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19 pages, 855 KiB  
Article
Comparative Analysis of Audio Features for Unsupervised Speaker Change Detection
by Alymzhan Toleu, Gulmira Tolegen, Rustam Mussabayev, Alexander Krassovitskiy and Bagashar Zhumazhanov
Appl. Sci. 2024, 14(24), 12026; https://doi.org/10.3390/app142412026 - 23 Dec 2024
Viewed by 512
Abstract
This study examines how ten different audio features, including MFCC, mel-spectrogram, chroma, and spectral contrast etc., influence speaker change detection (SCD) performance. The analysis is conducted using two unsupervised methods: Bayesian information criterion with Gaussian mixture model (BIC-GMM), a model-based approach, and Kullback-Leibler [...] Read more.
This study examines how ten different audio features, including MFCC, mel-spectrogram, chroma, and spectral contrast etc., influence speaker change detection (SCD) performance. The analysis is conducted using two unsupervised methods: Bayesian information criterion with Gaussian mixture model (BIC-GMM), a model-based approach, and Kullback-Leibler divergence with Gaussian Mixture Model (KL-GMM), a metric-based approach. Evaluation involved statistical analysis of feature changes in relation to speaker changes (vice versa), supported by comprehensive experimental validation. Experimental results show MFCC as the most effective feature, demonstrating consistently good performance across both methods. Features such as zero crossing rate, chroma, and spectral contrast also showed notable effectiveness within the BIC-GMM framework, while mel-spectrogram consistently ranked as the least influential feature in both approaches. Further analysis revealed that BIC-GMM exhibits greater stability in managing variations in feature performance, whereas KL-GMM is more sensitive to threshold optimization. Nevertheless, KL-GMM achieved competitive results when paired with specific features, such as MFCC and zero crossing rate. These findings offer valuable insights into the impact of feature selection on unsupervised SCD, providing guidance for the development of more robust and accurate algorithms for practical applications. Full article
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32 pages, 777 KiB  
Article
A Comprehensive Approach to Bias Mitigation for Sentiment Analysis of Social Media Data
by Jothi Prakash Venugopal, Arul Antran Vijay Subramanian, Gopikrishnan Sundaram, Marco Rivera and Patrick Wheeler
Appl. Sci. 2024, 14(23), 11471; https://doi.org/10.3390/app142311471 - 9 Dec 2024
Viewed by 1117
Abstract
Sentiment analysis is a vital component of natural language processing (NLP), enabling the classification of text into positive, negative, or neutral sentiments. It is widely used in customer feedback analysis and social media monitoring but faces a significant challenge: bias. Biases, often introduced [...] Read more.
Sentiment analysis is a vital component of natural language processing (NLP), enabling the classification of text into positive, negative, or neutral sentiments. It is widely used in customer feedback analysis and social media monitoring but faces a significant challenge: bias. Biases, often introduced through imbalanced training datasets, can distort model predictions and result in unfair outcomes. To address this, we propose a bias-aware sentiment analysis framework leveraging Bias-BERT (Bidirectional Encoder Representations from Transformers), a customized classifier designed to balance accuracy and fairness. Our approach begins with adapting the Jigsaw Unintended Bias in Toxicity Classification dataset by converting toxicity scores into sentiment labels, making it suitable for sentiment analysis. This process includes data preparation steps like cleaning, tokenization, and feature extraction, all aimed at reducing bias. At the heart of our method is a novel loss function incorporating a bias-aware term based on the Kullback–Leibler (KL) divergence. This term guides the model toward fair predictions by penalizing biased outputs while maintaining robust classification performance. Ethical considerations are integral to our framework, ensuring the responsible deployment of AI models. This methodology highlights a pathway to equitable sentiment analysis by actively mitigating dataset biases and promoting fairness in NLP applications. Full article
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22 pages, 1347 KiB  
Article
Semi-Empirical Approach to Evaluating Model Fit for Sea Clutter Returns: Focusing on Future Measurements in the Adriatic Sea
by Bojan Vondra
Entropy 2024, 26(12), 1069; https://doi.org/10.3390/e26121069 - 9 Dec 2024
Viewed by 467
Abstract
A method for evaluating Kullback–Leibler (KL) divergence and Squared Hellinger (SH) distance between empirical data and a model distribution is proposed. This method exclusively utilises the empirical Cumulative Distribution Function (CDF) of the data and the CDF of the model, avoiding data processing [...] Read more.
