Geometric and Spectral Alignment for Deep Neural Network II

This paper develops the angular and static-channel component of Geometric and Spectral Alignment for residual Jacobian chains. Starting from Cartan-coordinate rigidity and fitted effective-rank windows, we study how dominant singular subspaces are transported across adjacent layers and how the resulting finite matrices can be displayed in physical channel coordinates. The main results are deterministic, margin-verified results. We bound the error between full interface transport and its dominant-window truncation, add fitted-tail errors so that empirical spectra can be certified against the Gibbs--Cartan tail model, and distinguish source-mode incidence from fully physical input-output channel incidence. Given row groups and active supports, the Physical Alignment Matrix decomposes orthogonally as core plus overlap plus noise. Active-column gaps, pairwise overlap margins, and noise bounds combine into a static certificate radius under which the full transport and the truncated transport induce the same active supports, pairwise incidence graph, SRS sets, hub columns, and core/overlap/noise masks. The finer SC/SA/ST labels of the Invariant Channel Mapping require additional row-energy and profile-correlation margins, stated as explicit perturbation tests. The empirical section reports the matrices and block-energy heatmaps that measure these certificate quantities across CNNs, language models, and vision/diffusion backbones. The figures are interpreted as finite-dimensional measurements; complete membership in the Physical GSA certificate domain requires checking the numerical margin protocol stated in Section 10.

Variational Kernel Design for Internal Noise: Gaussian Chaos Noise, Representation Compatibility, and Reliable Deep Learning

Internal noise in deep networks is usually inherited from heuristics such as dropout, hard masking, or additive perturbation. We ask two questions: what correlation geometry should internal noise have, and is the implemented perturbation compatible with the representations it acts on? We answer these questions through Variational Kernel Design (VKD), a framework in which a noise mechanism is specified by a law family, a correlation kernel, and an injection operator, and is derived from learning desiderata. In a solved spatial subfamily, a quadratic maximum-entropy principle over latent log-fields yields a Gaussian optimizer with precision given by the Dirichlet Laplacian, so the induced geometry is the Dirichlet Green kernel. Wick normalization then gives a canonical positive mean-one gate, Gaussian Chaos Noise (GCh). For the sample-wise gate used in practice, we prove exact Gaussian control of pairwise log-ratio deformation, margin-sensitive ranking stability, and an exact expected intrinsic roughness budget; hard binary masks instead induce singular or coherence-amplified distortions on positive coherent representations. On ImageNet and ImageNet-C, GCh consistently improves calibration and under shift also improves NLL at competitive accuracy.

Generalized Multi-Level Replanning TAMP Framework for Dynamic Environment

Task and Motion Planning (TAMP) algorithms can generate plans that combine logic and motion aspects for robots. However, these plans are sensitive to interference and control errors. To make TAMP more applicable in real-world, we propose the generalized multi-level replanning TAMP framework(GMRF), blending the probabilistic completeness of sampling-based TAMP algorithm with the robustness of reactive replanning. GMRF generates an nominal plan from the initial state, then dynamically reconstructs this nominal plan in real-time, reorders robot manipulations. Following the logic-level adjustment, GMRF will try to replan a new motion path to ensure the updated plan is feasible at the motion level. Finally, we conducted real-world experiments involving stack and rearrange task domains. The result demonstrate GMRF's ability to swiftly complete tasks in scenarios with varying degrees of interference.