Fisher Information

Kolmogorov-Arnold Networks (KANs) are motivated in part by interpretability: their learned edge functions can be inspected, pruned, and reduced to symbolic structure. In a fixed-basis KAN, this makes a small or zero basis coefficient look like a certificate of simplicity, much as a dead rectified linear unit (ReLU) marks unused computation in a multilayer perceptron (MLP). Fisher nullity gives a precise statistical notion: a parameter direction is Fisher-simple exactly when perturbing it is invisible under the task distribution. We study when these architectural and Fisher notions agree. For a dead ReLU unit, they agree: the closed activation region makes the associated score directions vanish. For a fixed-basis KAN, they do not. In the single-layer Gaussian case, the coefficient Fisher matrix is a basis Gram matrix under the input distribution and is independent of the fitted coefficients. In a multilayer KAN, Fisher simplicity is graph-path based: the data must reach a basis atom and its perturbation must propagate through the downstream network. We encode these two conditions in an effective edge measure and, under local dictionary independence and effective-measure nondegeneracy, show that zero effective exposure exactly identifies Fisher-null directions within an edge. Controlled diagnostics confirm that zero coefficients can preserve rank while effective path disconnections remove the predicted directions. Coefficient magnitude alone is therefore not a Fisher-based pruning criterion for KANs.

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.

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.

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).

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.

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.

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.

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.

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.

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.

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.

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.

Parameter-efficient fine-tuning often selects trainable parameters before adaptation using architectural heuristics, without accounting for their varying importance during training. We introduce \textbf{FisherAdapTune}, which progressively selects parameter groups based on temporal drift in their Fisher information. Under a local Gaussian approximation, we bound the divergence between the fine-tuned posterior and pretrained prior by accumulated Fisher-weighted update costs, motivating curvature-aware selection. FisherAdapTune measures Jensen-Shannon distance between successive Fisher-value distributions and uses an adaptive threshold to freeze stabilized groups. Across VTAB-1k classification tasks, it achieves the highest macro Top-1 accuracy among the compared methods with a smaller average trainable set than full fine-tuning. Across four segmentation backbones, it maintains competitive in-distribution performance and improves zero-shot transfer in several settings. Its selections reveal architecture-dependent patterns where input, output, and normalization parameters can remain trainable as attention and MLP groups freeze at different rates. The results support Fisher structural drift as a task-dependent signal for allocating updates during adaptation. We release our \href{https://github.com/AtlasAnalyticsLab/FisherAdapTune}{code} publicly to enable further application of our proposed approach.

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/.

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.

We study the query complexity of obtaining a relative Fisher information guarantee for sampling from a log-smooth non-log-concave distribution; this is a sampling analog of finding an approximate stationary point in optimization. Our algorithm is based on the proximal sampler, which is an implicit discretization of the Langevin diffusion, and requires an implementation of the backward step known as the restricted Gaussian oracle (RGO). We show that by leveraging the recent results for log-concave sampling with high-accuracy guarantees in Rényi divergence, we can obtain an approximate RGO implementation that -- when used with the proximal sampler -- yields a complexity guarantee in relative Fisher information that inherits the same dimension dependence as log-concave sampling, and improves upon prior work for non-log-concave sampling. We also show a converse reduction that any improvement in the dimension dependence in relative Fisher information for non-log-concave sampling will yield an improved dimension dependence for high-accuracy log-concave sampling.

Evidential Deep Learning (EDL) enables single-pass uncertainty estimation by predicting Dirichlet evidence, but it can remain overconfident and poorly calibrated, and it often fails to represent multi-modal epistemic uncertainty. We introduce Gated Evidential Mixtures (GEM), a family of models that learns an in-model energy signal and uses it to gate evidential outputs end-to-end in a distance-informed manner. GEM-CORE learns a feature-level energy and maps it to a bounded gate that smoothly suppresses evidence when support is low. To capture epistemic multi-modality without multi-pass ensembling, GEM-MIX adds a lightweight mixture of evidential heads with learned routing weights while preserving single-pass inference. Finally, GEM-FI stabilizes mixture allocations via a Fisher-informed regularizer, reducing head collapse and producing smoother boundary uncertainty. Across image classification and OOD detection benchmarks, GEM improves calibration and ID/OOD separation with single-pass inference. On CIFAR-10, GEM-FI vs. DAEDL improves accuracy from 91.11 to 93.75 (+2.64 pp), reduces Brier x100 from 14.27 to 6.81 (-7.46), and also improves misclassification-detection AUPR from 99.08 to 99.94 (+0.86). For epistemic OOD detection, GEM-FI achieves AUPR/AUROC of 92.59/95.09 on CIFAR-10 to SVHN and 90.20/89.06 on CIFAR-10 to CIFAR-100, compared with 85.54/89.30 and 88.19/86.10 for DAEDL.

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.

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.

Industrial signal analysis is hindered by severe data heterogeneity, which we characterize as the M5 problem. Existing solutions rely on specialized models that lack robustness and scalability, while large-scale pre-training has rarely been investigated in this area. In this work, we derive a prioritized roadmap for the M5 problem and propose FISHER, a Foundation model for multi-modal Industrial Signal compreHEnsive Representation. To address the foremost multi-sampling-rate problem, FISHER utilizes a novel sub-band modeling approach that treats sampling rate increments as concatenated sub-band information, enabling the adaptive usage of full signal bandwidth without resampling. FISHER is pre-trained by teacher-student self-distillation over external audio and music data. We also establish the RMIS benchmark, comprising 19 datasets across four modalities. In the experiment, FISHER outperforms 24 state-of-the-art series encoders (up to 2B) with much smaller sizes (up to 16x), showcasing groundbreaking diagnostic accuracy and remarkable versatility. We further demonstrate that 1) seamless adaptation to variable sampling rates is the key to generalization 2) audio and music data provide better temporal variability, which is essential for pre-training. Both FISHER and RMIS are open-sourced.