A method for evaluating Kullback–Leibler (KL) divergence and Squared Hellinger (SH) distance between empirical data and a model distribution is proposed. This method exclusively utilises the empirical Cumulative Distribution Function (CDF) of the data and the CDF of the model, avoiding data processing such as histogram binning. The proposed method converges almost surely, with the proof based on the use of exponentially distributed waiting times. An example demonstrates convergence of the KL divergence and SH distance to their true values when utilising the Generalised Pareto (GP) distribution as empirical data and the K distribution as the model. Another example illustrates the goodness of fit of these (GP and K-distribution) models to real sea clutter data from the widely used Intelligent PIxel processing X-band (IPIX) measurements. The proposed method can be applied to assess the goodness of fit of various models (not limited to GP or K distribution) to clutter measurement data such as those from the Adriatic Sea. Distinctive features of this small and immature sea, like the presence of over 1300 islands that affect local wind and wave patterns, are likely to result in an amplitude distribution of sea clutter returns that differs from predictions of models designed for oceans or open seas. However, to the author’s knowledge, no data on this specific topic are currently available in the open literature, and such measurements have yet to be conducted. Full article
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14 pages, 3671 KiB  
Article
A YOLO Network Based on Depthwise Convolution Attention, Feature Fusion, and KL Divergence (DFK-YOLO): A Deep Learning Method for Infrared Small Target Detection Based on YOLOv7
by Peng Ji, Changhao Wu, Xiangyue Zhang, Hean Liu and Dongsheng He
Electronics 2024, 13(23), 4820; https://doi.org/10.3390/electronics13234820 - 6 Dec 2024
Viewed by 576
Abstract
Infrared imaging technology has a wide range of applications across various fields, with one of its most critical uses being the detection of small infrared targets. However, model-driven approaches often lack robustness in identifying these small targets, while current deep learning-based methods face [...] Read more.
Infrared imaging technology has a wide range of applications across various fields, with one of its most critical uses being the detection of small infrared targets. However, model-driven approaches often lack robustness in identifying these small targets, while current deep learning-based methods face challenges in effectively extracting and integrating features. Additionally, appropriate labeling strategies for small infrared targets remain underdeveloped. To address these limitations, this paper proposes a novel detection method based on YOLOv7. Specifically, an attention module leveraging Depthwise Convolution is incorporated into the backbone of YOLOv7. Furthermore, a new Feature Fusion Neck is designed to replace the original neck component of YOLOv7. Lastly, a novel label assignment strategy is introduced. The proposed method achieves a [email protected] of 99.5% and a [email protected] of 71.6% on a public dataset, surpassing the baseline YOLOv7 by 1% and 4.6%, respectively. Compared to state-of-the-art deep learning object detection methods, the proposed approach demonstrates superior performance. Full article
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22 pages, 4829 KiB  
Article
Robot Task-Constrained Optimization and Adaptation with Probabilistic Movement Primitives
by Guanwen Ding, Xizhe Zang, Xuehe Zhang, Changle Li, Yanhe Zhu and Jie Zhao
Biomimetics 2024, 9(12), 738; https://doi.org/10.3390/biomimetics9120738 - 3 Dec 2024
Viewed by 769
Abstract
Enabling a robot to learn skills from a human and adapt to different task scenarios will enable the use of robots in manufacturing to improve efficiency. Movement Primitives (MPs) are prominent tools for encoding skills. This paper investigates how to learn MPs from [...] Read more.
Enabling a robot to learn skills from a human and adapt to different task scenarios will enable the use of robots in manufacturing to improve efficiency. Movement Primitives (MPs) are prominent tools for encoding skills. This paper investigates how to learn MPs from a small number of human demonstrations and adapt to different task constraints, including waypoints, joint limits, virtual walls, and obstacles. Probabilistic Movement Primitives (ProMPs) model movements with distributions, thus providing the robot with additional freedom for task execution. We provide the robot with three modes to move, with only one human demonstration required for each mode. We propose an improved via-point generalization method to generalize smooth trajectories with encoded ProMPs. In addition, we present an effective task-constrained optimization method that incorporates all task constraints analytically into a probabilistic framework. We separate ProMPs as Gaussians at each timestep and minimize Kullback–Leibler (KL) divergence, with a gradient ascent–descent algorithm performed to obtain optimized ProMPs. Given optimized ProMPs, we outline a unified robot movement adaptation method for extending from a single obstacle to multiple obstacles. We validated our approach with a 7-DOF Xarm robot using a series of movement adaptation experiments. Full article
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15 pages, 355 KiB  
Article
Exact Expressions for Kullback–Leibler Divergence for Univariate Distributions
by Victor Nawa and Saralees Nadarajah
Entropy 2024, 26(11), 959; https://doi.org/10.3390/e26110959 - 7 Nov 2024
Viewed by 1185
Abstract
The Kullback–Leibler divergence (KL divergence) is a statistical measure that quantifies the difference between two probability distributions. Specifically, it assesses the amount of information that is lost when one distribution is used to approximate another. This concept is crucial in various fields, including [...] Read more.
