Entropy

20 pages, 409 KB

Open AccessArticle

Transformation Equivalence of Neural Networks

by Masaki Kobayashi

Entropy 2026, 28(9), 946; https://doi.org/10.3390/e28090946 (registering DOI) - 23 Aug 2026

Multilayer perceptrons (MLPs) are considered as a singular model of learning machines. Singularities cause local minima and plateaus in the learning process. I/O-equivalence, where two different MLPs are regarded as the same multivariable function, is an important concept for understanding singularities in neural [...] Read more.

Multilayer perceptrons (MLPs) are considered as a singular model of learning machines. Singularities cause local minima and plateaus in the learning process. I/O-equivalence, where two different MLPs are regarded as the same multivariable function, is an important concept for understanding singularities in neural networks. In this paper, I/O-equivalence is extended to T-equivalence, which is a concept where two MLPs yield the same results through a transformation of input and output. We provide constructive families and procedures for obtaining T-equivalent networks of real-, complex-, and quaternion-valued neural networks. In particular, T-equivalence of quaternion-valued neural networks is much more complicated than that of the others. Full article

29 pages, 2600 KB

Open AccessArticle

Fault-Tolerant Private Information Retrieval via Threshold Distributed Point Functions

by Dazeng Yuan, Xiheng Liu and Bin Liu

Entropy 2026, 28(9), 945; https://doi.org/10.3390/e28090945 (registering DOI) - 23 Aug 2026

Abstract

Multi-server private information retrieval (PIR) based on function secret sharing (FSS) has emerged as a prominent paradigm for achieving sublinear communication. However, standard FSS constructions require full server participation, making them highly vulnerable to single-node fail-stop faults. Existing fault-tolerant schemes mitigate this but [...] Read more.

Multi-server private information retrieval (PIR) based on function secret sharing (FSS) has emerged as a prominent paradigm for achieving sublinear communication. However, standard FSS constructions require full server participation, making them highly vulnerable to single-node fail-stop faults. Existing fault-tolerant schemes mitigate this but inevitably inflate the response overhead to scale with the database size N (e.g.,

O (\sqrt{N})

). To overcome this limitation, we propose a fault-tolerant PIR (FT-PIR) protocol based on a newly designed

(t, p)

-threshold distributed point function (FT-DPF). By introducing a hierarchical recursive patching mechanism, our scheme transforms rigid all-party evaluations into flexible t-out-of-p reconstructions. This architecture completely decouples the response communication from N and ensures efficient client-side reconstruction via lightweight XOR aggregations. Formal analysis proves that our stateless protocol guarantees

(t - 1)

-computational privacy under the semi-honest model. Theoretical analysis demonstrates that the proposed FT-PIR achieves a response complexity bounded by

O (F_{maxlevel} (t, p))

. Comprehensive experimental evaluations confirm that our implementation significantly reduces practical communication and computation overheads, outperforming the state-of-the-art scheme. Full article

(This article belongs to the Special Issue Private Information Retrieval and Its Applications)

22 pages, 3110 KB

Open AccessArticle

TriAIF-RWKV: A Physiology-Guided Spatiotemporal Framework for Robust Arterial Input Function Selection in CT Perfusion Imaging

by Lei Lei, Yu Shen, Dawei Wang, Feng Xi, Yixin He, Chaochao Wang and Jiandong Liu

Entropy 2026, 28(9), 944; https://doi.org/10.3390/e28090944 (registering DOI) - 22 Aug 2026

Abstract

Accurate delineation of infarct core and ischemic penumbra in acute ischemic stroke primarily relies on computed tomography perfusion (CTP), where the arterial input function (AIF) is essential for reliable perfusion quantification. However, reliable and fast AIF selection remains challenging in clinical practice due [...] Read more.

Accurate delineation of infarct core and ischemic penumbra in acute ischemic stroke primarily relies on computed tomography perfusion (CTP), where the arterial input function (AIF) is essential for reliable perfusion quantification. However, reliable and fast AIF selection remains challenging in clinical practice due to noise, vascular heterogeneity, and inter-patient variability in bolus dynamics. In this study, we propose TriAIF-RWKV, a three-stage framework for robust and automated AIF extraction. Specifically, ACSANet is first employed for spatial vascular localization using axial and channel-aware attention mechanisms, thereby narrowing the candidate arterial region and reducing the AIF search space. Then, a Dilated-RWKV network is introduced to model temporal intensity dynamics from a global sequence perspective, allowing robust identification of AIF-consistent patterns. Finally, a physiology-informed scoring strategy is used to select the optimal AIF by evaluating baseline stability, peak enhancement, and washout characteristics. Extensive experiments on CTP datasets were conducted from multiple perspectives, including AIF waveform fidelity, perfusion parameter estimation, and lesion-level analysis. The results demonstrate that the proposed method achieved high agreement with expert-selected AIFs, with a global waveform PCC of 0.973, peak correlation of 0.942, and TTP correlation of 0.973 with a mean error of 0.923 s. Furthermore, the proposed method provides more consistent downstream perfusion quantification, achieving higher consistency of CTP-derived parameters and improved lesion-to-normal tissue discrimination compared with existing approaches. These results highlight its potential for reliable clinical perfusion assessment. Full article

