Fisher Decorator: Refining Flow Policy via A Local Transport Map

Recent advances in flow-based offline reinforcement learning (RL) have achieved strong performance by parameterizing policies via flow matching. However, they still face critical trade-offs among expressiveness, optimality, and efficiency. In particular, existing flow policies interpret the $L_2$ regularization as an upper bound of the 2-Wasserstein distance ($W_2$), which can be problematic in offline settings. This issue stems from a fundamental geometric mismatch: the behavioral policy manifold is inherently anisotropic, whereas the $L_2$ (or upper bound of $W_2$) regularization is isotropic and density-insensitive, leading to systematically misaligned optimization directions. To address this, we revisit offline RL from a geometric perspective and show that policy refinement can be formulated as a local transport map: an initial flow policy augmented by a residual displacement. By analyzing the induced density transformation, we derive a local quadratic approximation of the KL-constrained objective governed by the Fisher information matrix, enabling a tractable anisotropic optimization formulation. By leveraging the score function embedded in the flow velocity, we obtain a corresponding quadratic constraint for efficient optimization. Our results reveal that the optimality gap in prior methods arises from their isotropic approximation. In contrast, our framework achieves a controllable approximation error within a provable neighborhood of the optimal solution. Extensive experiments demonstrate state-of-the-art performance across diverse offline RL benchmarks. See project page: https://github.com/ARC0127/Fisher-Decorator.

Geometric Metrics for MoE Specialization: From Fisher Information to Early Failure Detection

Expert specialization is fundamental to Mixture-of-Experts (MoE) model success, yet existing metrics (cosine similarity, routing entropy) lack theoretical grounding and yield inconsistent conclusions under reparameterization. We present an information-geometric framework providing the first rigorous characterization of MoE specialization dynamics. Our key insight is that expert routing distributions evolve on the probability simplex equipped with the Fisher information metric, enabling formal analysis via Riemannian geometry. We prove that standard heuristic metrics violate parameterization invariance (Theorem 1), establish that specialization corresponds to geodesic flow with quantified approximation bounds (Theorem 2), and derive a failure predictor with theoretical threshold justification (Theorem 3). The framework introduces two principled metrics: Fisher Specialization Index (FSI) achieving r=0.91+/-0.02 correlation with downstream performance, and Fisher Heterogeneity Score (FHS) predicting training failure at 10% completion with AUC=0.89+/-0.03 -- outperforming validation-loss-based early stopping by 23% while requiring 40x fewer compute cycles. We validate intervention protocols achieving 87% recovery rate when FHS>1 is detected. Comprehensive experiments across language modeling (WikiText-103, C4), vision MoE (ImageNet), and scaling studies (8-64 experts, 125M-2.7B parameters) validate our theoretical predictions.

On Higher-Order Geometric Refinements of Classical Covariance Asymptotics: An Approach via Intrinsic and Extrinsic Information Geometry

Classical Fisher-information asymptotics describe the covariance of regular efficient estimators through the local quadratic approximation of the log-likelihood, and thus capture first-order geometry only. In curved models, including mixtures, curved exponential families, latent-variable models, and manifold-constrained parameter spaces, finite-sample behavior can deviate systematically from these predictions. We develop a coordinate-invariant, curvature-aware refinement by viewing a regular parametric family as a Riemannian manifold \((Θ,g)\) with Fisher--Rao metric, immersed in \(L^2(μ)\) through the square-root density map. Under suitable regularity and moment assumptions, we derive an \(n^{-2}\) correction to the leading \(n^{-1}I(θ)^{-1}\) covariance term for score-root, first-order efficient estimators. The correction is governed by a tensor \(P_{ij}\) that decomposes canonically into three parts, an intrinsic Ricci-type contraction of the Fisher--Rao curvature tensor, an extrinsic Gram-type contraction of the second fundamental form, and a Hellinger discrepancy tensor encoding higher-order probabilistic information not determined by immersion geometry alone. The extrinsic term is positive semidefinite, the full correction is invariant under smooth reparameterization, and it vanishes identically for full exponential families. We then extend the picture to singular models, where Fisher information degenerates. Using resolution of singularities under an additive normal crossing assumption, we describe the resolved metric, the role of the real log canonical threshold in learning rates and posterior mean-squared error, and a curvature-based covariance expansion on the resolved space that recovers the regular theory as a special case. This framework also suggests geometric diagnostics of weak identifiability and curvature-aware principles for regularization and optimization.

