Fisher Information

Heterogeneous federated learning requires clients with diverse computational capacities to collaboratively train a global model, where each client trains a capacity-constrained submodel. Existing methods select submodel parameters using heuristic importance measures---most prominently parameter magnitude---without theoretical justification for why these measures support convergence. We identify a fundamental gap: existing parameter selection criteria lack theoretical grounding in the convergence framework, partial client participation introduces additional estimation effects in the Fisher scores. We propose \textbf{FedFIbOS}: Fisher Importance-based Optimal Submodelling for heterogeneous federated learning, using Fisher Information in a principled criterion derived from minimizing submodel masking error. %We formally establish when magnitude selection is equivalent to Fisher selection fail under non-IID heterogeneous federated learning. We theoretically formulate submodel selection through a Fisher-weighted quadratic masking surrogate and show that the raw Fisher top-$k$ rule implemented by FedFIbOS solves this surrogate under a Fisher-dominant ranking condition. The resulting method retains the convergence structure of the underlying masked federated optimization bound. Fisher scores are efficiently estimated from empirical diagonal Fisher information using squared gradients, enabling stable and adaptive parameter selection without additional optimization overhead. Experiments on CIFAR-10, CIFAR-100, and AGNews under pathological and Dirichlet non-IID settings show FedFIbOS achieves ${\approx}10\%$ higher accuracy than the state of the art, with improvements becoming more pronounced under stronger heterogeneity.

Clinical B-mode images are widely available as potential data sources for quantitative ultrasound (QUS) analysis for tissue characterization. However, standard clinical ultrasound devices apply unknown log-compression to RF envelope data before display and storage. Previous work has demonstrated estimation of the underlying RF envelope statistics in the presence of an unknown compression law. Using Fisher information analysis, we show that finite-offset log compression causes severe information loss when estimating the Rayleigh scale $σ$, which controls diffuse speckle. For a single image window, unknown compression raises the minimum achievable variance for unbiased estimation of $σ$ by a compression-independent factor of approximately $\FisherMinInflation$. When $M$ equal-sized windows share the same unknown compression settings, the excess variance decays as $1/M$; even in the most favorable regime, reducing the variance inflation factor below $1.1$ requires $\FisherBestCaseWindows$ windows. Our analysis treats the contrast parameter $a$ as unknown and the boundary offset $b$ as known; estimating $b$ experimentally shows even larger variance. We validate this theory using synthetic estimation experiments and demonstrate RF-scale recovery on real RF-envelope windows from the OASBUD dataset. Together, these results clarify the limitations of using routine B-mode images for QUS.

Large Language Models (LLMs) are now routinely trained using synthetic data, since high-quality human data has been exhausted by the ever increasing needs of larger and larger models. However, recursive training on synthetic data frequently induces model collapse, a degenerative feedback loop where models progressively forget the true underlying data distribution. Training on a mixture of synthetic and fresh human data is a logical countermeasure and can prevent model collapse. However, it is an open question as to what is the exact minimum required ratio of human-to-synthetic data to maintain training stability. In this paper, we establish rigorous theoretical guarantees on the minimum rate of human data required to prevent model collapse. Although previous work established a formal lower bound for this ratio, such bound can be vacuous for very high dimensions, as the analysis relies on the usual Euclidean metric in R^n and is not adapted to the space of categorical probability distributions. Instead, in this paper we explicitly leverage the information-geometric structure of the probability simplex by analyzing the dynamics of the process under the Fisher-Rao metric. We derive quantitative contraction and invariance bounds that are stable and do not become trivial as the dimensions increase. Thus, we show that the effective required data ratio to prevent model collapse is different than previously implied.

Some aspects of AI development resemble a population process in which models are specialised, retrained on the output of peers, or combined by averaging weights. These practices lead to generations of models, in the biological sense studied by population genetics. Here, I develop this parallelism and interpret multigenerational model populations in terms of sexual and asexual reproduction, formally recombining the two fields. I test these analogies in an exact inheritance model, in trained networks (recurrent, feedforward and variational autoencoder generators) and in large language models, and show that they hold generally, with some measurable architecture-specific biases. Training recursively on model output is known to lead to model collapse, a process previously described as akin to genetic drift; I develop all that follows. A minimal model of a learner retrained on its parent's output reproduces the Wright-Fisher process exactly; verified real data added to each generation play the role of immigration, with the surprising finding that the absolute number of real data samples matters, not their share, exactly as in population genetics. Training a child on the average of its parents' outputs cancels the benefit of having several parents, matching blending inheritance (and reviving Jenkin's objection to Darwin), whereas combining parents so that each keeps its strongest contribution preserves it; merged language-model specialists exceeded every parent across seeds (the Fisher-Muller effect); and lineages become reproductively isolated, losing the ability to merge at all, when they have learned conflicting conventions and not when they have merely drifted apart. As AI societies become societies in time as well as in space, a mathematical framework for their inheritance acquires predictive power. Remarkably, that framework can be adapted almost wholesale from biology.

Neural networks acquire internal representations through learning. In this work, we formulate stochastic gradient descent (SGD) as a Markovian stochastic process and derive a Fisher-information flow speed limit that bounds the rate at which trainable parameters can acquire information about latent variables in the data-generating process. The resulting inequality decomposes the information flow into drift and noise contributions, thereby quantifying the roles of deterministic learning forces and SGD-induced fluctuations from an information-theoretic perspective. We verify the bound in analytically tractable basis-function linear regression, where the information budget predicted by the bound reproduces the ordering and characteristic time scales with which different latent variables are encoded in the learned parameters. These results establish Fisher-information speed limits as a quantitative framework for diagnosing when and how different aspects of the data-generating mechanism are acquired during stochastic learning.

In this article, we propose an online electronic counter-countermeasure (ECCM) framework designed to conceal the strategic decision-making processes of a cognitive radar (CR) operating under adversarial surveillance. We model the CR under two distinct decision paradigms: a static constrained utility-maximizing behavior and a dynamic expected utility-maximizing behavior. The radar's utility function is modeled via a von Mises--Fisher (vMF) distribution, with the distributional parameter constituting the private information to be protected from adversarial inference. We adopt a distribution privacy framework to conceal this private information and provide formal distribution privacy guarantees for cognition masking. In this work, we develop cognition-hiding algorithms for both static constrained utility maximization (WDPCH-SU), and dynamic expected utility maximization (WDPCH-DU). Through rigorous mathematical analysis, we show that both WDPCH-SU and WDPCH-DU satisfy $ε$-distribution privacy ($ε$-DistP) against inference-based adversarial attacks and present the privacy--performance trade-off bounds, quantifying utility loss (in static setting) and expected utility deviation (in dynamic setting) as functions of $ε$. Numerical results show that WDPCH-SU gives about 15% improvement in utility loss at maximum privacy compared to the existing methodology while WDPCH-DU achieves a greater reduction in adversarial Fisher information without requiring explicit Fisher information constraints, at a moderate, analytically bounded utility deviation. These results are highly promising in many 6G communication scenarios such as network slicing for automated driving and swarm UAV coordination, where it is essential to keep the resource allocation policy robust against privacy attacks.

Model compression is key to mitigate deployment challenges of ever growing machine learning models. In this area of research, singular value decomposition (SVD)-based compression offers a compelling trade-off between computational efficiency and model accuracy. Fisher-weighted SVD in particular provides principled, loss-aware compression. However, we find that improving the fidelity of Fisher approximation used in the compression is poorly predictive of post-compression accuracy for Vision Transformers (ViTs). Motivated by this observation, we propose FACTS, a structured Fisher Approximation tailored to Compressing ViTs with Fisher-weighted SVD, which enforces token-local aggregation while preserving within-token activation-gradient dependence. Additionally, we introduce a fast Constrained Rank Search (CoRS), that optimizes layer-wise rank allocation while adhering to a fixed floating point operation (FLOP) constraint. Extensive experiments across ViTs and hybrid architectures demonstrate that FACTS consistently improves accuracy-efficiency trade-offs without requiring finetuning. Notably, it outperforms the strongest SVD baseline by up to +5.8 percentage points (p.p.) Top-1 on Swin-B, with further gains driven by our search method. Code is available at https://github.com/MoritzTho/FACTS.

Recent years have witnessed remarkable achievements of Large Language Models (LLMs) in multiple domains, while the excessive resource requirements of LLMs hinder the deployment on resource-constrained devices. Although model quantization stands out as an effective approach, conventional quantization approaches typically incur severe performance degradation due to uniform bit-width or simple heuristic sensitivity evaluation. In this paper, we propose a novel Fisher information-based Adaptive Mixed Precision Weight Quantization approach, i.e., FAMPWQ, which performs layer-adaptive weight quantization for effective LLM inference on commodity GPUs. First, we propose a system model with a novel Fisher information metric to measure the layer-wise sensitivity to quantization. Second, we propose a reinforcement learning-based bit-width allocator in FAMPWQ, which generates an adaptive bit-width allocation strategy based on the Fisher information sensitivity metric. Extensive experiments on 7 models and 5 benchmarks demonstrate that FAMPWQ significantly outperforms 7 baseline approaches in terms of PPL (up to 3.39 smaller), accuracy (up to 6.87% higher), and LLM-as-a-judge comparison (up to 76% win rate).

