Graph diffusion distance: Properties and efficient computation

We define a new family of similarity and distance measures on graphs, and explore their theoretical properties in comparison to conventional distance metrics. These measures are defined by the solution(s) to an optimization problem which attempts find a map minimizing the discrepancy between two graph Laplacian exponential matrices, under norm-preserving and sparsity constraints. Variants of the distance metric are introduced to consider such optimized maps under sparsity constraints as well as fixed time-scaling between the two Laplacians. The objective function of this optimization is multimodal and has discontinuous slope, and is hence difficult for univariate optimizers to solve. We demonstrate a novel procedure for efficiently calculating these optima for two of our distance measure variants. We present numerical experiments demonstrating that (a) upper bounds of our distance metrics can be used to distinguish between lineages of related graphs; (b) our procedure is faster at finding the required optima, by as much as a factor of 103; and (c) the upper bounds satisfy the triangle inequality exactly under some assumptions and approximately under others. We also derive an upper bound for the distance between two graph products, in terms of the distance between the two pairs of factors. Additionally, we present several possible applications, including the construction of infinite "graph limits" by means of Cauchy sequences of graphs related to one another by our distance measure.

Distance and similarity measures have recently been investigated in-depth within the context of hesitant fuzzy sets. By analyzing the existing studies concerning distance measures for hesitant fuzzy sets, we find that they have some limitations. To address the flaws, this study develops some novel distance measures for hesitant fuzzy sets, including the normalized Euclidean distance measure, the Hausdorff metric distance measure, the normalized generalized distance measure, and their corresponding weighted distance measures. The proposals of this study not only hold many ideal characteristics but also do not consider the lengths of hesitant fuzzy elements as well as the arrangement of their possible values. To deal with the situations where both of the universe of discourse and the weight of element are continuous, some continuous hesitant fuzzy distance measures are also investigated. Based on the relationship between distance measure and similarity measure, some novel similarity measures for hesitant fuzzy sets can be further deduced from the proposed distance measures. Finally, two numerical examples are given to demonstrate the applicability and validity of the proposed hesitant fuzzy distance measures.

Similarity measurement plays an important role to solve many pattern recognition problems such as classification, clustering and particularly the content based image retrieval problems. In this paper, a detailed comparison of 10 similarity measures has been presented for medical image retrieval. Different types of medical images from databases such as NEMA, OASIS and EXACT09 are used to evaluate the performance. Features of all the medical images are extracted using Log-Gabor wavelet. The effectiveness of all the distance metrics are investigated and tested on more than 800 medical images from the databases. It is observed from the experimental results that the retrieval performance can be improved by the distance measures like Bray- Curtis and Canberra distance metric as compared to other existing distance based approach.

Context. The development of effective distance metrics and similarity measures for categorical features is an important task in data analysis, machine learning, and decision theory since a significant portion of object properties is described by non-numerical values. Typically, the dependence between categorical features may be more complex than simply comparing them for equality or inequality. Such attributes can be relatively similar, and to construct an effective model, it is necessary to consider this similarity when calculating distance or similarity measures.
 Objective. The aim of the study is to improve the efficiency of solving practical data analysis problems by developing mathematical tools for determining the similarity of objects based on categorical ordered features.
 Method. A distance based on weighted Manhattan distance and a similarity measure for determining the similarity of objects based on categorical ordinal features (i.e. a linear order with scales of preference considering the problem domain can be specified on the attribute value set) are proposed. It is proven that the distance formula satisfies the axioms of non-negativity, symmetry, triangle inequality, and upper bound, and therefore is a distance metric in the space of ranked categorical features. It is also proven that the similarity measure presented in the study satisfies the axioms of boundedness, symmetry, maximum and minimum similarity, and is described by a decreasing function.
 Results. The developed approach has been implemented in an applied problem of determining the degree of similarity between objects described by ordered categorical features.
 Conclusions. In this study, mathematical tools were developed to determine similarity between structured data described by categorical attributes that can be ordered based on a specific priority in the form of a ranking system with preferences. Their properties were analyzed. Experimental studies have shown the convenience and “intuitive understanding” of the logic of data processing in solving practical problems. The proposed approach can provide the opportunity to conduct new meaningful research in data analysis. Prospects for further research lie in the experimental use of the proposed tools in practical tasks and in studying their effectiveness.

