FISHER: Multifaceted Framework in Stats & ML

• FISHER is a polysemous term that unifies methods in supervised subspace learning, estimation precision, forecasting, and reaction–diffusion dynamics via local sensitivity and separability.
• Deep learning formulations use Fisher criteria to enhance domain adaptation, auto-encoder robustness, and quantum metrology by optimizing class discriminability and latent representations.
• Fisher matrices operationalize parameter inference in areas like cosmological forecasting and quantum tomography, ensuring stability through robust numerical and design methods.

In current technical literature, Fisher is a polysemous term rather than a single formalism. It names a discriminant criterion for supervised subspace learning, an information measure governing estimation precision, a matrix approximation to parameter posteriors in forecasting, a consistency notion for statistical estimators, a reaction–diffusion equation class, and several recent machine-learning systems whose titles adopt FISHER as an acronym or framework name (Ghojogh et al., 2019, Tan et al., 2021, Yahia-Cherif et al., 2020, Xu et al., 2024, Fan et al., 22 Jul 2025). A plausible unifying interpretation is that these usages repeatedly operationalize local sensitivity or separability: between classes, between parameters and observables, between competing hypotheses, or between latent sessions of an experiment.

1. Fisher discriminant criteria and subspace methods

In classical Fisher Discriminant Analysis (FDA), the data are partitioned into classes $C_1,\dots,C_c$, with within-class scatter

$$S_W=\sum_{j=1}^c\sum_{x\in C_j}(x-\mu_j)(x-\mu_j)^T$$

and between-class scatter

$S_B=\sum_{j=1}^c n_j(\mu_j-\mu)(\mu_j-\mu)^T.$

The one-dimensional Fisher criterion is

$J(w)=\frac{w^T S_B w}{w^T S_W w},$

and maximizing it yields the generalized eigenproblem

$S_B w=\lambda S_W w.$

For a $p$-dimensional Fisher subspace, one solves $S_B U=S_W U\Lambda$ and projects by $U^T x$ (Ghojogh et al., 2019).

This formulation emphasizes simultaneous collapse of within-class variance and expansion of between-class variance. The tutorial literature also treats several structurally important extensions. When $d\gg n$ or $S_W$ is singular, robust FDA replaces small eigenvalues after an eigendecomposition $$S_W=\sum_{j=1}^c\sum_{x\in C_j}(x-\mu_j)(x-\mu_j)^T$$0, or equivalently regularizes with $$S_W=\sum_{j=1}^c\sum_{x\in C_j}(x-\mu_j)(x-\mu_j)^T$$1. Kernel FDA lifts the construction to a reproducing-kernel Hilbert space, where the discriminant lies in the span of training points and the generalized eigenproblem becomes $$S_W=\sum_{j=1}^c\sum_{x\in C_j}(x-\mu_j)(x-\mu_j)^T$$2 in kernel coordinates. The same literature proves that FDA and classical LDA are equivalent in the two-class Gaussian equal-covariance setting, introduces “Fisher forest” as an ensemble of Fisher subspaces over heterogeneous feature groups, and contrasts FDA with PCA by noting that PCA maximizes total scatter under orthonormality whereas FDA imposes a within-class constraint $$S_W=\sum_{j=1}^c\sum_{x\in C_j}(x-\mu_j)(x-\mu_j)^T$$3 (Ghojogh et al., 2019).

The practical meaning of the criterion is visible in the AT&T face-recognition example: with four subjects and six training images per subject, FDA yields a $$S_W=\sum_{j=1}^c\sum_{x\in C_j}(x-\mu_j)(x-\mu_j)^T$$4-dimensional Fisher subspace, while kernel FDA with linear, RBF, or cosine kernels further improves non-linear separability. The same example also illustrates a common misconception: FDA is not a reconstruction method analogous to PCA. Its reconstructions are poorer because separability, not reconstruction error, is the optimization target (Ghojogh et al., 2019).