Quantum technology has the potential to transform scientific discovery, but quantum advantages often require processing capabilities well beyond the reach of experimental platforms. We show that coupling a single controllable qubit to an otherwise conventional sensor can exponentially reduce the number of measurements required to learn classical signals. These rigorous quantum advantages apply to fundamental sensing tasks, including learning Fourier coefficients, extracting temporal correlations from time-varying signals, and estimating transformations of physical observables. Using a superconducting cavity--qubit architecture, we experimentally demonstrate $10^7$-fold reductions in the number of measurements required for Fourier-amplitude and time-varying signal learning. Our $\textit{quantum feature sensing}$ algorithms further enable orders-of-magnitude improvements in simulations of weak-signal dark matter detection and wireless communication applications. These quantum advantages are derived from Quantum Phase-Space Inference (Q$Ψ$), a unifying theory of quantum-enhanced experiments that simultaneously converts a set of experimental objectives and constraints into tight lower bounds and optimal quantum-enhanced learning algorithms while producing a certificate of quantum advantage. Q$Ψ$ extends beyond the regimes captured by quantum Fisher information and provides a framework for systematically identifying rigorous quantum advantages in practical experimental tasks. Together, our results establish that near-term quantum technology can exponentially enhance our ability to learn from classical signals.

A novel advective Fisher-Rao metric is introduced for optimization tasks on paths of probability measures governed by the continuity equation. This metric is shown to lead to optimal descent directions. It is then shown that this metric arises naturally from three different perspectives: As the rescaled zero-noise limit of the Fisher-Rao metric on path measures, as the expected value of the second variation of the Freidlin--Wentzell large deviation rate functional, and as the Hessian of the Benamou--Brenier action functional from dynamic optimal transport. We supplement this geometric construction with computational experiments. Here, we demonstrate empirically that the advective Fisher-Rao metric yields the desired optimal fitting of probability densities, whereas the Gauss--Newton method yields optimal fitting of velocity fields.

Training neural networks to jointly predict mean and uncertainty estimates from noisy observations can be unstable, prompting a series of independent stabilization efforts. We argue that these interventions highlight a common underlying issue where gradient steps are poorly aligned with the geometry of the loss landscape. To better align updates with local curvature, we derive Fisher8, an output-layer gradient correction that reorients and rescales updates using Fisher geometry rather than Euclidean geometry. Unlike past stabilizers, Fisher8 introduces no data-dependent hyperparameters beyond learning rate and admits an approximate KL trust radius between successive predictive distributions. We show that prior stabilizers converge on overlapping components of this geometric correction. Across multidimensional regression and representation-learning tasks, Fisher8 obtains superior likelihood--error tradeoffs, predicts calibrated uncertainty estimates, and learns rich uncertainty-aware feature spaces.

Model learning by an eavesdropper is treated as an estimation problem in a federated environment. The Fisher Information Matrix for the eavesdropper's estimation problem is driven to singularity through a signaling design; this ensures that the eavesdropper cannot learn the model. Herein, the innovation of prior designs is that model shifts are designed to maximize the difference in the model learned by Eve and the central server while satisfying a transmission power constraint for the agents. Two shift schemes are provided. MaxModShift outperforms a prior ModShift design while requiring lesser transmission power. Compared to a noise injection scheme, MaxModShift performs better while requiring a lower bandwidth secret channel and a reduced average power consumption.

Reliable hypothesis testing is the foundation of many empirical scientific claims. Large language model (LLM) agents are increasingly used to automate this process, as they can inspect datasets, generate code, and produce analyses end-to-end. However, we show that they frequently make subtle inferential errors that lead to incorrect conclusions despite correctly executed analyses. Existing benchmarks fail to capture this failure mode, as they rarely assess whether a reported p-value is statistically valid given the assumptions underlying the data. We address this gap by building P-Bench, a benchmark comprising 425 open-ended, realistic hypothesis-testing tasks spanning economics, biology, and medicine. Each task requires an agent to select a statistical method, compute a p-value, and draw a conclusion given only a scientific hypothesis and a dataset. We further introduce Fisher-R1, an open-weight LLM agent trained for rigorous hypothesis testing using synthetic tasks and reinforcement learning. On P-Bench, Fisher-R1-14B substantially improves over its backbone and outperforms strong proprietary and open-source baselines, including GPT-5.4 and DeepSeekV4-Pro, achieving a 21% average relative improvement in single-trial success over DeepSeek-V4-Pro, with gains up to 26% on the most challenging tasks. Our results demonstrate that current LLM agents lack reliable statistical reasoning for hypothesis testing and that reinforcement learning on tasks with verified statistical reward substantially improves reliability.

We adapt two classical statistical estimators for quantifying uncertainty to modern deep learning, in order to provide clearer insights into uncertainty attributable to two sources : aleatoric uncertainty, or locally scarce data. Our approach leverages recent advances in approximate Fisher Information Matrices, to enable scaling to actual architectures. Experimental results demonstrate how each test points is differentially impacted by both sources, highlighting the practical utility of our estimators in improving the robustness of real-world applications.

Deep parameterized quantum circuits may remain sensitive to a parameter change while the observables retained by a learning model barely respond. We study this separation for a fixed computational-basis measurement. For a pure-state tangent, we compare the quantum Fisher information $F_Q$, the Fisher information $F_{\rm full}$ in the complete bitstring distribution, and the largest variance-normalized response $\mathcal I_{\mathcal A}$ available to a diagonal readout space $\mathcal A$. If the joint state--tangent frame is Haar random, we prove that the two successive information fractions are independent Beta variables whose means are $1/2$ and $r/(2^n-1)$, where $r$ is the centered dimension of the readout. Consequently, even the joint span of all computational-basis Pauli strings through any fixed weight $k$ retain only $O(n^k2^{-n})$ of the full-record information. Exact-statevector experiments across six circuit families show increasing finite-size agreement with this hierarchy in five nonconserving ensembles as the circuit depth grows. A number-conserving family departs strongly from the isotropic prediction even after correcting the support and readout rank, showing that rank alone is insufficient without tangent isotropy.

Generative Flow Networks (GFlowNets) have emerged as a flexible framework for amortised inference over discrete and mixed discrete-continuous objects, requiring only an unnormalised target density specified through a reward. In this work, we formulate forward-policy training in GFlowNets through the information geometry of the induced trajectory sampler. Treating the forward policy as an induced trajectory sampler, we show that its intrinsic first-order geometry is given by the Fisher-Rao metric of the trajectory family, and that the associated natural gradient provides the canonical local update whenever the corresponding Fisher information is computable or accurately approximable. We derive an exact decomposition of the trajectory Fisher into per-step conditional second moments, which clarifies when temporal score interactions vanish and when dense couplings remain under shared parameterisation. This leads to three computational regimes: settings with tractable exact Fisher information, settings where Monte Carlo estimators of the expected Fisher are sufficient, and structure-exploitable settings in which target locality or factorisation yields accurate approximations of the Fisher expectation. In the latter case, graphical-model tools such as exact marginalisation, separator methods, and belief propagation provide principled surrogates for natural-gradient updates. The resulting framework turns target structure into optimisation geometry and yields a tractable route to structure-aware forward-policy training in GFlowNets. We illustrate the framework empirically through examples comparing convergence and exploration behaviour under Riemannian and Euclidean optimisation.

On-policy distillation (OPD) samples trajectories from the current student policy and minimizes token-level divergence between student and teacher next-token distributions at prefixes along those trajectories. This aligns the distillation states with the student's own generation distribution. However, it still assumes that the complete teacher distribution is an appropriate target across student capacities. In vision--language reasoning, teacher corrections can depend on visual distinctions that a compact student cannot represent. Our target-scaling study shows that, as the target approaches the complete teacher distribution, the student realizes less of the prescribed shift and obtains worse downstream performance. We therefore propose \emph{Fisher-Projected On-Policy Distillation} (FP-OPD), which distills only locally realizable teacher corrections. FP-OPD uses continuous visual perturbations to estimate the student's local visual tangent space and projects the centered teacher--student log-probability gap onto this space under the student's Fisher metric. The resulting capacity-aware target is optimized with full-vocabulary reverse KL on student trajectories, retaining the standard OPD framework. In 8B-to-2B distillation, FP-OPD improves all seven evaluated multimodal benchmarks. It raises the average score by 2.77 points over the pretrained student and by 1.60 points over standard OPD. These results demonstrate that locally realizable teacher corrections provide a more effective target for distilling compact vision--language models.