(This article belongs to the Special Issue Entropy in Image, Video and Signal Processing)

27 pages, 473 KB

Open AccessArticle

A Ranked Sparsity Extension to the Bayesian Information Criterion: A Tool for Selecting Variables from Multiple Data Modalities

by Ryan A. Peterson, Sarah M. Bird, Logan M. Harris, Patrick J. Breheny and Joseph E. Cavanaugh

Entropy 2026, 28(9), 943; https://doi.org/10.3390/e28090943 (registering DOI) - 22 Aug 2026

Abstract

The concept of ranked sparsity, originally introduced in the context of penalized regression, arises in modeling applications when an expected disparity exists in the quality of information between different feature sets. Its presence can cause traditional and modern model selection methods to fail [...] Read more.

The concept of ranked sparsity, originally introduced in the context of penalized regression, arises in modeling applications when an expected disparity exists in the quality of information between different feature sets. Its presence can cause traditional and modern model selection methods to fail because such procedures commonly presume “covariate equipoise”—that each potential parameter is equally worthy of entering into the final model. However, this presumption does not always hold, especially in the presence of derived variables or with highly disparate feature sets (i.e., multi-modal data). For instance, when all possible interactions are considered as candidate predictors, the sheer number of them grossly inflates the number of false discoveries, resulting in unnecessarily complex and difficult-to-interpret models with many (truly spurious) interactions. In this work, we motivate a ranked sparsity extension to the Bayesian Information Criterion (RBIC) that requires a stronger level of evidence in order to allow certain variables (e.g., interactions vs main effects and genetic vs clinical covariates) into a model. We compare the performance of RBIC relative to competing methods for selecting polynomials and interactions in a simulation study and in two applications, showing that stepwise selection guided by RBIC produces better-predicting, more transparent models (with fewer false interactions) compared to existing alternatives. Full article

(This article belongs to the Special Issue Statistical Planning, Inference, and Decision Making in High-Dimensional Data Analysis)

28 pages, 31709 KB

Open AccessArticle

An Exploratory Statistical Modeling Framework for National Rule-of-Law Profiles

by Sadullah Çelik, Muhammet Ali Köroğlu and Cemile Zehra Köroğlu

Entropy 2026, 28(9), 942; https://doi.org/10.3390/e28090942 (registering DOI) - 22 Aug 2026

Abstract

The rule of law can be considered as a multidimensional institutional phenomenon, which emerges through interplay between legal, governance and administrative institutions. The paper offers an exploratory statistical modeling approach to find empirical patterns in national rule-of-law profiles according to the 2024 World [...] Read more.

The rule of law can be considered as a multidimensional institutional phenomenon, which emerges through interplay between legal, governance and administrative institutions. The paper offers an exploratory statistical modeling approach to find empirical patterns in national rule-of-law profiles according to the 2024 World Justice Project (WJP) Rule of Law Index. Eight dimensions of the index are considered to identify differences between countries and similarities of their multidimensional institutional performance. Principal Component Analysis reveals strong associations between eight dimensions, which are structured along the same performance institutional scale; the first principal component explains 85.7% of the overall variation and two principal components explain 92.5% of it. K-Means, hierarchical and DBSCAN clustering methods are then used to examine the empirical similarities between countries. While the six-cluster solution of K-Means offers distinct group descriptions, low bootstrap stability of this solution suggests that these groups cannot be regarded as fixed rule-of-law regimes. In addition, the Random Forest analysis reveals Regulatory Enforcement, Absence of Corruption, and Criminal Justice as the three dimensions, which contribute to the empirical differentiation of the described profiles the most. In general, the results imply that international variations in rule-of-law performance are viewed as heterogeneous locations in a multidimensional institution space, rather than as stable and distinct legal systems. The above-presented methodology allows for an exploratory approach to analyze international variations in rule-of-law performance that considers the limitations of cross-section data and instability of clusters. Full article

(This article belongs to the Special Issue Statistical Approaches for Modeling Human Social Systems)

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18 pages, 849 KB

Open AccessArticle

Distributed Kalman Filter with Maximum Correlation Entropy Criterion and Consensus Weighted Term Fusion

by Xiaoliang Feng, Zhouliner Gao and Teng Liu

Entropy 2026, 28(8), 941; https://doi.org/10.3390/e28080941 - 21 Aug 2026

Abstract

Distributed maximum correntropy Kalman filters improve robustness to non-Gaussian noise, but existing variants generally introduce consensus through average or weighted fusion without explicitly separating the innovation residual from the state-disagreement residual in a dimensionally consistent objective. This paper proposes a Distributed Maximum Correntropy [...] Read more.