2. Deep-learning formulations

In deep domain adaptation, Fisher ideas are used to make latent embeddings not merely domain-invariant but also class-discriminative. “Fisher Deep Domain Adaptation” identifies a standard failure mode of adaptation networks: features learned from a labeled source domain may be separable in the source domain yet still exhibit large variance and class overlap in the unlabeled target domain. The proposed remedy is a Fisher loss on learned embeddings $$S_W=\sum_{j=1}^c\sum_{x\in C_j}(x-\mu_j)(x-\mu_j)^T$$5, with trainable class centers $$S_W=\sum_{j=1}^c\sum_{x\in C_j}(x-\mu_j)(x-\mu_j)^T$$6, instantiated either as

$$S_W=\sum_{j=1}^c\sum_{x\in C_j}(x-\mu_j)(x-\mu_j)^T$$7

or

$$S_W=\sum_{j=1}^c\sum_{x\in C_j}(x-\mu_j)(x-\mu_j)^T$$8

The full objective is

$$S_W=\sum_{j=1}^c\sum_{x\in C_j}(x-\mu_j)(x-\mu_j)^T$$9

where $S_B=\sum_{j=1}^c n_j(\mu_j-\mu)(\mu_j-\mu)^T.$0 may be MMD, CORAL, or an adversarial domain-discriminator loss, and $S_B=\sum_{j=1}^c n_j(\mu_j-\mu)(\mu_j-\mu)^T.$1 encourages confident target predictions so that unlabeled target features collapse toward class centers. The implementation uses a ResNet-50 pretrained on ImageNet, a 256-unit bottleneck layer, a linear classification head, SGD with momentum $S_B=\sum_{j=1}^c n_j(\mu_j-\mu)(\mu_j-\mu)^T.$2, weight decay $S_B=\sum_{j=1}^c n_j(\mu_j-\mu)(\mu_j-\mu)^T.$3, mini-batch size $S_B=\sum_{j=1}^c n_j(\mu_j-\mu)(\mu_j-\mu)^T.$4, and early stopping after no improvement in $S_B=\sum_{j=1}^c n_j(\mu_j-\mu)(\mu_j-\mu)^T.$5 steps (Zhang et al., 2020).

The reported gains are explicit. On Office-31, mean accuracy rises from $S_B=\sum_{j=1}^c n_j(\mu_j-\mu)(\mu_j-\mu)^T.$6 for ADA alone to $S_B=\sum_{j=1}^c n_j(\mu_j-\mu)(\mu_j-\mu)^T.$7 with Fisher-TD and $S_B=\sum_{j=1}^c n_j(\mu_j-\mu)(\mu_j-\mu)^T.$8 with Fisher-TR. On Office-Home, ADA alone yields $S_B=\sum_{j=1}^c n_j(\mu_j-\mu)(\mu_j-\mu)^T.$9, ADA+Fisher-TD $J(w)=\frac{w^T S_B w}{w^T S_W w},$0, and ADA+Fisher-TR $J(w)=\frac{w^T S_B w}{w^T S_W w},$1; on the $J(w)=\frac{w^T S_B w}{w^T S_W w},$2 task the improvement reaches $J(w)=\frac{w^T S_B w}{w^T S_W w},$3, and the mean gain across all $J(w)=\frac{w^T S_B w}{w^T S_W w},$4 Office-Home tasks is $J(w)=\frac{w^T S_B w}{w^T S_W w},$5. The paper connects this to the Ben-David target-risk bound by arguing that a more discriminative representation lowers the joint optimum term $J(w)=\frac{w^T S_B w}{w^T S_W w},$6, thereby tightening the adaptation bound (Zhang et al., 2020).

A distinct deep-learning use appears in “Fisher Auto-Encoders.” There the relevant object is the Fisher divergence

$J(w)=\frac{w^T S_B w}{w^T S_W w},$7

which depends only on score fields and therefore can be applied to unnormalized models. The Fisher AE minimizes the Fisher divergence between the true joint $J(w)=\frac{w^T S_B w}{w^T S_W w},$8 and the model joint $J(w)=\frac{w^T S_B w}{w^T S_W w},$9. The resulting loss decomposes into a posterior-matching term, a Hyvärinen-score reconstruction term, and an encoder-stability term $S_B w=\lambda S_W w.$0, which acts analogously to a contractive penalty. This permits unnormalized priors such as the Factorable Polynomial Exponential family. On CelebA, the reported FID values are $S_B w=\lambda S_W w.$1 for a VAE, $S_B w=\lambda S_W w.$2 for Fisher AE with Gaussian prior, $S_B w=\lambda S_W w.$3 for Fisher AE with exponential prior, and $S_B w=\lambda S_W w.$4 for WAE-GAN; the emphasis of the paper, however, is robustness under input noise rather than adversarial sample fidelity (Elkhalil et al., 2020).