Current advancements in Multimodal Anomaly Detection (MAD) are largely driven by enhancing multimodal fusion, particularly through the integration of RGB and Depth data for richer anomaly representation. However, less attention was devoted to analyzing the role of cross-modal fusion bias, a well-known challenge in multimodal learning, in MAD. This gap motivates a key question: can we overcome this bias to break the performance bottleneck of current work? In this paper, we first analyze the impact of cross-modal fusion bias in MAD via the Fisher Information Matrix. Then, grounded in these findings, we propose UCFB, a simple yet effective plug-and-play framework designed to mitigate cross-modal fusion bias in MAD. It achieves this by jointly employing Fisher-information-guided dynamic calibration to adjust modality-specific regularization weights and canonical similarity analysis to improve inter-modal interactions. Extensive experiments on the MVTec 3D-AD and Eyecandies datasets demonstrate that UCFB achieves consistent improvements in single-class, multi-class, and few-shot settings.

Understanding and exploiting the training dynamics of overparameterized deep neural networks remains a central challenge in modern machine learning. Recent evidence on Neural Collapse (NC) shows that class representations and classifiers exhibit highly structured geometry, while the Tunnel Effect suggests that only a subset of layers is essential for feature extraction. We combine these two perspectives and propose an NC-inspired training framework for simplifying deep networks during training. Our method monitors representation dynamics through the Inverse Fisher Criterion, a stable and efficient proxy for the variability collapse behavior, to identify both the split point between feature extraction and classification and the training stage at which simplification becomes viable. We then replace the trailing layers with a lightweight classification head and continue training the reduced model. Experiments on image-classification benchmarks across MLP, VGG, and ResNet architectures show that the proposed method achieves substantial parameter reductions while maintaining accuracy comparable to that of the full model. Code to reproduce the experiments can be found at: https://github.com/LorenzoSciandra/NNS.

Submodular Information Measures (SIMs) have recently emerged as a powerful framework for representation learning and multimodal learning. In particular, the SCORE framework~\cite{majee2024score} demonstrated that SIMs can serve as effective objectives for supervised contrastive learning. Despite their empirical success, however, the geometric and statistical properties induced by different submodular information measures remain poorly understood. In this work, we develop a unified theoretical framework connecting SIMs to classical concepts in representation learning and statistical pattern recognition. We show that Total Information (TI) objectives characterize intra-class structure: Graph Cut TI recovers within-class variance, LogDet TI recovers generalized variance and covariance volume, and Facility Location TI induces imbalance-aware separation that emphasizes rare and confusable classes. We further show that Mutual Information (MI) objectives capture complementary notions of inter-class structure: Graph Cut MI is closely related to centroid separation and Fisher-style discrimination, LogDet MI captures covariance-aware separation through Mahalanobis distance, and Facility Location MI measures nearest-mode representational overlap. We validate these theoretical characterizations using controlled synthetic experiments that independently vary variance, covariance, class imbalance, class separation, and multimodal overlap. Across all settings, the empirical behavior closely matches the proposed theory. Our results provide the first unified geometric and statistical understanding of submodular information measures and offer principled guidance for selecting and designing SIM-based objectives for representation learning.

Medical world models aim to learn a latent state of patient or organ physiology and a transition function that forecasts how that state evolves under interventions, supporting downstream tasks from imaging-based diagnosis to digital-twin treatment planning. Two failure modes threaten the reliability of such models in clinical deployment: (i)\emph{covariate shift}, because training data are fragmented across hospitals, scanners, and time, so the feature distribution seen by the latent-dynamics predictor differs across fragments and from the distribution at deployment; and (ii)\emph{confidence misalignment}, because multi-step forecasts are often overconfident exactly where clinical risk is highest. We argue that both problems admit a unified treatment via a single lightweight regularisation objective, \textbf{CalTwin}, which combines a Fisher-Information-based shift penalty adapted from our prior work on fragmented covariate-shift remediation~\cite{khan2025mitigating,khan2025causal} with a Confidence Misalignment Penalty adapted from our prior work on calibrated vision-language classification~\cite{khan2025confidence}, applied here to a GRU-based medical world model's latent transition predictor. We derive the combined objective, establish which proof steps transfer from the classification setting without modification and which require adaptation, and evaluate it on the PhysioNet 2019 Sepsis Challenge, treating the two hospital systems as sequential training fragments and the unseen system as an out-of-distribution test. CalTwin reduces OOD next-step latent-state MSE by 9.1% relative to the no-penalty baseline (FIM penalty alone accounts for 7.0%); the ECE reduction from the Confidence Misalignment Penalty is real but small (0.7% for CalTwin, 1.3% for CMP alone).

Item Response Theory (IRT) has recently been proposed as a framework for evaluating large language model (LLM) benchmarks by separating a model's latent ability from the properties of individual benchmark items. Existing neural IRT approaches, including PSN-IRT, estimate these quantities using point estimates, limiting uncertainty quantification and downstream statistical inference. We introduce Laplace-PSN-IRT, a post-hoc last-layer Laplace approximation that augments a trained PSN-IRT model with approximate Bayesian posterior inference, recovering calibrated uncertainty over model ability and item difficulty without retraining. The resulting posterior enables credible intervals, probabilistic comparisons between models, and propagation of parameter uncertainty into Fisher-information-based item selection. We show that most pairwise comparisons among 12 models on a standard LLM benchmark leaderboard are not statistically distinguishable despite differing point-estimate ranks. We further show that point-estimate Fisher information can become nearly zero for many benchmark items because it is evaluated at a single reference ability, whereas posterior-expected Fisher information remains substantially more stable across the ability range. Finally, posterior-expected Fisher information more accurately recovers full-benchmark ability rankings from small benchmark subsets in most experimental settings while matching point-estimate performance for the smallest subsets. We validate the calibration of the approximate posterior using held-out predictive coverage and find that modeling item difficulty as random while treating item discrimination as fixed produces well-calibrated uncertainty in this architecture.

Random Vector Functional Link (RVFL) networks provide an efficient randomized learning framework for classification. Existing multi-view RVFL methods utilize complementary information from multiple views. However, preserving view-specific geometric structure, limiting the influence of large prediction residuals, and modeling relationships between multiple views remain challenging. This paper proposes a Residual-Coupled Graph-Embedded Multi-View RVFL model with fleXi guardian loss (XGRVFL-MV) for multi-view classification. The proposed model constructs RVFL representation for each view, incorporates graph embedding with intrinsic and penalty graphs constructed using the Local Fisher Discriminant Analysis weighting scheme. It also uses the bounded and asymmetric FleXi Guardian (XG) loss for residual learning. A residual-coupling term is introduced to encourage consistency among view-specific prediction residuals while preserving view-specific representations. The resulting optimization problem is solved using an inversion-free first-order optimization procedure based on Nesterov accelerated gradient descent. We evaluate the proposed model on UCI, KEEL, AwA, and Corel5k benchmark datasets. Experimental results, together with statistical analyses and hyperparameter sensitivity analyses, show that XGRVFL-MV achieves competitive classification performance compared with the baseline methods across the evaluated benchmark datasets.

Personal and organizational planning systems maintain two records that drift apart: what was planned (a task's effort budget) and what was done (a logged action's duration and description). Existing systems bridge them with an exclusive, all-or-nothing link that strands genuinely related but unlinked effort and reports false stalls on active goals. We formulate the bridge as a quasi-linear Fisher market: planned tasks are budget-constrained buyers, performed actions are divisible goods, and a fused text/structural/temporal signal sets each buyer's valuation. Two market instruments - a seller reserve price and a buyer cash option - yield conservation, a hard budget cap, and a provable junk filter as theorems. We extend the market with a concave completion utility discounting progress as a task nears its plan; standard convergence theory for the market's algorithm does not transfer here, resolved by a satiation-threshold fixed point with existence (Brouwer) and local uniqueness under an explicit diagonal-dominance condition, validated empirically on random and adversarial instances. A de-circularized, multi-seed benchmark - observed affinity corrupted independently of the scored ground truth - surfaces a genuine weak spot: the market's sharp, zero-entropy equilibrium is more sensitive to affinity noise than entropy-regularized optimal transport's permanently smoothed one. We resolve this with a one-parameter entropy-regularized generalization unifying the two, plus a noise-adaptive rule for its regularization strength. We report full reproducibility parameters, discuss limitations candidly, and relate the result to multi-touch attribution, optimal transport, and online Fisher-market algorithms.

