In this paper, we theoretically and empirically characterize obstructions to training encoders with geometric latent spaces. We show that local optima can arise due to singularities (e.g.

The ε-sensitivity of distributions is defined below. Next, we provide the following lemma. Suppose that the distribution map D (θ) forms a location family (7) . Therefore, we must carry out the worst-case analysis on this term. With these two Lemmas, we are ready to prove Lemma 3. 3 Proof of Lemma 3. By Lemma C1, we have See E for the proof.

In one view, this theorem extends the universality of "complete-observation" GNNs in to "partial-observation".

We present, a novel definition inspired by Lipschitz continuity which aims to certify the robustness of neural networks across input domains (such as continuous and discrete domains in vision and language, respectively). Most existing approaches that seek to certify robustness, especially Lipschitz continuity, lie within the continuous domain with norm and distribution-dependent guarantees. In contrast, our proposed definition yields certification guarantees that depend only on the loss function and the intermediate learned metric spaces of the neural network. These bounds are independent of domain modality, norms, and distribution. We further demonstrate that the expressiveness of a model class is not at odds with its knowledge continuity. This implies that achieving robustness by maximizing knowledge continuity should not theoretically hinder inferential performance. Finally, to complement our theoretical results, we present several applications of knowledge continuity such as regularization, a certification algorithm, and show that knowledge continuity can be used to localize vulnerable components of a neural network.

We propose Hellinger-type loss functions for training Generative Adversarial Networks (GANs), motivated by the boundedness, symmetry, and robustness properties of the Hellinger distance. We define an adversarial objective based on this divergence and study its statistical properties within a general parametric framework. We establish the existence, uniqueness, consistency, and joint asymptotic normality of the estimators obtained from the adversarial training procedure. In particular, we analyze the joint estimation of both generator and discriminator parameters, offering a comprehensive asymptotic characterization of the resulting estimators. We introduce two implementations of the Hellinger-type loss and we evaluate their empirical behavior in comparison with the classic (Maximum Likelihood-type) GAN loss. Through a controlled simulation study, we demonstrate that both proposed losses yield improved estimation accuracy and robustness under increasing levels of data contamination.

Learning is most effective when it's connected to relevant, relatable examples that resonate with learners on a personal level. However, existing educational AI tools don't focus on generating examples or adapting to learners' changing understanding, struggles, or growing skills. We've developed ExaCraft, an AI system that generates personalized examples by adapting to the learner's dynamic context. Through the Google Gemini AI and Python Flask API, accessible via a Chrome extension, ExaCraft combines user-defined profiles (including location, education, profession, and complexity preferences) with real-time analysis of learner behavior. This ensures examples are both culturally relevant and tailored to individual learning needs. The system's core innovation is its ability to adapt to five key aspects of the learning context: indicators of struggle, mastery patterns, topic progression history, session boundaries, and learning progression signals. Our demonstration will show how ExaCraft's examples evolve from basic concepts to advanced technical implementations, responding to topic repetition, regeneration requests, and topic progression patterns in different use cases.

Large language models handle single-turn generation well, but multi-turn interactions still require the model to reconstruct user intent and task state from an expanding token history because internal representations do not persist across turns. This token-first paradigm leads to drift, inconsistent reasoning modes, and growing prompts as conversations deepen. We propose CORE, a concept-first interaction layer that improves multi-turn stability without modifying model weights. CORE combines a small library of universal cognitive operators with a persistent Local Concept--a compact semantic state capturing the task, constraints, preferences, and intermediate results. Each model call receives only this concept state, the user's latest instruction, and the selected operator, eliminating the need to replay full history. A preliminary prototype simulating CORE's behavior shows a ~42% reduction in cumulative prompt tokens, though this number reflects prototype conditions and should not be interpreted as a real-world performance estimate. CORE offers a model-agnostic mechanism that separates conceptual reasoning from language generation, suggesting a scalable direction for more stable multi-turn systems.

Crowd navigation has garnered considerable research interest in recent years, especially with the proliferating application of deep reinforcement learning (DRL) techniques. Many studies, however, do not sufficiently analyze the relative priorities among evaluation metrics, which compromises the fair assessment of methods with divergent objectives. Furthermore, trajectory-continuity metrics, specifically those requiring $C^2$ smoothness, are rarely incorporated. Current DRL approaches generally prioritize efficiency and proximal comfort, often neglecting trajectory optimization or addressing it only through simplistic, unvalidated smoothness reward. Nevertheless, effective trajectory optimization is essential to ensure naturalness, enhance comfort, and maximize the energy efficiency of any navigation system. To address these gaps, this paper proposes a unified framework that enables the fair and transparent assessment of navigation methods by examining the prioritization and joint evaluation of multiple optimization objectives. We further propose a novel reward-shaping strategy that explicitly emphasizes trajectory-curvature optimization. The resulting trajectory quality and adaptability are significantly enhanced across multi-scale scenarios. Through extensive 2D and 3D experiments, we demonstrate that the proposed method achieves superior performance compared to state-of-the-art approaches.