Quantization Dominates Rank Reduction for KV-Cache Compression

We compare two strategies for compressing the KV cache in transformer inference: rank reduction (discard dimensions) and quantization (keep all dimensions, reduce precision). At matched storage budgets across five models (124M-14B, MHA and GQA), we find that quantization consistently outperforms rank reduction by 4-364 PPL depending on model and compression level. The gap persists even when rank reduction is combined with quantization in hybrid baselines, and it grows with GQA aggressiveness. On LAMBADA, INT4 matches FP16 accuracy (+0.23 PPL on Mistral 7B, +0.58 on GPT-2) while rank-32 at identical storage collapses to 0.4%. We trace this gap to a structural asymmetry: under softmax attention routing, removing a dimension can flip which token is attended (a discrete failure), while quantization noise is bounded and typically preserves score ordering. We formalize this via a perturbation result showing projection damage exceeds quantization damage by 3 x 2^(2b) per direction under the softmax Fisher metric. A basis ablation confirms the finding is basis-independent (spread <0.4 PPL), establishing that the advantage comes from preserving dimensions, not from a better coordinate system. Joint K+V INT4 quantization achieves 75% total KV reduction at only +0.18 PPL on Mistral 7B.

Retrieval Is Not Enough: Why Organizational AI Needs Epistemic Infrastructure

Organizational knowledge used by AI agents typically lacks epistemic structure: retrieval systems surface semantically relevant content without distinguishing binding decisions from abandoned hypotheses, contested claims from settled ones, or known facts from unresolved questions. We argue that the ceiling on organizational AI is not retrieval fidelity but \emph{epistemic} fidelity--the system's ability to represent commitment strength, contradiction status, and organizational ignorance as computable properties. We present OIDA, a framework that structures organizational knowledge as typed Knowledge Objects carrying epistemic class, importance scores with class-specific decay, and signed contradiction edges. The Knowledge Gravity Engine maintains scores deterministically with proved convergence guarantees (sufficient condition: max degree $< 7$; empirically robust to degree 43). OIDA introduces QUESTION-as-modeled-ignorance: a primitive with inverse decay that surfaces what an organization does \emph{not} know with increasing urgency--a mechanism absent from all surveyed systems. We describe the Epistemic Quality Score (EQS), a five-component evaluation methodology with explicit circularity analysis. In a controlled comparison ($n{=}10$ response pairs), OIDA's RAG condition (3,868 tokens) achieves EQS 0.530 vs.\ 0.848 for a full-context baseline (108,687 tokens); the $28.1\times$ token budget difference is the primary confound. The QUESTION mechanism is statistically validated (Fisher $p{=}0.0325$, OR$=21.0$). The formal properties are established; the decisive ablation at equal token budget (E4) is pre-registered and not yet run.

LLM Evaluation as Tensor Completion: Low Rank Structure and Semiparametric Efficiency

Large language model (LLM) evaluation platforms increasingly rely on pairwise human judgments. These data are noisy, sparse, and non-uniform, yet leaderboards are reported with limited uncertainty quantification. We study this as semiparametric inference for a low-rank latent score tensor observed through pairwise comparisons under Bradley-Terry-Luce-type models. This places LLM evaluation in a new tensor completion setting with structured observations, non-uniform sampling, and pairwise contrasts. Our target is a smooth functional $ψ(T^\star)$, including linear estimands such as ability gaps and nonlinear ones such as win probabilities. We derive the information operator on the low-rank tangent space, the efficient influence function, and the semiparametric efficiency bound, then construct a one-step debiased estimator with asymptotic normality. A central challenge is that the information operator is anisotropic and does not commute with the tangent-space projection, creating a bottleneck absent from isotropic models. We introduce a score-whitening method that equalizes local Fisher information and restores stable inference at the optimal sample-complexity scale. Our results provide a principled framework for uncertainty quantification in LLM evaluation and more broadly for inference on low-rank structures from pairwise data.