We study Gaussian-width complexity on statistical manifolds through a pair of functionals: the primal Fisher width $w_G(T) = w(G^{1/2}T)$, induced by the Fisher metric, and the inverse-Fisher width $w_{G^{-1}}(T) = w(G^{-1/2}T)$, induced by the inverse Fisher metric. The two widths play complementary statistical roles. On the learning side, the Fisher width measures the size of local parameter fluctuations in the geometry induced by the Fisher information. For Fisher-regular losses, we prove that the scale $w_G(H_r)/\sqrt n$ is attained on sufficiently small Fisher balls. On the recovery side, the inverse-Fisher width captures the effect of anisotropic Gaussian measurements whose covariance is determined by the inverse Fisher information. For sparse recovery, the resulting geometry depends not only on sparsity but also on the position of the active coordinates in the Fisher spectrum. We obtain a two-sided estimate for the corresponding statistical dimension, together with support-sensitive recovery estimates and a natural ordering of supports with different curvature profiles. Finally, we establish a sharp relation between the primal and inverse-Fisher widths. On any common compact coordinate set $T$, they satisfy $$w_G(T)w_{G^{-1}}(T)\geq w(T)^2.$$ Thus, Fisher anisotropy may transfer complexity from one geometry to the other, but cannot reduce both widths relative to the Euclidean scale.

We establish a $\widetildeΩ(d^{5/4}\sqrt T)$ lower bound on the minimax expected regret of stochastic bandit convex optimization of $1$-Lipschitz functions on the Euclidean ball. This presents the first nontrivial regret lower bound that grows faster than $d\sqrt{T}$ for this problem, establishing that stochastic bandit convex optimization is fundamentally harder than linear bandits. The hard class of convex functions we construct takes the following form in dimension $2d$: for an action $a = (a^1,a^2) \in \mathbb{B}^{2d}_2$, each function is the scaled soft maximum of a "tube", $r^{-1} \| W^\star a^1 - \frac{r}{8\varepsilon} a^2 \|_2$ (hyperparameterized by $\varepsilon,r$), and a squared distance function, $\frac12 \| a^1 - u^\star \|_2^2 - \frac12 \| u^\star \|_2^2$. Here, $W^\star \in \mathbb{R}^{d \times d}$ is an unknown linear transformation, and $u^\star \in \mathbb{R}^{d}$ is an unknown vector which must be learned to minimize the function. Observations are informative about $u^\star$ only when the learner's action lies near the tube determined by $W^\star$, satisfying $a^2 \approx \frac{8\varepsilon}{r} W^\star a^1$: thus the learner must either find this tube without knowing $W^\star$, or spend observations learning useful directions of $W^\star$. Formally, our regret analysis exploits this tradeoff by bounding the posterior spread of Fisher information matrices obtained under an adaptive sequence of actions. Together, these ingredients give a sample complexity lower bound of $\widetildeΩ(d^{5/2}/\varepsilon^2)$ to find an $\varepsilon$-optimal action, which translates to an $\widetildeΩ (d^{5/4} \sqrt{T})$ regret lower bound. We also extend this lower bound to the unconstrained setting where the action space is $\mathbb{R}^d$.

Digital Twins rely on surrogate models to mirror physical systems in real time, yet these models can degrade as operating conditions evolve, a phenomenon known as concept drift. Maintaining surrogate fidelity under drift, particularly when models must also capture aleatoric uncertainty, remains an open challenge. Existing adaptive frameworks lack principled mechanisms for detecting when updates are needed, for efficiently adapting models from limited streaming data, and for certifying that updates genuinely improve predictive performance. Here we present an adaptive Digital Twin framework that integrates a Fisher score--based multivariate drift detector, Low-Rank Adaptation (LoRA) for parameter-efficient continual learning, and a Mann--Whitney $U$ test for online statistical validation. The framework monitors surrogate-model confidence via Fisher score vectors, triggers targeted fine-tuning of fewer than 1% of model parameters upon drift detection, and statistically certifies predictive improvement before deploying the updated surrogate. Applied to a stochastic linear system and a directed energy deposition additive manufacturing process as case studies, the framework successfully detects distributional shifts with short delays and restores both predictive accuracy and uncertainty quantification under abrupt and incremental drift. These results establish a statistically rigorous and computationally tractable pathway for sustaining the trustworthiness of neural-network--based Digital Twins throughout their operational life cycle.

Structured pruning compresses large language models (LLMs) by removing whole computational units, such as attention heads and feed-forward (FFN) channel groups. Most training-free methods, however, rank these units independently, implicitly treating the loss from pruning a set as the sum of its individual losses. This view fails for Transformers, whose sublayers are coupled through a shared residual stream. Two individually weak units can thus be jointly indispensable, yet independent scoring is blind to such dependence and removes them together. We introduce CoCurve (Cross-Module Co-Pruning Curvature), a calibration-only, fine-tuning-free method that prunes attention and FFN units jointly. A second-order Taylor expansion of the token-level KL between the frozen model and its masked copy yields a single Fisher matrix whose diagonal is classical node saliency and whose off-diagonal entries are co-pruning curvature edges: the extra damage of removing two units together. Under a single-ablation additivity approximation this matrix reduces to a Gram product of single-unit ablation features, so the full M x M interaction is recovered from M forward passes, with no pairwise sweeps or gradients. Pruning then reduces to one budgeted quadratic program, solved in a single shot under a shared attention--FFN budget, with no labels, fine-tuning, or recovery.

Quantum continual learning aims to train quantum models on sequential tasks without losing previously learned knowledge. However, variational quantum classifiers (VQCs) are prone to catastrophic forgetting under nonstationary task distributions. We propose quantum elastic weight consolidation (QEWC), a quantum Fisher information (QFI)-informed regularization method for mitigating forgetting. Unlike conventional elastic weight consolidation based on classical Fisher information (CFI), which measures parameter importance through measurement-dependent output statistics, QEWC uses QFI to quantify the intrinsic sensitivity of the parameterized quantum state. This gives an information-geometric view in which important parameters are identified by the local response of the quantum state manifold. We evaluate QEWC on VQCs trained on sequential binary classification tasks, including classical image-classification and quantum phase-classification tasks. Simulations show that sequential training without regularization causes severe forgetting, while both CFI-based EWC and QFI-based QEWC improve retention of previous tasks. Mechanistic analyses further show that the two methods impose different regularization geometries: CFI acts selectively on measurement-sensitive directions, whereas QFI imposes a denser state-geometric constraint over parameter space. Under depolarizing noise, CFI values are strongly suppressed by degraded measurement statistics, while QFI preserves a more stable sensitivity structure of the noisy parameterized quantum state. These results establish QEWC as a physically motivated approach for studying and mitigating forgetting in quantum continual learning through quantum-state geometry.

Deep networks trained with label noise often learn clean structure before memorizing corrupted labels. We show that this transition leaves a spectral signature in the centered scatter of per-example last-layer gradients. Its effective rank transiently expands during memorization and contracts after corrupted labels are fit. We call this phenomenon Fisher Rank Inflation. Corrupted labels increase effective rank by injecting spectral mass into low-energy or previously unused eigendirections, increasing the entropy of the gradient spectrum. We derive a first-order leave-one-out attribution formula, identify conditions under which corrupted examples contribute more strongly than clean examples, and explain why attribution signals weaken once the normalized Fisher-gradient spectrum stabilizes. We test these predictions on CIFAR-10, CIFAR-100, and CIFAR-10N using SmallCNN, ResNet18, and Vision Transformers. Across settings, Fisher effective rank exhibits a consistent inflation--collapse trajectory aligned with memorization. At peak-rank checkpoints, corrupted examples are enriched among the highest rank-contributing samples, with top-100 noisy fractions from $69.2\%$ to $96.2\%$ across five-seed synthetic-corruption experiments and $94.4\%\pm1.9\%$ on CIFAR-10N. First-order spectral attribution closely matches exact leave-one-out contributions in convolutional models and remains enriched in the Vision Transformer. Peak effective rank increases monotonically with corruption severity, from $28.88\pm1.95$ under clean training to $97.09\pm1.78$ at $60\%$ corruption. In several settings, the retrospectively identified onset of rank inflation precedes observable test degradation. These results establish Fisher Rank Inflation as a spectral signature connecting corrupted-example enrichment, corruption severity, and the transition from structure learning to memorization.

Spectral methods are widely used to construct representations from the geometry of data, but they often rely on a fixed kernel, graph Laplacian, or manually selected feature scaling. We propose Physics-Informed Eigenfunction Features with Learnable Scaling (PIEFS), a supervised neural representation-learning framework with a spectral inductive bias, based on a modified Dirichlet energy. In PIEFS, scalar coordinate maps are trained under empirical Gram orthogonality, a supervised linear readout, and a Dirichlet penalty in which the input gradient is transformed by a learnable metric $A(x)=Λ(x)U(x)$. The diagonal factor $Λ(x)$ controls anisotropic scaling, while the orthogonal factor $U(x)$ is parameterized by a structured product of Givens rotations. This construction yields task-adaptive Dirichlet-regularized coordinates rather than eigenfunctions of a fixed supervision-independent operator. Experiments on synthetic, tabular, and image-based benchmarks study the effect of identity, diagonal, and rotation-scaling metrics, and compare the resulting coordinates with classical baselines and NeuralEF. The results support PIEFS as a compact supervised spectral representation method and identify optimization stability, validation on explicit operator eigenproblems, and richer metric parameterizations as the main directions for future work.