The Kullback–Leibler divergence (KL divergence) is a statistical measure that quantifies the difference between two probability distributions. Specifically, it assesses the amount of information that is lost when one distribution is used to approximate another. This concept is crucial in various fields, including information theory, statistics, and machine learning, as it helps in understanding how well a model represents the underlying data. In a recent study by Nawa and Nadarajah, a comprehensive collection of exact expressions for the Kullback–Leibler divergence was derived for both multivariate and matrix-variate distributions. This work is significant as it expands on our existing knowledge of KL divergence by providing precise formulations for over sixty univariate distributions. The authors also ensured the accuracy of these expressions through numerical checks, which adds a layer of validation to their findings. The derived expressions incorporate various special functions, highlighting the mathematical complexity and richness of the topic. This research contributes to a deeper understanding of KL divergence and its applications in statistical analysis and modeling. Full article
(This article belongs to the Section Information Theory, Probability and Statistics)
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18 pages, 435 KiB  
Article
Some Improvements on Good Lattice Point Sets
by Yu-Xuan Lin, Tian-Yu Yan and Kai-Tai Fang
Entropy 2024, 26(11), 910; https://doi.org/10.3390/e26110910 - 27 Oct 2024
Viewed by 720
Abstract
Good lattice point (GLP) sets are a type of number-theoretic method widely utilized across various fields. Their space-filling property can be further improved, especially with large numbers of runs and factors. In this paper, Kullback-Leibler (KL) divergence is used to measure GLP sets. [...] Read more.
Good lattice point (GLP) sets are a type of number-theoretic method widely utilized across various fields. Their space-filling property can be further improved, especially with large numbers of runs and factors. In this paper, Kullback-Leibler (KL) divergence is used to measure GLP sets. The generalized good lattice point (GGLP) sets obtained from linear-level permutations of GLP sets have demonstrated that the permutation does not reduce the criterion maximin distance. This paper confirms that linear-level permutation may lead to greater mixture discrepancy. Nevertheless, GGLP sets can still enhance the space-filling property of GLP sets under various criteria. For small-sized cases, the KL divergence from the uniform distribution of GGLP sets is lower than that of the initial GLP sets, and there is nearly no difference for large-sized points, indicating the similarity of their distributions. This paper incorporates a threshold-accepting algorithm in the construction of GGLP sets and adopts Frobenius distance as the space-filling criterion for large-sized cases. The initial GLP sets have been included in many monographs and are widely utilized. The corresponding GGLP sets are partially included in this paper and will be further calculated and posted online in the future. The performance of GGLP sets is evaluated in two applications: computer experiments and representative points, compared to the initial GLP sets. It shows that GGLP sets perform better in many cases. Full article
(This article belongs to the Section Information Theory, Probability and Statistics)
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16 pages, 303 KiB  
Article
Asymptotic Properties of a Statistical Estimator of the Jeffreys Divergence: The Case of Discrete Distributions
by Vladimir Glinskiy, Artem Logachov, Olga Logachova, Helder Rojas, Lyudmila Serga and Anatoly Yambartsev
Mathematics 2024, 12(21), 3319; https://doi.org/10.3390/math12213319 - 23 Oct 2024
Viewed by 609
Abstract
We investigate the asymptotic properties of the plug-in estimator for the Jeffreys divergence, the symmetric variant of the Kullback–Leibler (KL) divergence. This study focuses specifically on the divergence between discrete distributions. Traditionally, estimators rely on two independent samples corresponding to two distinct conditions. [...] Read more.