3. Fisher information as an operational metric

In estimation theory, the classical Fisher information for a parameterized distribution $S_B w=\lambda S_W w.$5 is

$S_B w=\lambda S_W w.$6

For quantum statistical models, the classical Fisher information of a measurement is optimized over POVMs to define the quantum Fisher information (QFI), and for unitary encoding $S_B w=\lambda S_W w.$7 one obtains

$S_B w=\lambda S_W w.$8

In convex quantum resource theories, these quantities become universal witnesses: for any resourceful state $S_B w=\lambda S_W w.$9, there exists a parameter-encoding channel and measurement such that the Fisher information exceeds what any free state can achieve. The same work bounds the metrological advantage in terms of generalized robustness and extends the logic from states to quantum channels (Tan et al., 2021).

The same operational viewpoint appears in structured optics. If $p$0 is a normalized transverse intensity distribution, then

$p$1

For displacement estimation, this reduces to gradient energy of the beam profile; with $p$2, one may write $p$3. The paper derives closed forms for several beam families: $p$4 for Hermite–Gaussian modes, $p$5 for Laguerre–Gaussian modes, and approximately $p$6 for finite-energy Bessel–Gauss beams. A central result is that modes with nearly equal Shannon entropy can have markedly different Fisher information, because nodal structure and local curvature dominate displacement sensitivity (Sumaya-Martinez et al., 29 Dec 2025).

A third operational use is in locally optimum processing of weak signals in additive white noise. For a standardized even noise density $p$7, Fisher information sets upper bounds on three asymptotic performance metrics: maximum SNR gain $p$8, asymptotic relative efficiency $p$9, and cross-correlation gain $S_B U=S_W U\Lambda$0. Equality holds for the locally optimum processor $S_B U=S_W U\Lambda$1. The minimal Fisher information is $S_B U=S_W U\Lambda$2, attained by the Gaussian density, whereas any non-Gaussian even density has strictly larger Fisher information; in the limiting dichotomous-noise case the Fisher information diverges, and the corresponding nonlinearity removes the noise exactly for known weak signals (Duan et al., 2011).

4. Quantum Fisher structures and quenched certification

Quantum tomography and quantum metrology introduce further specialized Fisher constructions. A measurement on $S_B U=S_W U\Lambda$3 copies is Fisher symmetric at a state $S_B U=S_W U\Lambda$4 when its classical Fisher matrix is proportional to the QFI and saturates the Gill–Massar bound. For pure states, two-copy collective measurements built from complex-projective $S_B U=S_W U\Lambda$5-designs satisfy $S_B U=S_W U\Lambda$6, hence are universally Fisher symmetric for all pure states. In the qubit case, the minimal tight coherent measurement is a five-element POVM consisting of four SIC-derived symmetric outcomes together with the singlet projector, and it satisfies $S_B U=S_W U\Lambda$7 for all qubit states (Zhu et al., 2017).

“Fisher Glasses” redefines what it means to certify metrological performance in quenched environments, where slow latent variables remain frozen within a session but vary between sessions. The paper’s pipeline is explicit: condition on the latent session $S_B U=S_W U\Lambda$8, project nuisance directions by a Schur complement $S_B U=S_W U\Lambda$9, invert to obtain the session loss $U^T x$0, and tail-certify that loss via conditional value at risk,

$U^T x$1

A no-go theorem states that no averaged Fisher data can determine this certificate: ensembles may share the same averaged Fisher matrix and still have $U^T x$2 in one case and $U^T x$3 in another. Collapse is controlled by a Fisher-zero integrability transition: if the inverse-loss tail exponent satisfies $U^T x$4, then $U^T x$5 and the certified information vanishes. The paper formulates design laws—safe windows, nondegenerate portfolios, Fisher reserves, action separation, and Fisher cuts—and demonstrates them on a shallow-NV Ramsey case study. Under the same shot budget and latent ensemble, safe-window design recovers nearly three orders of magnitude in certified information relative to average-QFI optimization (Mansouri et al., 1 Jul 2026).

These quantum results clarify a frequent confusion between large average Fisher response and guaranteed operational precision. The former can coexist with catastrophic rare-session losses, whereas the latter requires explicit control of nuisance-projected inverse-Fisher tails (Mansouri et al., 1 Jul 2026).

