Dimension reduction (DR) is an important and widely studied technique in exploratory data analysis.

Misalignedrewards can lead to suboptimal behaviors [Ngo et al., 2022], undermining the potential benefits of RL in practical scenarios.

Traffic state estimation (TSE) becomes challenging when probe-vehicle penetration is low and observations are spatially sparse. Pure data-driven methods lack physical explanations and have poor generalization when observed data is sparse. In contrast, physical models have difficulty integrating uncertainties and capturing the real complexity of traffic. To bridge this gap, recent studies have explored combining them by embedding physical structure into Gaussian process. These approaches typically introduce the governing equations as soft constraints through pseudo-observations, enabling the integration of model structure within a variational framework. However, these methods rely heavily on penalty tuning and lack principled uncertainty calibration, which makes them sensitive to model mis-specification. In this work, we address these limitations by presenting a novel Physics-Embedded Gaussian Process (PEGP), designed to integrate domain knowledge with data-driven methods in traffic state estimation. Specifically, we design two multi-output kernels informed by classic traffic flow models, constructed via the explicit application of the linearized differential operator. Experiments on HighD, NGSIM show consistent improvements over non-physics baselines. PEGP-ARZ proves more reliable under sparse observation, while PEGP-LWR achieves lower errors with denser observation. Ablation study further reveals that PEGP-ARZ residuals align closely with physics and yield calibrated, interpretable uncertainty, whereas PEGP-LWR residuals are more orthogonal and produce nearly constant variance fields. This PEGP framework combines physical priors, uncertainty quantification, which can provide reliable support for TSE.

The classical formulation of PCA, dating back to Carl F. Gauss, is based on minimizing the sum of

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Recent work has discovered that large language models can develop broadly misaligned behaviors after being fine-tuned on narrowly harmful datasets, a phenomenon known as emergent misalignment (EM). However, the fundamental mechanisms enabling such harmful generalization across disparate domains remain poorly understood. In this work, we adopt a geometric perspective to study EM and demonstrate that it exhibits a fundamental cross-task linear structure in how harmful behavior is encoded across different datasets. Specifically, we find a strong convergence in EM parameters across tasks, with the fine-tuned weight updates showing relatively high cosine similarities, as well as shared lower-dimensional subspaces as measured by their principal angles and projection overlaps. Furthermore, we also show functional equivalence via linear mode connectivity, wherein interpolated models across narrow misalignment tasks maintain coherent, broadly misaligned behavior. Our results indicate that EM arises from different narrow tasks discovering the same set of shared parameter directions, suggesting that harmful behaviors may be organized into specific, predictable regions of the weight landscape. By revealing this fundamental connection between parametric geometry and behavioral outcomes, we hope our work catalyzes further research on parameter space interpretability and weight-based interventions.