Distributed maximum correntropy Kalman filters improve robustness to non-Gaussian noise, but existing variants generally introduce consensus through average or weighted fusion without explicitly separating the innovation residual from the state-disagreement residual in a dimensionally consistent objective. This paper proposes a Distributed Maximum Correntropy Kalman Filter with Innovation and Consensus Weighting Terms (DMCKF-IW-CWT). The innovation and consensus residuals are normalized separately and mapped by Gaussian kernels, after which the resulting information matrices are incorporated into a fixed-point local update. Posterior covariance intersection (CI) is then used to fuse neighboring estimates without requiring the unavailable cross-covariances. A sufficient contraction condition is given for the fixed-point iteration. In a five-node benchmark with 500 independent Monte Carlo runs and 1000 sampling steps, the proposed method obtains overall, transient, and steady-state MAEs of 0.172210, 0.188271, and 0.168195, respectively, corresponding to reductions of 0.254%, 0.526%, and 0.178% relative to DMCKF-W; the paired 95% confidence intervals of all three differences remain below zero. The consensus RMS is further reduced by 5.371%. Additional tests involving five noise families, packet loss and communication noise, a four-state nonlinear model, and systems with up to eight states and twenty nodes confirm the numerical convergence and extensibility of the framework. Full article

(This article belongs to the Section Information Theory, Probability and Statistics)

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65 pages, 721 KB

Open AccessArticle

Pseudo-Additive Tsallis Entropy and Non-Factorizing Joint Statistics in Product Sheffer Stroke Basic Algebras

by Ibrahim Senturk, Metin Bilge and Tahsin Oner

Entropy 2026, 28(8), 940; https://doi.org/10.3390/e28080940 - 21 Aug 2026

Abstract

This paper addresses the problem of formulating generalized, non-extensive information-theoretic measures on finite non-distributive algebraic structures equipped with Riečan states, with particular emphasis on product Sheffer stroke basic algebras. Our approach formalizes finite summations, admissible partitions, refinement relations, and Sheffer stroke joint refinement [...] Read more.

This paper addresses the problem of formulating generalized, non-extensive information-theoretic measures on finite non-distributive algebraic structures equipped with Riečan states, with particular emphasis on product Sheffer stroke basic algebras. Our approach formalizes finite summations, admissible partitions, refinement relations, and Sheffer stroke joint refinement candidates by using the primitive Sheffer stroke operation, with partition and marginalization properties imposed under the stated product and admissibility assumptions. By leveraging the state-theoretic properties of Riečan states, we construct baseline Shannon and logical entropies alongside algorithmic procedures for their computational evaluation. As the main result, we introduce and analytically characterize a parametric Tsallis entropy functional over these basic algebras. We prove its fundamental properties, including bounding inequalities, state concavity, monotonicity under refinement, subadditivity (for

α > 1

), conditional chain-type identities under the relevant joint refinement marginalization assumptions, and exact analytical convergence to the classical Shannon limit as the entropic index

α \to 1

. Furthermore, under a state-dependent statistical independence condition, we show that the joint Tsallis entropy satisfies a pseudo-additive relation. By defining the Tsallis mutual information and the associated pseudo-additive residual, we isolate the deviation of a joint Sheffer stroke refinement from the factorized model determined by its marginal Riečan-state distributions. This residual is intended as a state-dependent algebraic indicator of deviations from the factorized Tsallis pseudo-additive model; it is not claimed to be an operational contextuality witness, a contextuality inequality, an entanglement measure, or a physical implementation criterion. Full article

(This article belongs to the Special Issue Uncertainty and Fuzziness: Analysis and Applications)

38 pages, 1394 KB

Open AccessArticle

A Correlation-Decoupled Interval Belief Rule Base for Interpretable Cross-Condition Bearing Fault Diagnosis

by Xingchi Yan, Yan Yu and Ning Li

Entropy 2026, 28(8), 939; https://doi.org/10.3390/e28080939 - 21 Aug 2026

Abstract

Cross-condition bearing fault diagnosis requires models that remain reliable under load-induced distribution shifts while providing transparent and traceable reasoning. Conventional belief rule bases (BRBs) may repeatedly use correlated vibration evidence during inference, and their Cartesian-product rule construction can rapidly increase rule-base complexity. This [...] Read more.

