImageNet-1k, MDM reduces NF by up to 46% and improves accuracy under analog distortion by an average of 3.6% in ResNets. Overall, it provides a lightweight, spatially informed method for scaling CIM DNN accelerators.

--The phenomenon of linear mode connectivity (LMC) links several aspects of deep learning, including training stability under noisy stochastic gradients, the smoothness and generalization of local minima (basins), the similarity and functional diversity of sampled models, and architectural effects on data processing. In this work, we experimentally study LMC under data shifts and identify conditions that mitigate their impact. We interpret data shifts as an additional source of stochastic gradient noise, which can be reduced through small learning rates and large batch sizes. These parameters influence whether models converge to the same local minimum or to regions of the loss landscape with varying smoothness and generalization. Although models sampled via LMC tend to make similar errors more frequently than those converging to different basins, the benefit of LMC lies in balancing training efficiency against the gains achieved from larger, more diverse ensembles. Code and supplementary materials will be made publicly available at https://github.com/DLR-KI/LMC in due course. ODE connectivity refers to a phenomenon, when stochastic gradient descent (SGD) solutions or modes are connected via a path of low loss in neural networks parameter space [1], [2]. So every solution along such path exhibits similar performance and generalization as those solutions, between which the path is constructed. Moreover, such paths were shown to be embedded in a multi-dimensional manifold of low loss [3]. When a connecting path is linear the phenomenon is referred to as linear mode connectivity (LMC) [4]. LMC was investigated under different perspectives: (1) conditions affecting LMC [4], [5], (2) connectivity of layers, features or different types of solutions [6], [7], [8] and (3) so-called "re-basin" approaches, that "transport" a solution from one local minimum From a practical view point, LMC is expected to improve ensemble methods, in particular in federated learning setting, robustness of fine-tuned models, distributed optimization and model pruning [13], [9]. This work focuses on LMC from the perspective of data shifts [14], which are ever-present in real world applications. In particular, when training is performed on multiple training datasets separately and ensembles of models are employed.

This suggests that the low-rank fast attention only works for functions approx-imable by polynomials.

Path planning for autonomous robots presents a fundamental trade-off between optimality and safety. While conventional algorithms typically prioritize one of these objectives, we introduce the Unified Path Planner (UPP), a unified framework that simultaneously addresses both. UPP is a graph-search-based algorithm that employs a modified heuristic function incorporating a dynamic safety cost, enabling an adaptive balance between path length and obstacle clearance. We establish theoretical sub-optimality bounds for the planner and demonstrate that its safety-to-optimality ratio can be tuned via adjustable parameters, with a trade-off in computational complexity. Extensive simulations show that UPP achieves a high success rate, generating near-optimal paths with only a negligible increase in cost over traditional A*, while ensuring safety margins that closely approach those of the classical Voronoi planner. Finally, the practical efficacy of UPP is validated through a hardware implementation on a TurtleBot, confirming its ability to navigate cluttered environments by generating safe, sub-optimal paths.

Manipulation surfaces indirectly control and reposition objects by actively modifying their shape or properties rather than directly gripping objects. These surfaces, equipped with dense actuator arrays, generate dynamic deformations. However, a high-density actuator array introduces considerable complexity due to increased degrees of freedom (DOF), complicating control tasks. High DOF restrict the implementation and utilization of manipulation surfaces in real-world applications as the maintenance and control of such systems exponentially increase with array/surface size. Learning-based control approaches may ease the control complexity, but they require extensive training samples and struggle to generalize for heterogeneous objects. In this study, we introduce a simple, precise and robust PID-based linear close-loop feedback control strategy for heterogeneous object manipulation on MANTA-RAY (Manipulation with Adaptive Non-rigid Textile Actuation with Reduced Actuation density). Our approach employs a geometric transformation-driven PID controller, directly mapping tilt angle control outputs(1D/2D) to actuator commands to eliminate the need for extensive black-box training. We validate the proposed method through simulations and experiments on a physical system, successfully manipulating objects with diverse geometries, weights and textures, including fragile objects like eggs and apples. The outcomes demonstrate that our approach is highly generalized and offers a practical and reliable solution for object manipulation on soft robotic manipulation, facilitating real-world implementation without prohibitive training demands.

We introduce a new linear-algebraic tool based on group representation theory, and use it to address three key problems in machine learning.1. Past researchers have proposed fast attention algorithms for LLMs by approximating or replace softmax attention with other functions, such as low-degree polynomials. The key property of these functions is that, when applied entry-wise to the matrix QK {\top}, the result is a low rank matrix when Q and K are n \times d matrices and n \gg d . This suggests a natural question: what are all functions f with this property? If other f exist and are quickly computable, they can be used in place of softmax for fast subquadratic attention algorithms.

This paper introduces feature subset weighting using monotone measures for distance-based supervised learning. The Choquet integral is used to define a distance metric that incorporates these weights. This integration enables the proposed distances to effectively capture non-linear relationships and account for interactions both between conditional and decision attributes and among conditional attributes themselves, resulting in a more flexible distance measure. In particular, we show how this approach ensures that the distances remain unaffected by the addition of duplicate and strongly correlated features. Another key point of this approach is that it makes feature subset weighting computationally feasible, since only $m$ feature subset weights should be calculated each time instead of calculating all feature subset weights ($2^m$), where $m$ is the number of attributes. Next, we also examine how the use of the Choquet integral for measuring similarity leads to a non-equivalent definition of distance. The relationship between distance and similarity is further explored through dual measures. Additionally, symmetric Choquet distances and similarities are proposed, preserving the classical symmetry between similarity and distance. Finally, we introduce a concrete feature subset weighting distance, evaluate its performance in a $k$-nearest neighbors (KNN) classification setting, and compare it against Mahalanobis distances and weighted distance methods.