We investigate the asymptotic properties of the plug-in estimator for the Jeffreys divergence, the symmetric variant of the Kullback–Leibler (KL) divergence. This study focuses specifically on the divergence between discrete distributions. Traditionally, estimators rely on two independent samples corresponding to two distinct conditions. However, we propose a one-sample estimator where the condition results from a random event. We establish the estimator’s asymptotic unbiasedness (law of large numbers) and asymptotic normality (central limit theorem). Although the results are expected, the proofs require additional technical work due to the randomness of the conditions. Full article
(This article belongs to the Special Issue Mathematical Modeling and Applications in Industrial Organization)
19 pages, 2448 KiB  
Article
MM-IRSTD: Conv Self-Attention-Based Multi-Modal Small and Dim Target Detection in Infrared Dual-Band Images
by Junyan Yang, Zhihui Ye, Jian Lin, Dongfang Chen, Lingbian Du and Shaoyi Li
Remote Sens. 2024, 16(21), 3937; https://doi.org/10.3390/rs16213937 - 23 Oct 2024
Viewed by 1102
Abstract
Infrared multi-band small and dim target detection is an important research direction in the fields of modern remote sensing and military surveillance. However, achieving high-precision detection remains challenging due to the small scale, low contrast of small and dim targets, and their susceptibility [...] Read more.
Infrared multi-band small and dim target detection is an important research direction in the fields of modern remote sensing and military surveillance. However, achieving high-precision detection remains challenging due to the small scale, low contrast of small and dim targets, and their susceptibility to complex background interference. This paper innovatively proposes a dual-band infrared small and dim target detection method (MM-IRSTD). In this framework, we integrate a convolutional self-attention mechanism module and a self-distillation mechanism to achieve end-to-end dual-band infrared small and dim target detection. The Conv-Based Self-Attention module consists of a convolutional self-attention mechanism and a multilayer perceptron, effectively extracting and integrating input features, thereby enhancing the performance and expressive capability of the model. Additionally, this module incorporates a dynamic weight mechanism to achieve adaptive feature fusion, significantly reducing computational complexity and enhancing the model’s global perception capability. During model training, we use a spatial and channel similarity self-distillation mechanism to drive model updates, addressing the similarity discrepancy between long-wave and mid-wave image features extracted through deep learning, thus improving the model’s performance and generalization capability. Furthermore, to better learn and detect edge features in images, this paper designs an edge extraction method based on Sobel. Finally, comparative experiments and ablation studies validate the advancement and effectiveness of our proposed method. Full article
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20 pages, 2184 KiB  
Article
A Stealthy Communication Model for Protecting Aggregated Results Integrity in Federated Learning
by Lu Li, Xuan Sun, Ning Shi, Xiaotian Ci and Chen Liang
Electronics 2024, 13(19), 3870; https://doi.org/10.3390/electronics13193870 - 29 Sep 2024
Cited by 1 | Viewed by 926
Abstract
Given how quickly artificial intelligence technology is developing, federated learning (FL) has emerged to enable effective model training while protecting data privacy. However, when using homomorphic encryption (HE) techniques for privacy protection, FL faces challenges related to the integrity of HE ciphertexts. In [...] Read more.
Given how quickly artificial intelligence technology is developing, federated learning (FL) has emerged to enable effective model training while protecting data privacy. However, when using homomorphic encryption (HE) techniques for privacy protection, FL faces challenges related to the integrity of HE ciphertexts. In the HE-based privacy-preserving FL framework, the public disclosure of the public key and the homomorphic additive property of the HE algorithm pose serious threats to the integrity of the ciphertext of FL’s aggregated results. For the first time, this paper employs covert communication by embedding the hash value of the aggregated result ciphertext received by the client into the ciphertext of local model parameters using the lossless homomorphic additive property of the Paillier algorithm. When the server receives the ciphertext of the local model parameters, it can extract and verify the hash value to determine whether the ciphertext of the FL’s aggregated results has been tampered with. We also used chaotic sequences to select the embedding positions, further enhancing the concealment of the scheme. The experimental findings demonstrate that the suggested plan passed the Welch’s t-test, the K–L divergence test, and the K–S test. These findings confirm that ciphertexts containing covert information are statistically indistinguishable from normal ciphertexts, thereby affirming the proposed scheme’s effectiveness in safeguarding the integrity of the FL’s aggregated ciphertext results. The channel capacity of this scheme can reach up to 512 bits per round, which is higher compared to other FL-based covert channels. Full article
(This article belongs to the Section Networks)
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22 pages, 47104 KiB  
Article
Salp Swarm Algorithm-Based Kalman Filter for Seamless Multi-Source Fusion Positioning with Global Positioning System/Inertial Navigation System/Smartphones
by Jin Wang, Xiyi Dong, Xiaochun Lu, Jin Lu, Jian Xue and Jianbo Du
Remote Sens. 2024, 16(18), 3511; https://doi.org/10.3390/rs16183511 - 21 Sep 2024
Viewed by 905
Abstract
With the rapid development of high-precision positioning service applications, there is a growing demand for accurate and seamless positioning services in indoor and outdoor (I/O) scenarios. To address the problem of low localization accuracy in the I/O transition area and the difficulty of [...] Read more.