Integrated sensing and communication (ISAC) enables intelligent wireless infrastructure but raises growing regulatory concern as fine-grained personal trajectory histories become a byproduct of sensing. General Data Protection Regulation (GDPR) Articles 5(1)(c) and 5(1)(f) require that personal data be limited to what is necessary and protected through appropriate technical measures against unauthorised reconstruction. This paper addresses both requirements through a Fisher information density (FID)-constrained trajectory sharing scheme for robot collision avoidance, where sensing estimates are perturbed according to local information content before sharing. Experiments on real pedestrian traces show that FID-controlled sharing achieves a strictly better privacy-utility tradeoff than fixed-error perturbation: at matched missed-conflict rates, reconstruction leakage and sustained exposure lengths are consistently lower, establishing information-aware perturbation as a principled technical measure aligned with GDPR data minimisation and integrity requirements.

We give a descent-free, alignment-free measurement of singular structure on trained networks. At a single frozen checkpoint the read recovers the order $k$ of each dead direction from the directional-Fisher rate, the master invariant from which the per-direction learning coefficient $1/(2k)$ follows exactly, in whatever basis the optimizer left. The same read classifies each direction, separating a genuine singularity, whose order the architecture fixes, from a flat gauge symmetry; the directional-Fisher magnitude settles the cases the order cannot. A pluggable detector supplies the directions for transformer, convolutional, and normalisation layers. The read recovers the architecture-predicted order across constructed cells and trained networks, including a fine-tuned vision transformer whose dead structure is the LayerNorm-kernel gauge and a from-scratch one whose compressed MLP forms a node-death at its activation order. Where the singular structure enumerates, the per-direction orders assemble, through the typed intersection of the loci, into the global coefficient $(λ, m)$ matching the closed form. The method removes the canonical-alignment and descent preconditions of the underlying rate result, turning order-recovery into a deterministic, architecture-general reading. We then map its reach into the Watanabe triple: the order determines the universal singular fluctuation $ν(k)$, though a trained network's realized $ν$ falls below it as the live structure absorbs the dead direction's data fluctuation, and the multiplicity recovers from the dominant structure under a single-locus assumption.

Graph-based narrative extraction relies on a coherence function to score transitions between events, but the coherence metrics in current use are defined operationally and lack an information-theoretic foundation. We study the composite metric $C=\sqrt{A\cdot T}$, where $A$ is the angular similarity of document embeddings and $T=1-d_{\mathrm{JS}}$ is a topic proximity from the Jensen-Shannon distance of soft memberships, and give it an information-geometric reading together with an axiomatic characterization of the geometric-mean combinator. On the product manifold $\mathbb{S}^{d-1}\timesΔ^{K-1}$, the negative log-coherence decomposes additively into an angular and a topic cost. Because the Riemannian metric tensor induced by the Jensen-Shannon distance on the simplex is proportional to the Fisher information matrix, the topic component is locally consistent with the Fisher-Rao metric singled out by Chentsov's theorem. Within the compensability spectrum of combinators, the geometric mean is the unique one consistent with four natural axioms (a boundary/veto condition, symmetry, log-additivity, normalization), and the construction motivates a proper product metric $d_\times$. Experiments on four corpora, three embedding families, and three topic models are consistent with the framework: the Fisher identity holds ($R\ge0.99$), the geometric mean tracks $d_\times$ closely ($ρ=0.999$), and a downstream LLM-as-judge check finds it is not dominated by any alternative combinator or single-channel baseline. Sweeping the spectrum, the bottleneck-coherence gap between extracted and random storylines splits into a symmetric component, maximized at the geometric mean across five corpora, and a displacement term; a cross-modal image-narrative case study reproduces the effect. These results justify the composite coherence metric and articulate when the geometric mean is the natural choice.

In general, an ensemble classifier is more accurate than a single classifier. In this study, we propose an ensemble classifier called the kernel Fisher discriminant analysis forest (KFDA Forest), which is a tree-based ensemble method that applies KFDA. To promote diversity, bootstrap is used, and variable sets are randomly divided into K subsets. KFDA is performed on each subset to increase classification accuracy. KFDA maximizes the distance between classes while minimizing the distance within classes. KFDA can also be applied to classification problems in a nonlinear data structure using the kernel trick because it can transform the input space into a kernel feature space, commonly named a rotation, rather than performing a dimensionality reduction. Because new feature axes and KFDA projections are parallel, decision trees are used as a base classifier. To compare the proposed method with existing ensemble methods, we apply these to real datasets from the UCI and KEEL repositories.

Federated Learning (FL) emerged as a promising distributed machine learning paradigm. However, extending FL to the class incremental learning scenarios introduces unique challenges: 1) Capacity conflict and catastrophic forgetting from the shared model overloading, 2) Heterogeneity from Non-Independent and Identically Distributed (Non-IID) data, and 3) Synchronized class misalignment. In this paper, we propose \textbf{F}isher-Routed \textbf{M}i\textbf{X}ture of Experts for \textbf{Fed}erated Class-Incremental Learning (\textsc{FedFMX}), a novel framework to address these challenges via adaptive expert specialization across clients. The crucial insight is to route each sample to an expert subset that jointly optimizes knowledge acquisition and retention. Specifically, we introduce a Fisher-Routed Expert Scoring (FRES) module to estimate expert importance via Fisher-based stability cost and gradient-based plasticity gain. Then, we design an Adaptive Expert Selection (AES) module by quantifying marginal contributions for adaptive expert subset determination. Finally, by the routing-aware regularization (RAR), we achieve load balance and efficient FL training. We theoretically prove the $\mathcal{O}(T^{-1})$ convergence rate. Extensive experiments on multiple benchmarks compared with state-of-the-art methods demonstrate the superiority of \textsc{FedFMX}.

We study the spectral perturbation of the empirical Fisher Information Matrix (FIM) of a parametric statistical model under two structured perturbations: departure of the input from a reference (in-distribution) ensemble, and finite-precision (quantized) perturbation of the model's parameters. For the first, under an explicit local curvature-monotonicity hypothesis on the dominant eigenvalue lambda_max of the FIM, we show departure from a reference manifold provably elevates lambda_max relative to a calibration baseline (Proposition 3.2), and discuss why this hypothesis is required, since curvature need not increase monotonically under every perturbation. Our principal result is a directional eigenvalue perturbation bound, via Weyl's inequality, showing lambda_max under a quantization noise perturbation is lower bounded by its unperturbed value up to a third-order remainder, and, under a mild genericity condition, strictly exceeds it at leading order (Theorem 4.3). We give two tractable approximations to lambda_max -- one heuristic, one with a rigorous two-sided bound -- and a completeness result for a threshold-based partition of an augmented state space. These results motivate using sigma_t = lambda_max(F_t)/lambda_base as a runtime monitoring statistic for deployed language models: the quantization result offers a mechanism for an empirical observation of our own, where a calibration threshold for this statistic was approximately 244 times larger than a preliminary full-precision estimate on a 4-bit quantized model, a single measurement rather than a value derived in closed form. We report supporting measurements (twelve models, n=1,080 trajectories) broadly consistent with our predictions, discuss the scope and limitations of every result, and state as an open problem the closed-form prediction of the quantization inflation magnitude our bound does not supply.

Singular learning theory characterises the complexity of a deep network through the geometry of its loss singularities. The local learning coefficient (LLC), the standard estimator of Watanabe's real log canonical threshold (RLCT, $λ$), reads this geometry as an integrated Bayesian scalar through SGLD, which needs per-task calibration and $10^4$-$10^6$ forward-backward passes per checkpoint. We introduce Dead-Direction Signatures (DDS), a family of cheap closed-form spectral readings of singular structure: each reads a network's activation matrix or per-sample-gradient Fisher-Gram at a chosen layer, replacing the SGLD posterior chain with spectral linear algebra. The readings rest on a dead-direction framework that predicts a structural correlation between activation- and Fisher-side spectra at any singular minimum, and a rank-multiplicative volume identity that single-eigenvalue monitors cannot produce: the active-volume $\log\det^{+}(G)$ slope counts the dead directions, tracking the rank-deficit $r$ across $r \in \{1,2,3,4\}$ (slope ratios $2.0, 3.1, 4.0$ at $r{=}2,3,4$ against the predicted $2,3,4$), where the smallest eigenvalue is rank-blind. On reduced-rank regression with closed-form $λ$, calibrated LLC recovers $λ$ at $99\%$ mean and the DDS observables rank-track it at the framework-predicted sign; on a non-linear modular-addition transformer DDS separates $d_{\mathrm{model}}$ across eighteen orders of magnitude where calibrated LLC at the protocol budget is rank-flat. Complementary to LLC's integrated posterior reading, DDS gives a directional, layer-local handle on a network's dead directions, read in closed form from its activation and gradient spectra.