Cross-condition bearing fault diagnosis requires models that remain reliable under load-induced distribution shifts while providing transparent and traceable reasoning. Conventional belief rule bases (BRBs) may repeatedly use correlated vibration evidence during inference, and their Cartesian-product rule construction can rapidly increase rule-base complexity. This study proposes a correlation-decoupled interval belief rule base (CD-IBRB) for cross-condition bearing fault diagnosis. Seven diagnostically relevant time-domain features are selected using XGBoost and transformed into a less-correlated feature space through a Kendall-rank-correlation-guided matrix estimated exclusively from the source training data. Attribute-wise referential points and intervals are then constructed from the transformed training attributes, allowing the rule base to grow additively rather than combinatorially. Initial belief distributions are obtained from interval-level class distributions. The projection covariance matrix adaptation evolution strategy (P-CMA-ES) jointly optimizes the belief degrees, rule reliabilities, and rule weights, while evidential reasoning aggregates the activated interval rules to produce the final diagnostic result. In the primary cross-load bearing experiment, CD-IBRB achieved an accuracy of 0.9702 and a macro-averaged F1 score of 0.9703. It outperformed the strongest BRB variant and data-driven baseline by 7.70 and 6.10 percentage points in accuracy, respectively. Ablation experiments confirmed that removing parameter optimization or attribute decoupling reduced accuracy to 0.9053 and 0.9303, respectively. Additional cross-load and noise-injection experiments further demonstrated the stability of CD-IBRB under load shifts and input disturbances. Across five public multiclass datasets, CD-IBRB achieved a mean accuracy of 0.9004 and consistently outperformed the compared BRB variants. These results demonstrate that CD-IBRB provides a compact, uncertainty-aware, and traceable framework for cross-condition bearing fault diagnosis. Full article

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21 pages, 6389 KB

Open AccessArticle

Fixed-Candidate Reliability Auditing for Closed-Set Binary Function Retrieval Under Known-Source Cross-Compilation Protocols

by Yiming An, Yanshu Yu, Weidong Li and Orest Kochan

Entropy 2026, 28(8), 938; https://doi.org/10.3390/e28080938 - 21 Aug 2026

Abstract

Binary code similarity detection (BCSD) ranks candidates but does not quantify the reliability of an already selected Top-1 match. We study this post-retrieval problem in a known-source, closed-set protocol: the target Top-1 is frozen before same-source cross-compilation views are queried, so auxiliary evidence [...] Read more.

Binary code similarity detection (BCSD) ranks candidates but does not quantify the reliability of an already selected Top-1 match. We study this post-retrieval problem in a known-source, closed-set protocol: the target Top-1 is frozen before same-source cross-compilation views are queried, so auxiliary evidence audits cannot replace it. A frozen 34-variable map feeds a low-capacity logistic model with project-grouped cross-fitting, Platt calibration, and training-side threshold selection. On 413 families from 16 projects, cross-view evidence improved discrimination over target score/margin features. GCC-O0 was a dominant-anchor regime: Full showed no statistically resolved ROC-AUC gain over Primary-anchor, whereas Clang-O0 benefited from complementary non-primary evidence. On 240 project-identity-disjoint families from 55 projects, the design-locked structural branch accepted 75/240 GCC and 99/240 Clang candidates (31.3%/41.3% coverage) with no observed family-level errors. Correspondence mismatch reduced discrimination toward chance. Corrected TF-IDF remained supportive because correction followed label access. The contribution of this paper is a versioned candidate-preserving audit interface with explicit evidence and deployment boundaries, but not a universal retrieval improvement or distribution-free guarantee. Full article

(This article belongs to the Section Information Theory, Probability and Statistics)

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26 pages, 4577 KB

Open AccessArticle

Structural Evolution and Cascading Propagation of Supply Risk in the Global Nickel Industry Chain: A Multilayer Network Approach

by Yi Liang, Xiaoduo Wang, Han Liu and Hao Wang

Entropy 2026, 28(8), 937; https://doi.org/10.3390/e28080937 - 21 Aug 2026

Abstract

Geopolitical conflicts, resource-protection policies, and unexpected disruptions have heightened supply-security concerns across the global nickel industry chain. This study constructs a multilayer trade network based on complex network theory to characterize structural evolution across the upstream, midstream, and downstream segments and applies a [...] Read more.

Geopolitical conflicts, resource-protection policies, and unexpected disruptions have heightened supply-security concerns across the global nickel industry chain. This study constructs a multilayer trade network based on complex network theory to characterize structural evolution across the upstream, midstream, and downstream segments and applies a cascading-failure model to simulate the propagation of supply risks. There are four main findings: (1) The global nickel trade network exhibits pronounced layer heterogeneity, with the midstream layer acting as the principal amplifier of cascading failure risks. (2) Nodes with high centrality and broad cross-layer participation largely coincide with the countries that generate the largest systemic risks. (3) A small group of countries controls most trade flows and dominates risk transmission. (4) The center of systemic risk is shifting from traditional industrial and trading economies toward resource suppliers and countries that integrate resource extraction with processing. These findings support a risk-governance strategy based on diversified supply sources, dynamic monitoring of critical nodes, improved resilience in midstream smelting and refining, strategic resource stockpiling, and the circular utilization of nickel resources. Full article