5. Fisher matrices in forecasting and inference

A Fisher matrix is defined generally by

$U^T x$6

and, under a Gaussian-likelihood approximation, its inverse is used as an approximate parameter covariance matrix. In cosmological forecasting this formalism is central for survey design. For stage IV spectroscopic surveys, the Fisher matrix integrates derivatives of the observed redshift-space power spectrum over $U^T x$7, $U^T x$8, and $U^T x$9, weighted by the effective survey volume. The inverse Fisher yields the covariance, and the DETF figure of merit for $d\gg n$0 is $d\gg n$1 (Yahia-Cherif et al., 2020).

The technical difficulty is numerical stability. “Validating the Fisher approach for stage IV spectroscopic surveys” argues that the condition-number bound $d\gg n$2 is often too stringent in practice for large Fisher matrices, and proposes a direct perturbation test: multiply each Fisher entry by $d\gg n$3, invert the perturbed matrix, compute the perturbed FoM, and determine the largest $d\gg n$4 that keeps $d\gg n$5 within a prescribed tolerance for at least $d\gg n$6 of random realizations. The same study recommends a 5-point centered stencil for numerical derivatives and reports that, with optimal step sizes, Fisher uncertainties agree with MCMC to $d\gg n$7 in an $d\gg n$8 cosmological parameterization and $d\gg n$9 in a $S_W$0 one (Yahia-Cherif et al., 2020).

A complementary note on weak-lensing forecasts emphasizes derivative tuning, cross-code validation, and post-inversion diagnostics. It recommends scanning finite-difference steps to locate a plateau in a forecast figure of merit, checking $S_W$1, using the Newman norm-bound test, rescaling parameters such as $S_W$2, and reducing dimensionality when $S_W$3 or larger. The note exhibits how $S_W$4 entrywise differences in $S_W$5 can become order-unity differences in $S_W$6 under poor conditioning (Bhandari et al., 2021).

The general review literature broadens the formalism in several directions. If both $S_W$7 and $S_W$8 measurements are uncertain, marginalization over latent variables yields an effective covariance $S_W$9 and a standard Gaussian Fisher expression with $$S_W=\sum_{j=1}^c\sum_{x\in C_j}(x-\mu_j)(x-\mu_j)^T$$00. If some parameters are fixed incorrectly, the resulting bias in retained parameters is given by a Fisher block formula involving $$S_W=\sum_{j=1}^c\sum_{x\in C_j}(x-\mu_j)(x-\mu_j)^T$$01. DALI extends the log-likelihood beyond quadratic order with third- and fourth-derivative tensors, and Laplace-approximated evidences provide Bayes-factor forecasts for model selection (Heavens, 2016). Public implementations such as the Python package FARO expose these ideas for galaxy clustering, weak lensing, and cross-correlation forecasts while parameterizing the observables through $$S_W=\sum_{j=1}^c\sum_{x\in C_j}(x-\mu_j)(x-\mu_j)^T$$02, $$S_W=\sum_{j=1}^c\sum_{x\in C_j}(x-\mu_j)(x-\mu_j)^T$$03, $$S_W=\sum_{j=1}^c\sum_{x\in C_j}(x-\mu_j)(x-\mu_j)^T$$04, $$S_W=\sum_{j=1}^c\sum_{x\in C_j}(x-\mu_j)(x-\mu_j)^T$$05, and $$S_W=\sum_{j=1}^c\sum_{x\in C_j}(x-\mu_j)(x-\mu_j)^T$$06 (Resco et al., 2020).

6. Additional statistical and dynamical constructions

In statistical quantification under dataset shift, Fisher consistency means population-level unbiasedness: an estimator $$S_W=\sum_{j=1}^c\sum_{x\in C_j}(x-\mu_j)(x-\mu_j)^T$$07 of test-set class prevalence is Fisher consistent on a family $$S_W=\sum_{j=1}^c\sum_{x\in C_j}(x-\mu_j)(x-\mu_j)^T$$08 when $$S_W=\sum_{j=1}^c\sum_{x\in C_j}(x-\mu_j)(x-\mu_j)^T$$09 for every $$S_W=\sum_{j=1}^c\sum_{x\in C_j}(x-\mu_j)(x-\mu_j)^T$$10. Under prior probability shift, Adjusted Classify & Count and the EM algorithm satisfy this criterion, while CDE-Iterate does not. The EM argument is stronger: its population maximum-likelihood equation remains Fisher consistent under the more general invariant density-ratio shift. The paper’s binormal counterexample shows that CDE-Iterate can converge to $$S_W=\sum_{j=1}^c\sum_{x\in C_j}(x-\mu_j)(x-\mu_j)^T$$11 even under pure prior shift (Tasche, 2017).