Optimized distance metrics for differential evolution based nearest prototype classifier

Many applications require measuring the distance between mixture distributions. For example in the content-based image retrieval (CBIR) systems and audio speech identification a distance measure between mixture models are often required. This is also an important element for multisensor tracking and fusion where different types of state representations employed by distributed agents need to be correlated. Various distance metrics have been developed to serve this purpose. The performance of these metrics can be evaluated by comparing probabilities of correct correlation verses false detection as a function of a pre-determined threshold on the calculated distance. In this paper, we compare several distance metrics for mixtures distributions. Specifically, we focus on three such distance measures, namely the Integral Square Error distance, the Bhattacharyya distance and the Kullback Leibler distance. To ensure that these techniques can be applied for general distributions, not just for Gaussian mixture model (GMM), we use these techniques in conjunction with a specific distance metric designed for mixture type, called general mixture distance (GMD). For evaluation purpose, we use GMM in the simulation as a test example of mixture models.

Content Based Image Retrieval (CBIR) is a process in which for a given query image similar images will be retrieved based on the image content similarity. Image content refers to its visual features, which are mathematical representations of a digital image. The image retrieval task primarily depends on image feature extraction and similarity measurement between the feature vectors. The performance of CBIR process not only decided by the optimum features extracted from the image but also on the proper choice of the similarity/dissimilarity measures (distance metrics). As the image features broadly diversified in terms of color, texture and shape based, using the same distance metric may not work well for all of these features. In this paper, first we presented overview of geometric and statistical distance metrics used in CBIR along with the comparative analysis of these measures on color and texture features. Color features extracted by computing color histograms in HSV space and texture features by wavelet decompositions. Geometrical distances such as Manhattan, Chebyshev, Euclidean, statistical distance metrics such as Cosine Similarity, Chi-square, KullbackLeibler, Jeffrey and cumulative statistical distance metrics of Kolmogorov-Smirnov, Cramer von Mises and Earthmover's distances were analyzed for feature similarity. We gave certain conclusions on the performance of all these distance metrics in terms of Mean Average Precision (MAP) and Recall rates with color and texture features respectively.

This study introduces distance metrics for quantized-channel-based precoding in multiuser multiantenna systems, aiming to enhance spectral efficiency in dense cellular networks. Traditional metrics, such as the chordal distance, face limitations when dealing with scenarios involving limited feedback and multiple receive antennas. We address these challenges by developing distance measures that more accurately reflect network conditions, including the impact of intercell interference. Our distance measures are specifically designed to approximate the instantaneous rate of each user by estimating the unknown components during the quantization stage. This approach enables the associated users to efficiently estimate their achievable rates during the quantization process. Our distance measures are specifically designed for block diagonalization precoding, a method known for its computational efficiency and strong performance in multi-user multiple-input and multiple-output systems. The proposed metrics outperform conventional distance measures, particularly in environments where feedback resources are constrained, as is often the case in 5G and emerging 6G networks. The enhancements are especially significant in dense cellular networks, where accurate channel state information is critical for maintaining high spectral efficiency. Our findings suggest that these new distance measures offer a robust solution for improving the performance of limited-feedback-based precoding in cellular networks.

Distance and similarity measures are fundamentally important in a variety of scientific fields such as clustering analysis, pattern recognition and decision making, etc. In this paper, by analyzing the existing distance and similarity measures between hesitant fuzzy sets, we show that they are not reasonable in some situations. To this end, we propose a novel concept of hesitancy index of hesitant fuzzy set to measure the hesitancy degree among the possible values in each hesitant fuzzy element of the hesitant fuzzy set. By taking their hesitancy indices into account, new methods for measuring the distances between hesitant fuzzy sets are proposed and their properties are discussed. According to the relationship between the distance measure and the similarity measure, two novel similarity measures for hesitant fuzzy sets are further developed. Afterwards, we propound a novel hesitant fuzzy clustering algorithm on the basis of the novel similarity measures for classifying objects with hesitant fuzzy sets. At length, a real-life example is given to illustrate the detailed implementation process of the proposed clustering approach, and a comparative study on the same example is conducted.

In this paper, we propose a neural network based distance metric learning method for a better discrimination in the sequence-matching based keyword search (KWS). In this technique, we conduct a version of Dynamic Time Warping (DTW) based similarity search on the speaker independent posteriorgram space. With this, we aim to compensate for the scarcity of the resources and overcome the out-of-vocabulary (OOV) term problem, which is one of the main issues for KWS on low-resource languages. This distance measure is then used in the DTW-based similarity search, as an alternative and in comparison to the widely and generally used distance metrics. The experiments ran on IARPA Babel Program's Turkish search data show that, the proposed system outperforms the baseline by 6.3% and when combined with the baseline system, the improvement reaches 44.9%.