(This article belongs to the Special Issue Analysis of Critical Behavior in Complex Systems)

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14 pages, 901 KB

Open AccessArticle

Directed Interband Response at Null Biorthogonal Quantum-Geometric Components

by Xinyi Xie, Jia-Ning Zhu and Bo Wan

Entropy 2026, 28(8), 936; https://doi.org/10.3390/e28080936 - 21 Aug 2026

Abstract

Biorthogonal quantum geometry is often read through scalar tensor components. In non-Hermitian bands, however, the biorthogonal contraction can lose the ordering of the left-right interband matrix elements from which a scalar component is formed. We study this information loss for spectrally separated, diagonalizable [...] Read more.

Biorthogonal quantum geometry is often read through scalar tensor components. In non-Hermitian bands, however, the biorthogonal contraction can lose the ordering of the left-right interband matrix elements from which a scalar component is formed. We study this information loss for spectrally separated, diagonalizable two-band Bloch Hamiltonians. For a specified control parameter, the Hamiltonian variation defines a local response vertex. In the instantaneous biorthogonal eigenbasis, the interband part of this vertex is completely specified by two ordered matrix elements, whereas the corresponding equal-parameter scalar QGT component retains only their product. This separation leads to a local classification of interband vertices into no-interband, Hermitian-locked, generic complex-transverse, and complex-null cases. On a complex-null branch, the scalar component can vanish even though one ordered interband matrix element remains nonzero. We identify this as a local chiral-vertex mechanism in a vertex-resolved geometric response kernel, distinct from generic non-Hermiticity or exceptional-point proximity. Nonreciprocal SSH, a two-dimensional complex-spin–orbit lattice, and a

k_{z}

-only chiral ladder stack realize the same mechanism in one, two, and three dimensions, while diagonal and gain–loss-like vertices provide nonselective comparisons. Full article

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21 pages, 6760 KB

Open AccessArticle

An Evaluation Method for Influential Nodes Based on Multi-Attribute Neighbor Contributions in Complex Networks

by Na Zhao, Chao Dai, Guolin Yang, Ting Luo, Nifei Xiong and Jian Wang

Entropy 2026, 28(8), 935; https://doi.org/10.3390/e28080935 - 21 Aug 2026

Abstract

Accurately identifying influential nodes is essential for analyzing network structures and optimizing information propagation. Existing methods predominantly rely on single indicators such as degree, H-index, or k-shell, inherently limiting their ability to capture a node’s true influence. Recent hybrid centrality approaches attempt to [...] Read more.

Accurately identifying influential nodes is essential for analyzing network structures and optimizing information propagation. Existing methods predominantly rely on single indicators such as degree, H-index, or k-shell, inherently limiting their ability to capture a node’s true influence. Recent hybrid centrality approaches attempt to address this by combining multiple local and global attributes; however, they typically integrate features through simple weighting or superposition, failing to characterize the intrinsic synergy among structural properties. Furthermore, they often quantify neighbor contributions too coarsely, overlook the regulatory role of edge strength, and some suffer from high computational complexity, limiting scalability. To overcome these deficiencies, we propose WKDH, a novel influential node identification method based on multi-attribute neighbor contributions. WKDH fuses local structural attributes (degree and H-index) with global structural attributes (k-shell) via a multiplicative weighted synergy model, simultaneously capturing local connection “quantity,” local connection “quality,” and global core-layer position. By transforming neighbors’ comprehensive characteristics into regulated contribution degrees, WKDH mitigates excessive self-attribute interference and accurately reflects the actual propagation potential of edges. Notably, the method achieves linear computational complexity of

O (m)

. Experimental results on nine real-world and six artificial networks demonstrate that WKDH outperforms nine established indicators in terms of node influence ranking, identification of high-influence nodes, and measuring propagation capability. Moreover, WKDH exhibits strong universality across diverse network structures, as it operates without parameter tuning. Full article

(This article belongs to the Special Issue Advances in Complex Networks and Their Applications, from COMPLEX NETWORKS 2025)

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26 pages, 4633 KB

Open AccessArticle

Event-Triggered Prescribed Performance Control for Maglev Systems Subject to Multiple Constraints

by Chenglong Zhu, Xiaolong Chen, Xinming Guo and Wei Sun

Entropy 2026, 28(8), 934; https://doi.org/10.3390/e28080934 - 20 Aug 2026

Abstract

Maglev trains are susceptible to various types of operational challenges, including track irregularities, load variations, and actuator faults. It is evident that these factors can compromise suspension performance and even pose a serious risk to operational safety. This paper proposes a prescribed performance [...] Read more.