3D Gaussian Splatting (3DGS) has emerged as a promising technique for novel view synthesis. However, 3DGS requires dense input views to achieve high-quality rendering. In sparse-view scenarios, 3DGS often prones to overfitting, resulting in noticeable artifacts and degraded rendering quality. Previous methods explore to address this issue by introducing additional priors (e.g. depth priors) or integrating regularization techniques (e.g. Dropout). However, these methods are often applied without principled guidance. In particular, prior-based augmentation typically samples novel viewpoints randomly, while Dropout-based regularization randomly removes Gaussians. The compounded randomness introduces uncertainty and instability, limiting the fidelity of novel view synthesis. In this paper, we propose a novel method for sparse-view 3DGS that incorporates Fisher Information to quantitatively guide the utilization of geometric priors and regularization. Specifically, our method comprises two key components: (1) Stereo augmentation with Fisher Information. By leveraging Fisher Information, we actively select most informative supporting views and use depth priors to curate reliable pseudo ground truths, which reduces randomness in augmentation and improves stability and rendering fidelity; (2) Uncertainty-aware regularization. We reduce the instability of Dropout-based regularization by using Fisher Information to quantitatively measure the uncertainty of each 3D Gaussian, and adaptively adjust the removal probability, leading to more stable and effective regularization. With these two components, our method effectively mitigates overfitting and improves the stability of optimization in sparse-view 3DGS, resulting in superior rendering fidelity. Extensive experiments show that our method achieves state-of-the-art performance in sparse-view novel view synthesis benchmarks.

The score matching problem is a central training objective in modern generative modeling, diffusion models, fitting unnormalized statistical models, and inverse problems. A standard approach is to minimize the forward Fisher divergence, where the expectation is taken with respect to the teacher distribution. However, recent results show that even in simple Gaussian mixture model settings, this objective can lead to undesirable and initialization-dependent convergence behavior. In this paper, we study an alternative objective: the reverse Fisher divergence, where the expectation is taken with respect to the student distribution. We analyze gradient descent (GD) for fitting Gaussian mixture models and show that this change in the objective leads to significantly better optimization properties. First, when the teacher distribution is a single Gaussian and the student is a Gaussian mixture model with fixed weights and identity covariances, we prove the global convergence of GD from arbitrary initializations. Second, we extend the analysis to the case where the teacher is also a Gaussian mixture model and prove global convergence guarantees under a global random initialization scheme and a $\widetildeΩ(1)$-separation assumption on the target means. In particular, with high probability, each student component converges near its closest teacher component, and we provide conditions under which the student distribution converges in total variation distance. Our proofs rely on a new Lyapunov-based analysis of the gradient descent dynamics, showing that the reverse Fisher divergence has a much more favorable optimization landscape than the forward Fisher divergence.

Pretrained transformers sit near singular minima of the loss, where the Fisher information metric degenerates along dead directions: directions in parameter space along which the directional Fisher vanishes. Locating such a direction normally needs a forward pass and an eigendecomposition of activations, or a sampling-based complexity estimate; none returns a direction computable from the network's parameters alone. We give one, for LayerNorm transformers. The inverse-scale direction $γ^{-1}/\|γ^{-1}\|$ of the LayerNorm affine is an exact algebraic kernel of the post-final-norm centred activation covariance, for any input distribution, and induces a corresponding dead direction in parameter space. It is read from the LN scale parameter alone, with no forward or backward pass and no eigensolve: the cheapest dead-direction read, specific to LayerNorm. We test it on $14$ pretrained transformers ($9$ LayerNorm, $5$ RMSNorm; $160$M-$35$B; language and vision objectives). At random initialisation the predicted direction matches the measured bottom singular direction (one forward pass, direct SVD) to four decimal places on $9/9$ LayerNorm models, and is correctly absent on $5/5$ RMSNorm models, which lack the mean-subtraction projector that creates it. On the trained checkpoint the covariance eigenvalue along this direction deepens by ${\sim}10^3\times$ and further dead directions open; the random-init-to-trained gap is a one-forward-pass, per-checkpoint readout of singular structure along the predicted coordinate. Two consequences follow in closed form: the residual stream's smallest singular value is preserved block-to-block on $13/14$ transformers measured on their own input distribution, the one exception (Gemma$4$-$31$B) a genuine dead direction the same read pinpoints; and the kernel direction's presence classifies a transformer's normalisation from the parameters alone.

Gaussian width is a central geometric complexity measure in high-dimensional probability, compressed sensing, convex optimization, and learning theory. It quantifies the average extent of a set along random directions, thereby capturing the effective dimension of constraint sets, hypothesis classes, and descent cones. However, this notion is intrinsically Euclidean. Statistical models instead carry a natural Riemannian geometry induced by the Fisher information metric, where directions are scaled according to statistical distinguishability rather than ambient Euclidean length. We introduce Fisher width, a Fisher-geometric analogue of Gaussian width for statistical manifolds. At a parameter point $θ$, Fisher width replaces the Euclidean identity by the local metric tensor $G(θ)^{1/2}$, measuring the Gaussian width of the Fisher-rescaled set. This makes the resulting quantity sensitive to local statistical curvature and invariant under smooth reparameterizations. We develop the basic theory of Fisher width, showing that it retains key structural features of Gaussian width, including concentration, metric perturbation stability, and spectral comparison bounds with the Euclidean baseline, while also capturing anisotropic geometric effects invisible to Euclidean measures. As an application, we prove a generalization bound for Fisher-Lipschitz hypothesis classes and propose computable estimators, which we evaluate empirically on MNIST across three model classes. Fisher width is to statistical manifolds what Gaussian width is to Euclidean convex bodies. This work lays the foundation for studying complexity and learning on curved statistical manifolds.

Neural operator models trained on simulation data often lose accuracy when applied to experimental measurements due to the sim-to-real gap. Standard fine-tuning with limited real data can reduce this gap, but it may also damage the core physics-relevant representations learned during pretraining. Although knowledge-preserving adaptation has been widely investigated in vision or language tasks, it remains unclear whether these methods are suitable for neural operators whose architectures and protected knowledge are fundamentally different. Neural operators need to preserve core-scale physical structures rather than semantic or visual features. We propose PhysGuard, a physics-preserving framework for accurate sim-to-real adaptation of neural operators. Specifically, PhysGuard uses the empirical Fisher Information Matrix computed on simulation data to identify physics-critical parameter directions, then restricts fine-tuning updates to directions that do not interfere with them. A layer-wise Gram-matrix formulation makes this efficient for models with millions of parameters, while an adaptive threshold automatically determines the protected subspace size. A spectral probe experiment shows that the dominant Fisher directions are strongly associated with low-frequency output structures. Experiments on benchmark across four neural operator architectures and different physical systems show that PhysGuard performs strongly on most evaluation metrics compared to baselines. The benefits are most evident under severe domain shift, where it reduces low-frequency error by up to 32% compared to standard fine-tuning while maintaining adaptability. Our code is available at https://github.com/ZhouChaunge/PhysGuard.

Deep neural networks have achieved strong performance in medical image classification, but often work like black-box. Commonly used post-hoc interpretation methods often provide heuristic visualizations whose relationship to the classifier's predictive distribution is indirect. This work introduces a local sensitivity analysis framework based on the input-dependent Fisher Information Matrix (iFIM) of a trained classifier. The iFIM characterizes how the classifier's predictive distribution changes under infinitesimal perturbations of the input image. By using a Gram-matrix formulation, the nonzero eigenspectrum of the iFIM can be recovered without explicitly forming the full image-dimensional Fisher matrix. The leading iFIM eigenspace is then used to project an input image into a high local-sensitivity component and its orthogonal component. These components provide a model-intrinsic description of local predictive sensitivity, rather than a conventional pixel-wise attribution heatmap or a causal segmentation of task-relevant anatomy. The framework is evaluated on controlled and clinical medical image classification tasks using multiple classifier architectures. Perturbation-based experiments show that high-sensitivity iFIM components are more strongly coupled to changes in predictive confidence and classification performance than lower-sensitivity complementary components. The results support the iFIM framework as a principled tool for analyzing local decision sensitivity and for complementing existing attribution-based interpretability methods in medical imaging.