The similarity metric between videos is integral to several key tasks, including video retrieval, classification and recommendation. Since there is no standard criterion for the similarity measurement between videos except measuring manually, it is difficult to collect large training dataset for distance metric learning algorithms. Moreover, the existing distance metric learning (DML) methods for multimedia data suffer from two critical limitations: (1) they typically attempt to learn a distance function on the single label setting, in which each item is only labeled with single label; (2) they are often designed for learning distance metrics on low-level features, which ignore the semantic similarity of the multimedia data. To address these problems, in this paper, we propose a novel framework of Intermediate Semantics based Distance Learning (ISDL) for video clips, which aims to integrate semantics of multiple modals optimally for distance metric learning. In particular, the proposed framework: (1) generates the training pairs automatically; (2) defines multi-modal concepts for similarity measure among videos; (3) learns the distance metric for video clips based on the intermediate semantics. We conduct an extensive set of experiments to evaluate the performance of the proposed algorithms, and the results validate the effectiveness of our proposed approach.

This paper represents a new approach for face recognition that incorporates Prewitt edge detection, Gabor filter and Zernike moments to transform the image into a unified domain. On this joint domain, five distance metrics are constructed using Schoenberg transform for the purpose of defining efficient similarity measures for holistic face recognition. The proposed Schoenberg similarity applies Schoenberg transform to the logarithm of five existing distance metrics: Minkowski, City-Block, Euclidean, Soergel and Lorentzian metrics. The constructed Schoenberg logarithmic metrics are called SL-Minkowski, SL-City-Block, SL-Euclidean, SL-Soergel and SL-Lorentzian. These distance metrics are utilized as similarity measures after being normalized over the range [0,1] for fair comparison with existing measures. The proposed Schoenberg system can resist three problems: Change in illumination, pose and facial expression. Simulation results show that the proposed distance measures have superior performance as compared to the classical metrics: Structural Similarity Index Measure (SSIM) and Feature-based Similarity Measure (FSIM). Performance criteria are the recognition rate and the recognition confidence, defined as the similarity difference between the best match and the second-best match in the face database.

Dual hesitant fuzzy set, first proposed by Zhu et al. (Dual hesitant fuzzy sets, J Appl Math, 1–13, 2012) as an extension of hesitant fuzzy sets, which encompass fuzzy sets, intuitionistic fuzzy sets, hesitant fuzzy sets, and fuzzy multisets as a special case. Dual hesitant fuzzy sets consist of two parts, that is, the membership and nonmembership degrees, which are represented by two sets of possible values. Therefore, in accordance with the practical demand, these sets are more flexible than the existing fuzzy sets, and provide much more information about the situation. In this paper, the axiom definition of distance and similarity measures between dual hesitant fuzzy sets is introduced. Some new distance and similarity measures based on the geometric distance model, the set-theoretic approach, and the matching functions are proposed. The proposed distance measures are then applied to the multiple-attribute decision making under dual hesitant fuzzy environments. Through the distance measure between each alternative and the ideal alternative, the ranking order of all alternatives can be determined and the best alternative can be easily identified as well. Finally, a practical example of investment alternatives is given to demonstrate the effectiveness of the developed measures. The advantages of the proposed distance measure over existing measures have been presented.

Distance metric is widely used in similarity estimation. The smaller the distance is, the greater the similarity is. The Minkowski distance metric are usually chosen as the similarity measure in the conventional similarity metrics. In this paper, the most popular Euclidean and Manhattan distance, and the morphology similarity distance we have proposed are compared as similarity estimation measure distance. We cluster thirty random datasets using the fuzzy c-mean algorithm, recognize the Iris data from the UCI repository, and the experiment results are compared and analyzed.

CSI fingerprinting based localization system is one of the most promising techniques for indoor positioning. CSI phase sanitization and distance measure between the fingerprint and the test point are two important operations, both of which have significant influence on the localization performance. However, there are potential disadvantages of the two operations which limit the system performance in conventional algorithms. In this paper, we propose a novel CSI phase sanitization method and a new distance metric. The two methods are both tested using the CSI measured in office environment. Compared with the conventional methods, they achieve much better performance.