With the rapid development of high-precision positioning service applications, there is a growing demand for accurate and seamless positioning services in indoor and outdoor (I/O) scenarios. To address the problem of low localization accuracy in the I/O transition area and the difficulty of achieving fast and accurate I/O switching, a Kalman filter based on the salp swarm algorithm (SSA) for seamless multi-source fusion positioning of global positioning system/inertial navigation system/smartphones (GPS/INS/smartphones) is proposed. First, an Android smartphone was used to collect sensor measurement data, such as light, magnetometer, and satellite signal-to-noise ratios in different environments; then, the change rules of the data were analyzed, and an I/O detection algorithm based on the SSA was used to identify the locations of users. Second, the proposed I/O detection service was used as an automatic switching mechanism, and a seamless indoor–outdoor localization scheme based on improved Kalman filtering with K-L divergence is proposed. The experimental results showed that the SSA-based I/O switching model was able to accurately recognize environmental differences, and the average accuracy of judgment reached 97.04%. The localization method achieved accurate and continuous seamless navigation and improved the average localization accuracy by 53.79% compared with a traditional GPS/INS system. Full article
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21 pages, 1387 KiB  
Article
Trust-Based Detection and Mitigation of Cyber Attacks in Distributed Cooperative Control of Islanded AC Microgrids
by Md Abu Taher, Mohd Tariq and Arif I. Sarwat
Electronics 2024, 13(18), 3692; https://doi.org/10.3390/electronics13183692 - 18 Sep 2024
Cited by 1 | Viewed by 1069
Abstract
In this study, we address the challenge of detecting and mitigating cyber attacks in the distributed cooperative control of islanded AC microgrids, with a particular focus on detecting False Data Injection Attacks (FDIAs), a significant threat to the Smart Grid (SG). The SG [...] Read more.
In this study, we address the challenge of detecting and mitigating cyber attacks in the distributed cooperative control of islanded AC microgrids, with a particular focus on detecting False Data Injection Attacks (FDIAs), a significant threat to the Smart Grid (SG). The SG integrates traditional power systems with communication networks, creating a complex system with numerous vulnerable links, making it a prime target for cyber attacks. These attacks can lead to the disclosure of private data, control network failures, and even blackouts. Unlike machine learning-based approaches that require extensive datasets and mathematical models dependent on accurate system modeling, our method is free from such dependencies. To enhance the microgrid’s resilience against these threats, we propose a resilient control algorithm by introducing a novel trustworthiness parameter into the traditional cooperative control algorithm. Our method evaluates the trustworthiness of distributed energy resources (DERs) based on their voltage measurements and exchanged information, using Kullback-Leibler (KL) divergence to dynamically adjust control actions. We validated our approach through simulations on both the IEEE-34 bus feeder system with eight DERs and a larger microgrid with twenty-two DERs. The results demonstrated a detection accuracy of around 100%, with millisecond range mitigation time, ensuring rapid system recovery. Additionally, our method improved system stability by up to almost 100% under attack scenarios, showcasing its effectiveness in promptly detecting attacks and maintaining system resilience. These findings highlight the potential of our approach to enhance the security and stability of microgrid systems in the face of cyber threats. Full article
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23 pages, 7245 KiB  
Article
Evolution and Quantitative Characterization of Stress and Displacement of Surrounding Rock Structure due to the Multiple Layers Backfill Mining under Loose Aquifers
by Jiawei Liu and Wanghua Sui
Water 2024, 16(18), 2574; https://doi.org/10.3390/w16182574 - 11 Sep 2024
Viewed by 643
Abstract
Backfill mining is an important means of ensuring the high efficiency and safety of the coal mining under thin bedrock and loose aquifers. Based on the case study of Taiping Coalmine, the theoretical analysis of entropy and numerical modeling methods are adopted to [...] Read more.