Sampling from high-dimensional, non-log-concave distributions with unnormalized densities is a fundamental challenge in machine learning, particularly when the exact gradient of the potential is unavailable and must be approximated via stochastic gradients that exhibit high variance under a fixed budget of gradient computations per iteration. Although variance reduction techniques such as SGD with momentum, STORM, and PAGE have demonstrated improved convergence properties in non-convex optimization, their implications for sampling from non-log-concave distributions remain largely unexplored. In this work, we develop the first unified analysis of these estimators for sampling from non-log-concave distributions. We establish improved non-asymptotic convergence rates in $\varepsilon$-relative Fisher information and, under a Poincaré inequality assumption, in squared total variation distance, and further prove weak convergence to the target distribution. We extend our analysis to solving inverse problems with score-based generative priors. We empirically validate our theory and demonstrate that, under a fixed gradient computations per iteration, variance-reduction techniques consistently improve sample quality in two standard imaging applications.

Finding D-optimal designs for generalized linear models (GLMs) is challenging due to the dependence of the Fisher information matrix on unknown parameters and the lack of closed-form solutions, particularly when input factors include both discrete and continuous variables. Although classical algorithms and recent metaheuristic approaches have offered partial solutions, there remains a need for robust and computationally efficient methods. In this paper, we propose a penalized Particle Swarm Optimization (PSO) approach, named $p$-PSO. Here we introduce a new, general-purpose penalty formulation for constrained optimization and demonstrate its effectiveness in optimal design problems. The formulation is algorithm-agnostic and applicable to a broad class of black-box optimization methods. Results show that the method is highly efficient, with its primary contribution being a penalty formulation that enables the direct use of an off-the-shelf PSO algorithm and extends naturally to more general constrained optimization tasks.

Parameter-efficient fine-tuning (PEFT) aims to adapt pretrained models with a small trainable parameter subset, however, most existing methods choose this subset from fixed architectural heuristics rather than using dynamic, task-aware criteria. We introduce \textbf{FisherAdapTune}, a Fisher-guided Adaptive Fine-Tuning framework that progressively selects parameter groups by tracking the temporal drift of their Fisher geometry. Starting from a PAC-Bayesian view of fine-tuning, we decompose the generalization error bound into Fisher-weighted update costs and show that parameter groups whose curvature contribution has stabilized can be frozen to reduce the error bound without interrupting the remaining adaptation dynamics. FisherAdapTune formulates this criterion with a scale-invariant Jensen-Shannon distance between consecutive Fisher distributions, yielding an adaptive active parameter set. We evaluate our approach on a downstream segmentation task, and results show FisherAdapTune improves the in-distribution performance and zero-shot transfer in multiple settings, validating that Fisher structural drift is a useful signal for efficient, task-aware adaptation. We release our \href{https://github.com/AtlasAnalyticsLab/FisherAdapTune}{code} publicly to enable further application of our proposed approach.

Physics-Informed Neural Networks inherently suffer from task interference because they rely on a shared parameter space to satisfy both governing differential equations and boundary conditions. We analyze this structural conflict using the Fisher Information Matrix to quantify the effective degrees of freedom ($d_{eff}$) in a physics-constrained model. Unlike the classical $d_{eff}$ which measures how many parameter directions are informed by data against a statistical prior, our $d_{eff}$ measures the dimension of the parameter directions unconstrained by the differential operator. For operators with finite-dimensional kernel, we show that $d_{eff}$ converges to the kernel dimension exactly, independent of network width, depth, or activation function, recasting it from a fit diagnostic into a structural invariant of the underlying continuous operator. For operators with infinite-dimensional kernel, $d_{eff}$ instead measures the network's finite-dimensional representational bandwidth for that kernel rather than recovering an integer invariant. Importantly, $d_{eff}$ also serves as an a priori structural diagnostic. Driving $d_{eff}$ of a well-posed problem to zero certifies that the physics and boundary constraints have absorbed the network's free directions. Building on this characterization, we introduce subspace projection strategies for boundary adaptation. Rather than retraining from scratch, we project parameter updates into the null space of the pre-trained physics operator so that new boundary conditions are satisfied without disturbing the learned physics. Gradient-based fine-tuning can match or exceed this but needs more wall-clock time and tuning, whereas subspace projection delivers near-equivalent quality in seconds to minutes. We validate on linear and nonlinear operators, demonstrating accurate adaptation to initial and boundary shifts and unencountered constraint types.

Mixture-of-Experts (MoE) models achieve strong performance through conditional computation, but their large parameter footprint poses deployment challenges. Prior MoE compression approaches catastrophically fail when evaluated on general-purpose benchmarks beyond commonsense reasoning. We trace this failure to the granularity of compression: important capabilities are distributed across experts but concentrated in FFN sparse intermediate dimensions. To identify these dimensions, we use Fisher importance which outperforms activation-, router-score-, and magnitude-based alternatives, and identifies tiny sets of task-critical dimensions: in Qwen1.5-MoE, removing as few as 12 of 1.35M routed-FFN intermediate dimensions collapses GSM8K accuracy while largely preserving factual-knowledge performance. Building on this, we propose Fisher-MoE, which operates within FFN to remove intermediate dimensions ranked by Fisher importance. At the same 50% MoE compression ratio, Fisher-MoE preserves model capability, while reducing weight memory by ~45% and improving inference throughput by 21%. These findings suggest intermediate dimension granularity is an effective unit for both compression and ranking where capability concentrates in MoE models.

The robustness of deep neural networks is crucial for safety-critical deployments, yet existing evaluation methods are often attack-dependent and lack interpretability. We propose a principled, attack-agnostic robustness metric based on the spectral norm of the Fisher Information Matrix (FIM), which quantifies the worst-case sensitivity of the model's output distribution to input perturbations. Theoretically, we establish that the FIM equals the variance of the input Jacobian and derive closed-form spectral bounds for common architectures, including VGG, ResNet, DenseNet, and Transformer, providing the first theoretical robustness ranking. To enable scalable evaluation, we develop efficient algorithms, including power iteration and Hutchinson-based estimation, that support both white-box and black-box settings. Extensive experiments across multiple datasets, including CIFAR, ImageNet, and medical images, and across multiple architectures show a strong correlation between our metric and adversarial vulnerability. Our framework serves as an interpretable diagnostic tool that complements attack-based evaluations, offering insights into architectural sensitivity and guiding the design of more robust models. Code is available at: https://github.com/franz-chang/SRP/.

Large machine learning models benefit substantially from multimodal inputs that provide a complementary view of the same example. We introduce QUIVER (QUantum-Informed Views for Enhanced Representations, a paradigm that enriches classical data-driven features with a quantum Fisher view: a geometrically motivated, basis-independent summary of higher-order correlations captured by a variational quantum circuit (VQC) trained to perform the same task. Unlike classical feature augmentation, the quantum Fisher information matrix encodes the intrinsic geometry of the learned quantum state manifold. While this feature map, motivated by quantum information theory, is ordinarily non-trivial to model classically, it can surface statistical structure that additional classical data or model capacity finds difficult to learn. This makes the quantum Fisher view a genuinely complementary modality rather than a redundant one. We demonstrate that QUIVER improves standard performance metrics on two benchmark datasets from very different fields: QM9 for predicting molecule properties, and JetClass for predicting jet flavor at the Large Hadron Collider (LHC). The core contribution, however, is domain-agnostic: the quantum Fisher view can be fused into a broad class of model architectures via targeted modifications to the base architecture, to incorporate information about the quantum geometry of the problem. These results demonstrate that quantum-geometric features, extracted from simulated variational circuits, can deliver measurable value for standard machine learning tasks, well before the advent of fault-tolerant quantum hardware.

Sampling from high-dimensional, non-log-concave distributions with unnormalized densities remains a fundamental challenge in machine learning, particularly in black-box settings where gradient information is inaccessible or computationally prohibitive. While Langevin dynamics provides a principled framework for sampling when gradients are accessible, its extension to the black-box settings suffers from high variance and lacks non-asymptotic convergence guarantees for non-log-concave sampling. To address these limitations, we propose a variance-reduced zeroth-order Langevin sampling method. Our method employs a gradient estimator that substantially reduces the variance of the classical batched zeroth-order estimator and eliminates the unfavorable dimensional dependence of the batch size required for accurate estimation, enabling practical and stable sampling. We establish the first non-asymptotic convergence guarantees for zeroth-order non-log-concave sampling in terms of $\varepsilon$-relative Fisher information, and, under a Poincaré inequality assumption, squared total variation distance. We further propose ZO-APMC, a posterior sampling algorithm for black-box inverse problems with pre-trained score-based generative priors, establishing the first non-asymptotic convergence guarantees for such methods. We validate our theory through synthetic experiments and demonstrate strong empirical performance on practical linear and nonlinear inverse problems.