In high-dimensional covariance testing, Fisher’s classical p-value combination becomes a scale-invariant power-enhancement device. For the two-sample null $$S_W=\sum_{j=1}^c\sum_{x\in C_j}(x-\mu_j)(x-\mu_j)^T$$12, one computes a quadratic-form statistic $$S_W=\sum_{j=1}^c\sum_{x\in C_j}(x-\mu_j)(x-\mu_j)^T$$13 and a maximum-form statistic $$S_W=\sum_{j=1}^c\sum_{x\in C_j}(x-\mu_j)(x-\mu_j)^T$$14, converts them into $$S_W=\sum_{j=1}^c\sum_{x\in C_j}(x-\mu_j)(x-\mu_j)^T$$15 and $$S_W=\sum_{j=1}^c\sum_{x\in C_j}(x-\mu_j)(x-\mu_j)^T$$16, and combines them by

$$S_W=\sum_{j=1}^c\sum_{x\in C_j}(x-\mu_j)(x-\mu_j)^T$$17

The key theoretical result is the asymptotic independence of $$S_W=\sum_{j=1}^c\sum_{x\in C_j}(x-\mu_j)(x-\mu_j)^T$$18 and $$S_W=\sum_{j=1}^c\sum_{x\in C_j}(x-\mu_j)(x-\mu_j)^T$$19, which yields $$S_W=\sum_{j=1}^c\sum_{x\in C_j}(x-\mu_j)(x-\mu_j)^T$$20 under the null. The combined test inherits power against both dense alternatives, where quadratic-form methods excel, and sparse alternatives, where maximum-form methods excel (Yu et al., 2020).

The term also appears in mathematical physics through the Fisher equation. In open-string field theory with a linear dilaton background, the tachyon dynamics reduce to a non-local, delayed Fisher-type reaction–diffusion equation with diffusion term $$S_W=\sum_{j=1}^c\sum_{x\in C_j}(x-\mu_j)(x-\mu_j)^T$$21, linear reaction term $$S_W=\sum_{j=1}^c\sum_{x\in C_j}(x-\mu_j)(x-\mu_j)^T$$22, Gaussian nonlocal smoothing $$S_W=\sum_{j=1}^c\sum_{x\in C_j}(x-\mu_j)(x-\mu_j)^T$$23, and time delay $$S_W=\sum_{j=1}^c\sum_{x\in C_j}(x-\mu_j)(x-\mu_j)^T$$24. Linearization around the unstable vacuum gives a travelling-wave dispersion relation $$S_W=\sum_{j=1}^c\sum_{x\in C_j}(x-\mu_j)(x-\mu_j)^T$$25, with selected minimal speed $$S_W=\sum_{j=1}^c\sum_{x\in C_j}(x-\mu_j)(x-\mu_j)^T$$26 (Ghoshal, 2011).

Two further applications treat Fisher information as a structural variational principle. In finance, extremizing

$$S_W=\sum_{j=1}^c\sum_{x\in C_j}(x-\mu_j)(x-\mu_j)^T$$27

under normalization and moment constraints yields a Schrödinger equation $$S_W=\sum_{j=1}^c\sum_{x\in C_j}(x-\mu_j)(x-\mu_j)^T$$28. Depending on the imposed constraints, this maps financial return densities onto harmonic-oscillator, anharmonic-oscillator, finite-well, or $$S_W=\sum_{j=1}^c\sum_{x\in C_j}(x-\mu_j)(x-\mu_j)^T$$29-well models, producing Gaussian or Laplace-type densities as special cases (Nastasiuk, 2015). In Big Data time-series analysis, Fisher information is computed over windowed state occupancies by the discrete approximation

$$S_W=\sum_{j=1}^c\sum_{x\in C_j}(x-\mu_j)(x-\mu_j)^T$$30

Applied to U.S. GDP per capita and total population from 1960 to 2013, the resulting Fisher-information time series remains nearly constant from 1967 to 2013, which the paper interprets as a stable, orderly dynamic regime (Ahmad et al., 2015).