RPM-Net Reciprocal Point MLP Network for Unknown Network Security Threat Detection

Effective detection of unknown network security threats in multi-class imbalanced environments is critical for maintaining cyberspace security. Current methods focus on learning class representations but face challenges with unknown threat detection, class imbalance, and lack of interpretability, limiting their practical use. To address this, we propose RPM-Net, a novel framework that introduces reciprocal point mechanism to learn "non-class" representations for each known attack category, coupled with adversarial margin constraints that provide geometric interpretability for unknown threat detection. RPM-Net++ further enhances performance through Fisher discriminant regularization. Experimental results show that RPM-Net achieves superior performance across multiple metrics including F1-score, AUROC, and AUPR-OUT, significantly outperforming existing methods and offering practical value for real-world network security applications. Our code is available at:https://github.com/chiachen-chang/RPM-Net

Inversion-Free Natural Gradient Descent on Riemannian Manifolds

The natural gradient method is widely used in statistical optimization, but its standard formulation assumes a Euclidean parameter space. This paper proposes an inversion-free stochastic natural gradient method for probability distributions whose parameters lie on a Riemannian manifold. The manifold setting offers several advantages: one can implicitly enforce parameter constraints such as positive definiteness and orthogonality, ensure parameters are identifiable, or guarantee regularity properties of the objective like geodesic convexity. Building on an intrinsic formulation of the Fisher information matrix (FIM) on a manifold, our method maintains an online approximation of the inverse FIM, which is efficiently updated at quadratic cost using score vectors sampled at successive iterates. In the Riemannian setting, these score vectors belong to different tangent spaces and must be combined using transport operations. We prove almost-sure convergence rates of $O(\log{s}/s^α)$ for the squared distance to the minimizer when the step size exponent $α>2/3$. We also establish almost-sure rates for the approximate FIM, which now accumulates transport-based errors. A limited-memory variant of the algorithm with sub-quadratic storage complexity is proposed. Finally, we demonstrate the effectiveness of our method relative to its Euclidean counterparts on variational Bayes with Gaussian approximations and normalizing flows.

The Silicon Mirror: Dynamic Behavioral Gating for Anti-Sycophancy in LLM Agents

Large Language Models (LLMs) increasingly prioritize user validation over epistemic accuracy - a phenomenon known as sycophancy. We present The Silicon Mirror, an orchestration framework that dynamically detects user persuasion tactics and adjusts AI behavior to maintain factual integrity. Our architecture introduces three components: (1) a Behavioral Access Control (BAC) system that restricts context layer access based on real-time sycophancy risk scores, (2) a Trait Classifier that identifies persuasion tactics across multi-turn dialogues, and (3) a Generator-Critic loop where an auditor vetoes sycophantic drafts and triggers rewrites with "Necessary Friction." In a live evaluation across all 437 TruthfulQA adversarial scenarios, Claude Sonnet 4 exhibits 9.6% baseline sycophancy, reduced to 1.4% by the Silicon Mirror - an 85.7% relative reduction (p < 10^-6, OR = 7.64, Fisher's exact test). Cross-model evaluation on Gemini 2.5 Flash reveals a 46.0% baseline reduced to 14.2% (p < 10^-10, OR = 5.15). We characterize the validation-before-correction pattern as a distinct failure mode of RLHF-trained models.

Information Geometry of Absorbing Markov-Chain and Discriminative Random Walks

Discriminative Random Walks (DRWs) are a simple yet powerful tool for semi-supervised node classification, but their theoretical foundations remain fragmentary. We revisit DRWs through the lens of information geometry, treating the family of class-specific hitting-time laws on an absorbing Markov chain as a statistical manifold. Starting from a log-linear edge-weight model, we derive closed-form expressions for the hitting-time probability mass function, its full moment hierarchy, and the observed Fisher information. The Fisher matrix of each seed node turns out to be rank-one, taking the quotient by its null space yields a low-dimensional, globally flat manifold that captures all identifiable directions of the model. Leveraging the geometry, we introduce a sensitivity score for unlabeled nodes that bounds, and in one-dimensional cases attains, the maximal first-order change in DRW betweenness under unit Fisher perturbations. The score can lead to principled strategies for active label acquisition, edge re-weighting, and explanation.