Integrated sensing, communication, and computation (ISCC) provides a promising framework for indoor human-centric applications. In these applications, short-term human pose prediction facilitates continuous human tracking and resource allocation in advance. In this paper, we propose a Cramer-Rao bound (CRB) guided resource allocation framework for indoor mmWave ISCC systems to minimize the human pose prediction error under communication, latency, and energy constraints. We characterize the impact of sensing power on range-estimation uncertainty and point-cloud perturbation based on the CRB. To capture the impact of computation resources on prediction performance, we adopt an adaptive-depth Mamba-based pose prediction model, where lightweight prediction heads are attached after every layer to enable inference with different model depths. With this unified sensing-computation modeling, we establish a quantitative relationship among sensing power, model depth, and prediction error. Furthermore, we formulate a joint resource allocation problem to minimize the pose prediction error. To solve this problem efficiently, we develop an alternating optimization (AO)-based algorithm, where closed-form solutions are derived for the sensing power and model depth update steps. Simulation results show that the proposed scheme significantly reduces pose prediction error compared with baseline methods, validating its effectiveness for resource-constrained indoor human-centric ISCC systems.

Parameter-efficient fine-tuning(PEFT) has largely focused on LoRA and its accuracy-oriented variants, leaving the original goal of reducing trainable parameters has receivedcomparatively little attention. We introduce FoRA, which revisits this goal by reducing the number of adapted layers rather than adapter rank. FoRA selects task-informative layers via a single-pass diagonal Fisher score (under 1% of training cost) and trains the LoRA down-projection at selected layers on the Stiefel manifold, preserving column orthonormality and effective rank. FoRA consistently outperforms LoRA and DoRA at half their parameter budget, and falls within 0.7-0.8 accuracy points of AdaLoRA at one-quarter its parameter count, across five LLaMA-family backbones. Cross-architecture experiments on twelve backbones from the LLaMA, Qwen3, and Gemma families confirm consistent gains from 270M to 32B parameters. The two components combine super-additively: Fisher selection alone matches rank reduction at the same budget, while the Stiefel constraint provides the decisive additional gain.

Existing score-based methods for inverse problems often resort to approximate minimization of the KL divergence between the inversion distribution and the Bayesian posterior. Such an approximation leads to severe mode collapse and unreliable uncertainty quantification. In this paper, we propose Principled Posterior Matching (PPM), a framework that returns to the fundamentals of variational inference, rather than using tricky approximations. Instead of relying on heuristic approximations, we rigorously formulate the exact optimization of the KL divergence via the integration of Fisher divergence. We derive a tractable, equivalent gradient form of this integral, enabling precise optimization without the biases introduced by prior approximations. Our analysis clearly reveals that the mode collapse in previous methods stems directly from this approximation gap. Supported by our theoretical solution, PPM unifies two complementary paradigms: (1) In variational inference, PPM adopts mass-covering divergences that significantly improve the inversion diversity and uncertainty quantification; (2) In amortized inference, it enables the training of an efficient reconstruction network for rapid, single-step reconstruction. Furthermore, our formulation naturally extends to a broader family of divergence measures by generalizing the integral of the Fisher divergence. We validate PPM across challenging computational imaging tasks, including inpainting, super-resolution fluorescent microscopy, and radio interferometric black-hole imaging. In all experiments, PPM achieves superior reconstruction fidelity, faithful multimodal posterior recovery, and well-calibrated uncertainty estimates, establishing a robust framework for scientific imaging.

Of $1{,}536$ Gaussian release covariances we tested for single-layer hidden-state privacy, zero achieve both moderate utility and moderate privacy against an adaptive retrieval attacker. We prove a complementary Fisher-ball lower bound: every full-rank Gaussian release at $O(1)$ Fisher utility admits a direction whose Mahalanobis signal grows linearly in hidden width, ruling out uniform Gaussian safety in the class and matching the empirical empty middle. The diagonal inverse-Fisher release $Σ^\star_{\mathrm{diag}}(\mathcal{K}) = (2\mathcal{K}/d)\,\mathrm{diag}(1/F_{ii})$ is the unique minimax-optimal diagonal mechanism at first-order KL budget $\mathcal{K}$ and the only release with worst-attacker top-1 $\le 0.001$ at every point of a 32 model-layer grid, but it sits on a privacy/utility edge rather than filling the middle. A generalized-eigen mechanism reaching $13\times$ Pareto reduction under Euclidean retrieval collapses to $100\%$ top-1 under the adaptive Mahalanobis attacker, and a full-trajectory sequence inverter recovers $94\%$ of clean GPT-2 prefixes but $0\%$ under $Σ_{\mathrm{diag}}$. A split-memory transformer trained from scratch reaches $G_{\mathrm{Mah}} \in [20, 33]$ at 90M and maintains a $6$--$24\times$ advantage over same-budget GPT baselines from 30M to 1B at a fixed-token language-modeling loss penalty; pretrained models top out at 9.3. These results reframe hidden-state release from mechanism-design within the Gaussian class to architecture or release co-design.

We develop a principled framework for analyzing and designing noise schedules in diffusion models. We show that one can recast this design problem as an optimal control problem, whose state is the Fisher information of the diffusion process which evolves according to an ODE and the control input is the noise schedule. The objective of the optimal control problem is a functional involving the Fisher information, which is shown to be an upper bound on the Kullback-Leibler sampling error. By solving this optimal control problem, we obtain sufficient conditions on noise schedules under which state-of-the-art $\tilde{\mathcal{O}} (d/n)$ sampling error is achievable, where $d$ is the data dimension and $n$ is the number of discretization steps. While existing theoretical work also prove that $\tilde{\mathcal{O}}(d/n)$ sampling error bounds are achievable, these results hold for specific noise schedules, which do not include the schedules used in practice. Under a further parametric assumption on the data distribution, we show that one can obtain closed-form expressions for the noise schedules. These noise schedules generalize standard empirical schedules such as exponential and sigmoid schedules by allowing additional parameters that can be tuned. Systematically tuning the parameters of these schedules yields new schedules that achieve superior FID scores on image generation benchmarks.

Noisy-label methods often estimate sample reliability from forward-space signals such as loss, confidence, or entropy. These signals indicate whether a sample is difficult to predict, but they do not directly test whether its observed label induces a reliable parameter update. This gap matters because hard clean samples and mislabeled samples can have similar loss while inducing different updates. We recast reliability estimation as diagnosis of the observed-label update. The sample-wise empirical Fisher trace gives a backward-space measure of update energy: for the classifier layer, it factorizes into a prediction-residual term and a feature-sensitivity term, so it captures information beyond scalar loss. Trace, however, is still a radial magnitude signal and cannot decide whether a large update is useful or harmful. We therefore propose Relative Geometric Conflict (RGC), which compares the observed-label gradient with a reference gradient induced by an EMA teacher. The conflict term helps distinguish large but aligned hard-clean updates from large conflicting updates caused by corrupted labels. Across synthetic and real-world noisy-label benchmarks, RGC improves hard-clean preservation and accuracy under our evaluation protocol.

The Quantum Fourier Transform (QFT) is employed by hidden subgroup problem (HSP) algorithms, including Shor's algorithm for factoring. The circuit depth of the QFT remains challenging for near-term hardware. To find shallower alternatives we identify two properties that are exploited by the QFT to enable HSP. Firstly, the shift invariance of the QFT allows for the removal of a random overall shift. Secondly, the QFT retains information about the hidden subgroup generator accessible in the measurement outcomes. We quantify that information via the discrete Fisher information. We construct a family of shallow circuits using Hadamards and controlled-Phase gates, HP-$L$ circuits, that we prove preserve shift invariance. Numerical analysis shows these circuits retain exponentially growing Fisher information. The $\mathcal{O}(n)$ HP-$1$ is employed in place of the $\mathcal{O}(n^2)$ QFT in our numerical implementation of Shor's algorithm. An efficient neural network is used for the corresponding classical post-processing.

Low-rank adaptation (LoRA) assigns a uniform rank to every adapted weight matrix - a practical convenience that ignores a fundamental reality: different layers contribute unequally to task adaptation. We address this with a lightweight engineering solution: before fine-tuning begins, run eight calibration backward passes, compute the gradient variance of each LoRA-B matrix as a proxy for layer informativeness, and redistribute the rank budget proportionally. The resulting adapter is a standard LoRA with a per-layer rank pattern - no new parameters, no training overhead, no changes to serving infrastructure. We implement this via an efficient approximation of the empirical Fisher Information Matrix (eFIM) diagonal, restricted to LoRA adapter matrices only, which reduces memory cost by approximately 256x compared to full-model Fisher estimation. On GLUE with DeBERTa-v3-base, FIM-LoRA matches LoRA (88.6 vs. 88.7) at the same parameter budget, and on commonsense reasoning with LLaMA-3-8B reaches 68.5 vs. 68.7 for LoRA. The per-layer rank maps are interpretable: value projections and early-to-middle layers consistently receive higher rank, consistent with established findings on transformer layer roles.