Maglev trains are susceptible to various types of operational challenges, including track irregularities, load variations, and actuator faults. It is evident that these factors can compromise suspension performance and even pose a serious risk to operational safety. This paper proposes a prescribed performance event-triggered fault-tolerant control method for the electromagnetic suspension system of a maglev train subject to multiple constraints. A projection-based adaptive extended state observer is designed to estimate the unknown gain caused by actuator faults and load variations, as well as the external disturbance. In light of the disparity in upper and lower safety margins inherent to the suspension gap error, arising from track irregularities, an asymmetric prescribed performance function and an error transformation are devised to ensure that the gap tracking error perpetually complies with the asymmetric prescribed performance constraint. In addressing the issue of rapid variations in the suspension gap, the vertical velocity is also constrained through the implementation of prescribed performance, resulting in a joint constraint framework that encompasses both the gap tracking error and the vertical motion. A dynamic event-triggered mechanism has been incorporated into the backstepping design with a view to reducing unnecessary control updates under limited communication resources, while Zeno behavior has been excluded from the closed-loop system. Within this framework, a dynamic gain adjustment mechanism with an explicitly bounded rate of variation is further developed to achieve smoother gain adaptation. The uniform ultimate boundedness of all closed-loop signals is demonstrated through Lyapunov stability analysis under the prescribed multiple constraints. The efficacy of the proposed method is demonstrated through comparative simulation results. Full article

(This article belongs to the Special Issue Information Theory in Control Systems, 3rd Edition)

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42 pages, 4656 KB

Open AccessArticle

Parameter-Independent Feature Ranking with Volume-Integrated Sharma–Mittal Entropy: Kernel-Based Estimation, Theoretical Properties and Empirical Validation

by Nida Oruç Ünal, Muzaffer Göztaş and Doğan Yıldız

Entropy 2026, 28(8), 933; https://doi.org/10.3390/e28080933 - 20 Aug 2026

Abstract

Feature selection is a critical step in regression problems where a large number of continuous explanatory variables explain the same target through different dependency structures. Classical filters may remain sensitive to a single form of dependence, a single scale, or a specific discretization [...] Read more.

Feature selection is a critical step in regression problems where a large number of continuous explanatory variables explain the same target through different dependency structures. Classical filters may remain sensitive to a single form of dependence, a single scale, or a specific discretization scheme; generalized entropy measures, on the other hand, typically require the parameters to be fixed at a single point. This study proposes a framework that evaluates the Sharma–Mittal entropy volumetrically across a two-dimensional parameter region rather than for a single parameter pair. For the continuous target and explanatory variables, the marginal, joint, and conditional densities are obtained using a Gaussian kernel density estimation; the conditional entropy and information gain surfaces are integrated across the region Ω = [0.05, 0.95]² in the α-β plane to define three indices: PICSME, which measures the conditional uncertainty volume; PIGSME, which measures the gain volume; and NIGSME, which is the ratio of this gain to the total entropy volume of the target. The method is supported by bandwidth consistency and the renormalization of conditional densities; thus, the issue of negative gain that can occur in the continuous variables is resolved, yielding positive and interpretable scores across all six datasets. It is formally demonstrated that the fact that the three indices produce the same ranking is not an empirical observation but rather the result of a monotonicity relationship valid under a fixed target entropy volume. The method is compared with Pearson and Spearman correlations, the Shannon information gain, mutual information, and random forest variable importance across six regression datasets (Airfoil Self-Noise, AirQualityUCI, BodyFat, Meteorology, Concrete, and WineQualityWhite) that differ in their sample size, dimensions, and application domain. The evaluation is not limited to ranking consistency; the out-of-sample prediction performance is measured using least-squares models on the top-k subsets, with rankings calculated from the training partition. The findings show that NIGSME exhibits a performance comparable to that of built-in filters, outperforms them on the Concrete and Meteorology datasets, and never ranks as the weakest method on any dataset. The results demonstrate that volumetric entropy metrics defined across the entire parameter space provide a feature-ranking tool that is independent of parameter selection for continuous variables. Full article

(This article belongs to the Special Issue Insight into Entropy)

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39 pages, 497 KB

Open AccessArticle

Invariant Boltzmann-Shannon Entropy for Black-Holes: A Manifestly-Covariant Canonical Quantum-Gravity Approach

by Claudio Cremaschini, Ramesh Radhakrishnan and Gerald Cleaver

Entropy 2026, 28(8), 932; https://doi.org/10.3390/e28080932 - 20 Aug 2026

Abstract

A novel theoretical study of Boltzmann-Shannon entropy arising in information-statistic theory applied to black-hole physics is proposed. The invariant setting implemented is represented by the manifestly-covariant quantum-gravity theory expressed in canonical Hamiltonian form. In such a framework the appropriate statistical interpretation relies on [...] Read more.