7. Systems explicitly named FISHER

Several recent machine-learning papers use FISHER as the proper name of a specific framework rather than as a generic Fisher-theoretic object.

System	Domain	Core formulation
FISHER	Multi-AUV underwater target tracking	Two-stage learning from demonstrations: MADAC, then MAIGDT
FISHER	Industrial signal representation	Teacher-student SSL on STFT sub-bands

In multi-AUV underwater target tracking, FISHER is a two-stage learning-from-demonstrations framework. Stage I uses a Multi-Agent Discriminator-Actor-Critic (MADAC), extending discriminator-actor-critic imitation learning to the multi-agent setting through a centralized discriminator and SAC-style updates. Stage II uses a Multi-Agent Independent Generalized Decision Transformer (MAIGDT), which conditions on latent representations of future high-quality states rather than on a hand-designed reward. The same work proposes a simulation-to-simulation demonstration-generation pipeline that combines simplified point-mass planning with point-to-point RL in a realistic underwater simulator. Across sparse- and dense-obstacle scenarios, varying $$S_W=\sum_{j=1}^c\sum_{x\in C_j}(x-\mu_j)(x-\mu_j)^T$$31 AUVs and different target motions, the reported findings are that MADAC converges stably where MAIDAC and GAIL+PPO do not, MAIGDT outperforms CQL in stability and multi-task behavior, and the full FISHER pipeline attains near-optimal min-distance to target, obstacle margin around $$S_W=\sum_{j=1}^c\sum_{x\in C_j}(x-\mu_j)(x-\mu_j)^T$$32, zero danger time, and expert-level connectivity in hard settings (Xu et al., 2024).

In industrial health management, “FISHER: A Foundation Model for Multi-Modal Industrial Signal Comprehensive Representation” addresses what the paper calls the M$$S_W=\sum_{j=1}^c\sum_{x\in C_j}(x-\mu_j)(x-\mu_j)^T$$33 heterogeneity problem: multi-modal, multi-sampling-rate, multi-scale, multi-task, and mini-fault data. The model treats STFT sub-bands as the basic modeling unit so that higher sampling rates correspond to concatenated extra sub-band information. Pre-training uses a teacher-student self-distillation framework with a ViT-style student encoder, a shallow CNN decoder, and an EMA teacher. The total loss is the sum of an $$S_W=\sum_{j=1}^c\sum_{x\in C_j}(x-\mu_j)(x-\mu_j)^T$$34 band-level and patch-level distillation term. FISHER is released in tiny ($$S_W=\sum_{j=1}^c\sum_{x\in C_j}(x-\mu_j)(x-\mu_j)^T$$35M), mini ($$S_W=\sum_{j=1}^c\sum_{x\in C_j}(x-\mu_j)(x-\mu_j)^T$$36M), and small ($$S_W=\sum_{j=1}^c\sum_{x\in C_j}(x-\mu_j)(x-\mu_j)^T$$37M) versions and is evaluated on the RMIS benchmark. The reported overall RMIS score for FISHER-small is $$S_W=\sum_{j=1}^c\sum_{x\in C_j}(x-\mu_j)(x-\mu_j)^T$$38, exceeding the best prior SSL baseline by $$S_W=\sum_{j=1}^c\sum_{x\in C_j}(x-\mu_j)(x-\mu_j)^T$$39 percentage points; on fault diagnosis, FISHER-tiny already outperforms all baselines by about $$S_W=\sum_{j=1}^c\sum_{x\in C_j}(x-\mu_j)(x-\mu_j)^T$$40 points, while on anomaly detection the model is competitive and second to BEATs by about $$S_W=\sum_{j=1}^c\sum_{x\in C_j}(x-\mu_j)(x-\mu_j)^T$$41 points (Fan et al., 22 Jul 2025).

Taken together, these systems illustrate a final sense in which FISHER functions in current literature: not only as a mathematical criterion or information measure, but also as a framework label for architectures that emphasize representation quality, transfer across modalities or tasks, and robustness under limited or indirect supervision (Xu et al., 2024, Fan et al., 22 Jul 2025).