SKG in Practice: OOD Generalization Cases Across FourDomains

This paper proposes the SKG (Symplectic-Kernel Gradient) reasoning engine for law-induced inference and induction. Laws are a priori; data provide instances. Inference is geometric evolution under law constraints. We organize symplectic geometry, GENERIC, kernel methods, RKHS, group-equivariant networks, and constrained mechanics into a unified, analyzable, implementable, verifiable framework. We propose the SKG octuple S = (M, g, omega, H, S, G, Delta, Pi), whose four components correspond to four levels of laws: conservation laws, constitutive relations, symmetry groups, and axiom systems. Main results: (1) Invariants enter metric construction via the Jacobian: g = J^T Sigma^-1 J + epsilon_G I. For constant cases, we establish inverse reconstruction theory: solvability rank condition (Thm 3.9), consistency criterion (Thm 3.11), and uniqueness theorem (Thm 3.14). (2) The kernel bandwidth matrix is uniquely determined by Fisher information and satisfies task-independence (Thm 3.19). (3) The metric reconstruction error decomposes into four terms (Thm 3.21), with squared condition number kappa(Sigma^*)^2 as the dominant amplification factor (Thm 3.22). (4) For state-dependent cases, we establish function-space reconstruction theory on matrix-valued RKHS (Thms 4.8, 4.9). (5) Three-layer projections handle axiom, symplectic-step, and conservation constraints. (6) The residual equation yields the observable formula, four-dimensional rank condition, and noise propagation, while g_qq is structurally unidentifiable. (7) Kernel parameterization yields the RKHS extrapolation bound (Thm 4.15) and strict OOD gain theorem in group directions (Thm 8.15). (8) The learning-decoupling theorem (Thm 9.1) proves the sole object of induction is the parameters of the task energy. Validation: 1D damped oscillator and 2D rotation-invariant system; 6 unit tests, 10 experiments (paired t-test, p < 0.01).

Out-of-distribution (OOD) detection identifies test samples that fall outside a model's training distribution, a capability critical for safe deployment in high-stakes applications. Standard OOD detectors are trained on a specific in-distribution (ID) dataset and detect deviations from that single domain. In contrast, we study few-shot cross-domain OOD detection: given a single pre-trained model, can we perform OOD detection on arbitrary new ID-OOD task pairs using only a handful of ID samples at inference time, with no additional training? We propose UFCOD, a unified framework that achieves this goal through information-geometric analysis of diffusion trajectories. Our key insight is that diffusion noise predictions are score functions (gradients of log-density), and we extract two energy features: Path Energy (integrated score magnitude) and Dynamics Energy (score smoothness), that form a discrete Sobolev norm capturing how samples interact with the learned diffusion process. The central contribution is a train-once, deploy-anywhere paradigm: a diffusion model trained on a single dataset (e.g., CelebA) serves as a universal feature extractor for OOD detection across semantically unrelated domains (e.g., CIFAR-10, SVHN, Textures). At deployment, each new task requires only $\sim$100 unlabeled ID samples for inference: no retraining, no fine-tuning, no task-specific adaptation. Using 100 ID samples per task, UFCOD achieves 93.7% average AUROC across 12 cross-domain benchmarks, competitive with methods trained on 50k--163k samples, demonstrating $\sim$500$\times$ improvement in sample efficiency. See our code in https://github.com/lili0415/UFCOD.

The practical deployment gap -- transitioning from controlled multi-view 3D skeleton capture to unconstrained monocular 2D pose estimation -- introduces a compound domain shift whose safety implications remain critically underexplored. We present a systematic study of this severe domain shift using a novel Gym2D dataset (style/viewpoint shift) and the UCF101 dataset (semantic shift). Our Skeleton Transformer achieves 63.2% cross-subject accuracy on NTU-120 but drops to 1.6% under zero-shot transfer to the Gym domain and 1.16% on UCF101. Critically, we demonstrate that high Out-Of-Distribution (OOD) detection AUROC does not guarantee safe selective classification. Standard uncertainty methods fail to detect this performance drop: the model remains confidently incorrect with 99.6% risk even at 50% coverage across both OOD datasets. While energy-based scoring (AUROC >= 0.91) and Mahalanobis distance provide reliable distributional detection signals, such high AUROC scores coexist with poor risk-coverage behavior when making decisions. A lightweight finetuned gating mechanism restores calibration and enables graceful abstention, substantially reducing the rate of confident wrong predictions. Our work challenges standard deployment assumptions, providing a principled safety analysis of both semantic and geometric skeleton recognition deployment.