A novel theoretical study of Boltzmann-Shannon entropy arising in information-statistic theory applied to black-hole physics is proposed. The invariant setting implemented is represented by the manifestly-covariant quantum-gravity theory expressed in canonical Hamiltonian form. In such a framework the appropriate statistical interpretation relies on the configuration-space quantum expectation value of physical observables over the

4 -

scalar quantum-gravity probability density function (PDF). A representation for the black-hole Boltzmann-Shannon entropy is obtained for a Gaussian PDF profile and by establishing simultaneously a relationship between the black-hole invariant energy-content and the mean value of the quantum-gravity nonlinear Bohm potential. This yields a non-trivial functional dependence of the Boltzmann-Shannon entropy on the black-hole surface area, to be interpreted as a quantum statistical entropy counting black-hole bulk quantum-gravity states. The mathematical setting is shown to preserve manifest covariance and be self-contained within quantum-gravity realm. Comparisons with literature treatments dealing with thermodynamic or kinetic-statistical entropies that lead to the Bekenstein-Hawking black-hole surface entropy linear relation or its proposed quantum modifications are discussed. Full article

(This article belongs to the Special Issue Hamiltonian Dynamics in Fundamental Physics)

12 pages, 1625 KB

Open AccessArticle

Geometric Phase-Induced Stückelberg Interference in an Optical Lattice Clock

by Wei-Xin Liu, Zhan-Peng Lu and Tao Wang

Entropy 2026, 28(8), 931; https://doi.org/10.3390/e28080931 - 20 Aug 2026

Abstract

We theoretically investigate geometric Stückelberg interferometry in a doubly driven optical lattice clock (OLC). By tuning the relative phase between the two driving fields, we control the relative sign of the effective coupling strengths at the avoided crossings. Within the adiabatic-impulse model, we [...] Read more.

We theoretically investigate geometric Stückelberg interferometry in a doubly driven optical lattice clock (OLC). By tuning the relative phase between the two driving fields, we control the relative sign of the effective coupling strengths at the avoided crossings. Within the adiabatic-impulse model, we analyze the time evolution of the two-level system, where nonadiabatic transitions occur only near the crossing points and adiabatic evolution takes place between them. We show that, besides the usual dynamical phase and the Stokes phase, a gauge-invariant noncyclic geometric phase contributes to the final transition probability. This geometric contribution yields a stable

π

-phase shift in the Stückelberg interference fringes. Moreover, we demonstrate that, under realistic experimental conditions, this geometric Stückelberg interferometer remains insensitive to inhomogeneities in atom-light coupling arising from the finite temperature of the atomic ensemble. Our results provide a general framework for engineering and detecting geometric phases on the OLC platform. Full article

(This article belongs to the Section Multidisciplinary Applications)

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25 pages, 1147 KB

Open AccessArticle

KEMFF: A Knowledge-Enhanced and Multidimensional Feature Fusion Model for Aspect-Based Sentiment Analysis

by Shuangshuang Yang, Peilun Liu and Wenlong Zhu

Entropy 2026, 28(8), 930; https://doi.org/10.3390/e28080930 - 19 Aug 2026

Abstract

Aspect-based Sentiment Analysis (ABSA) is a fine-grained sentiment classification task that aims to predict the sentiment polarity associated with aspect terms in sentences. Traditional methods based on syntactic and semantic dependency trees are insufficient for capturing contextual sentence features. To address this, we [...] Read more.

Aspect-based Sentiment Analysis (ABSA) is a fine-grained sentiment classification task that aims to predict the sentiment polarity associated with aspect terms in sentences. Traditional methods based on syntactic and semantic dependency trees are insufficient for capturing contextual sentence features. To address this, we propose a Knowledge-Enhanced and Multidimensional Feature Fusion (KEMFF) model for ABSA, which captures sentiment feature representations across multiple dimensions, including syntax, semantics, and knowledge. First, the pre-trained model RoBERTa is used to obtain embeddings of sentences and aspect terms. Then, a syntactic dependency parser and a graph convolutional network are utilized to learn syntactic features. Meanwhile, an Abstract Meaning Representation (AMR)-based parser is employed to construct semantic relations, and axial attention is used to aggregate incoming and outgoing semantic dependencies. Furthermore, external knowledge is embedded, and an attention mechanism is employed to obtain aspect-specific knowledge representations, thereby complementing syntactic and semantic representations with external lexical knowledge. Finally, multidimensional features are fused and passed to a softmax classifier for predicting sentiment polarities. Unlike previous models that mainly focus on either syntax–semantic fusion or knowledge-enhanced graph propagation, KEMFF explicitly models syntax, semantics, and lexical knowledge in three parallel branches and aligns them into a unified aspect-level representation. Experiments on Laptop14, Restaurant14, and Twitter datasets show that KEMFF achieves the best performance among the compared baselines on Laptop14 and Restaurant14, and it obtains competitive results on Twitter. Full article