Backfill mining is an important means of ensuring the high efficiency and safety of the coal mining under thin bedrock and loose aquifers. Based on the case study of Taiping Coalmine, the theoretical analysis of entropy and numerical modeling methods are adopted to establish the visualization model of temporal–spatial cube of stress and displacement induced by the multiple layers backfill mining. Moreover, the quantitative characterization and measurement framework of symmetric KL-divergence is established based on information entropy and mutual information. The results show that: (1) The non-uniformity of stress and displacement is enhanced due to the multiple layers backfill mining, showing certain fluctuation characteristics. (2) The KL-divergence of stress to displacement is slightly greater than that of displacement to stress, and the hotspot distribution law of stress–displacement related efficiency is consistent with KL-divergence. (3) The hotspots of stress entropy and the gap between stress entropy and displacement entropy in multiple layers backfill mining decrease obviously. (4) Stress plays a main role in displacement, and displacement is a linkage response to stress due to the coordinated deformation. Multiple layers backfill mining results in an enhanced correlation degree and more chaotic state between stress and displacement. The results will provide engineering geological basis for optimal design and safe production of backfill mining under loose aquifers. Full article
(This article belongs to the Section Hydrogeology)
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34 pages, 574 KiB  
Article
Optimum Achievable Rates in Two Random Number Generation Problems with f-Divergences Using Smooth Rényi Entropy
by Ryo Nomura and Hideki Yagi
Entropy 2024, 26(9), 766; https://doi.org/10.3390/e26090766 - 6 Sep 2024
Viewed by 631
Abstract
Two typical fixed-length random number generation problems in information theory are considered for general sources. One is the source resolvability problem and the other is the intrinsic randomness problem. In each of these problems, the optimum achievable rate with respect to the given [...] Read more.
Two typical fixed-length random number generation problems in information theory are considered for general sources. One is the source resolvability problem and the other is the intrinsic randomness problem. In each of these problems, the optimum achievable rate with respect to the given approximation measure is one of our main concerns and has been characterized using two different information quantities: the information spectrum and the smooth Rényi entropy. Recently, optimum achievable rates with respect to f-divergences have been characterized using the information spectrum quantity. The f-divergence is a general non-negative measure between two probability distributions on the basis of a convex function f. The class of f-divergences includes several important measures such as the variational distance, the KL divergence, the Hellinger distance and so on. Hence, it is meaningful to consider the random number generation problems with respect to f-divergences. However, optimum achievable rates with respect to f-divergences using the smooth Rényi entropy have not been clarified yet in both problems. In this paper, we try to analyze the optimum achievable rates using the smooth Rényi entropy and to extend the class of f-divergence. To do so, we first derive general formulas of the first-order optimum achievable rates with respect to f-divergences in both problems under the same conditions as imposed by previous studies. Next, we relax the conditions on f-divergence and generalize the obtained general formulas. Then, we particularize our general formulas to several specified functions f. As a result, we reveal that it is easy to derive optimum achievable rates for several important measures from our general formulas. Furthermore, a kind of duality between the resolvability and the intrinsic randomness is revealed in terms of the smooth Rényi entropy. Second-order optimum achievable rates and optimistic achievable rates are also investigated. Full article
(This article belongs to the Section Information Theory, Probability and Statistics)
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22 pages, 5458 KiB  
Article
A Maximum Value for the Kullback–Leibler Divergence between Quantized Distributions
by Vincenzo Bonnici
Information 2024, 15(9), 547; https://doi.org/10.3390/info15090547 - 6 Sep 2024
Viewed by 1123
Abstract
The Kullback–Leibler (KL) divergence is a widely used measure for comparing probability distributions, but it faces limitations such as its unbounded nature and the lack of comparability between distributions with different quantum values (the discrete unit of probability). This study addresses these challenges [...] Read more.
The Kullback–Leibler (KL) divergence is a widely used measure for comparing probability distributions, but it faces limitations such as its unbounded nature and the lack of comparability between distributions with different quantum values (the discrete unit of probability). This study addresses these challenges by introducing the concept of quantized distributions, which are probability distributions formed by distributing a given discrete quantity or quantum. This study establishes an upper bound for the KL divergence between two quantized distributions, enabling the development of a normalized KL divergence that ranges between 0 and 1. The theoretical findings are supported by empirical evaluations, demonstrating the distinct behavior of the normalized KL divergence compared to other commonly used measures. The results highlight the importance of considering the quantum value when applying the KL divergence, offering insights for future advancements in divergence measures. Full article
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