(This article belongs to the Section Multidisciplinary Applications)

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16 pages, 332 KB

Open AccessArticle

BATS Code Decoding Method Based on Left-Nullspace-Guided Rank-Completion Feedback

by Juan Yang, Jingjing Lu, Jianbo Ji and Tao Wang

Entropy 2026, 28(8), 929; https://doi.org/10.3390/e28080929 - 19 Aug 2026

Abstract

BP decoding of BATS codes may stop when no residual batch satisfies the full-row-rank condition. To resume BP decoding, this paper proposes an Important-Packet-Guided Left-Nullspace Rank-Completion (LNRC) method. LNRC first identifies the unrecovered source packet that connects to the largest number of undecoded [...] Read more.

BP decoding of BATS codes may stop when no residual batch satisfies the full-row-rank condition. To resume BP decoding, this paper proposes an Important-Packet-Guided Left-Nullspace Rank-Completion (LNRC) method. LNRC first identifies the unrecovered source packet that connects to the largest number of undecoded batches, denotes it as the Important Packet, and uses it as a guidance packet to locate the undecoded batches containing it as repair candidates. For a selected batch with rank deficit one, the destination computes a nonzero left-null vector and selects a local repair coordinate that provides the missing independent direction. The destination sends the corresponding global source-packet index and finite-field coefficient through a reliable reverse feedback-control link, and the source returns the scaled repair packet through a reliable forward repair-data link. The associated completion column increases the target residual transfer-matrix rank by one and makes the batch BP-decodable. Thus, LNRC exploits the column-space structure of the target undecoded batch to select the repair coordinate, rather than selecting the Important Packet solely by the number of connected undecoded batches. Under equal encoding redundancy, simulations show that LNRC achieves a lower packet error rate (PER) compared with conventional Important Packet feedback, with average relative PER reductions of approximately 0.40–12.17% across the evaluated settings. Full article

(This article belongs to the Special Issue Information Theory for Future Communication Systems)

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29 pages, 8782 KB

Open AccessReview

Hamiltonian Dynamics and Fundamental Phenomena in Biophysics: A Review

by Matteo Gori, Roberto Franzosi, Giulio Pettini and Marco Pettini

Entropy 2026, 28(8), 928; https://doi.org/10.3390/e28080928 - 19 Aug 2026

Abstract

We review a theoretical and experimental programme with the aim of understanding two intimately related fundamental phenomena in biophysics: (i) the classical analogue of Fröhlich phonon condensation in macromolecules driven out of thermal equilibrium and (ii) the consequent activation of long-range resonant electrodynamic [...] Read more.

We review a theoretical and experimental programme with the aim of understanding two intimately related fundamental phenomena in biophysics: (i) the classical analogue of Fröhlich phonon condensation in macromolecules driven out of thermal equilibrium and (ii) the consequent activation of long-range resonant electrodynamic intermolecular forces. Both phenomena are underpinned by explicit Hamiltonian models. The first is derived by applying the time-dependent variational principle (TDVP) to the quantum Wu–Austin model, producing a fully classical Hamiltonian in action-angle variables whose nonlinear rate equations exhibit a nonequilibrium phase transition: the channelling of supplied energy into the lowest-frequency collective mode. The second is grounded in a classical electrodynamic Hamiltonian for two coupled oscillating dipoles whose normal-mode structure predicts long-range (∼

1 / r^{3}

) resonant interactions, absent at thermal equilibrium but activated by out-of-equilibrium collective oscillations. We also discuss a complementary Hamiltonian approach that connects Fröhlich’s rate equations directly to Hamilton’s equations of motion, clarifying the role of bath-mediated nonlinear coupling and the conditions for strong condensation at room temperature. In addition, the TDVP is applied to a Davydov–Holstein–Fröhlich Hamiltonian describing electron–phonon motion along the backbone of a specific DNA sequence and its cognate restriction enzyme, EcoRI: the time-domain Fourier cross-spectrum of the resulting electron currents exhibits a sharp co-resonance peak for the canonical recognition sequence that disappears upon randomisation, providing a sequence-specific electrodynamic signature of DNA–protein recognition. Experimental evidence from THz near-field spectroscopy, fluorescence correlation spectroscopy, and direct observation of protein clustering is reviewed in relation to these theoretical predictions. The results establish a coherent physical picture suggesting that metabolic energy supply can play a role in driving macromolecules into coherently oscillating states that activate selective, distance-reaching electrodynamic forces capable of contributing to the organisation of biochemical reactions in living matter. Full article

(This article belongs to the Special Issue Hamiltonian Dynamics in Fundamental Physics)

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