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The notion of set-valued means is investigated. It is proved that set-valued counterparts of the classical arithmetic, quasi-arithmetic and Lagrangian means are Schur-concave. [ABSTRACT FROM AUTHOR]

The Jensen inequality is of fundamental importance because of its influential and interesting consequences. In recent years, the Jensen inequality has been supposed to be the most engaging source for research. We present interesting improvements to the continuous version of Jensen's inequality through the application of the concept of 6-convexity. For real visualization and comparison to other results, some numerical experiments were provided. With the aid of the acquired results, improvements for the Hermite-Hadamard and Hölder inequalities were presented. Some relationships between the means were granted as applications of established improvements. In addition, some estimations of the Csiszár divergence and its associated cases were received as further applications of the obtained results. The major techniques employed in formulating the proposed improvements included the Jensen inequality and the concept of convexity. [ABSTRACT FROM AUTHOR]

The Jensen inequality is of fundamental importance because of its influential and interesting consequences. In recent years, the Jensen inequality has been supposed to be the most engaging source for research. We present interesting improvements to the continuous version of Jensen's inequality through the application of the concept of 6-convexity. For real visualization and comparison to other results, some numerical experiments were provided. With the aid of the acquired results, improvements for the Hermite-Hadamard and Hölder inequalities were presented. Some relationships between the means were granted as applications of established improvements. In addition, some estimations of the Csiszár divergence and its associated cases were received as further applications of the obtained results. The major techniques employed in formulating the proposed improvements included the Jensen inequality and the concept of convexity.

Motivated by a paper of Dragomir, we give new refinements for both discrete and integral Jensen inequalities using the Jensen's gap. As applications, we give refinements of various inequalities verifiable by Jensen's inequality. Topics covered: norms, quasi-arithmetic means, Hölder's inequality and f-divergences in information theory. [ABSTRACT FROM AUTHOR]

Exponential families are statistical models which are the workhorses in statistics, information theory, and machine learning, among others. An exponential family can either be normalized subtractively by its cumulant or free energy function, or equivalently normalized divisively by its partition function. Both the cumulant and partition functions are strictly convex and smooth functions inducing corresponding pairs of Bregman and Jensen divergences. It is well known that skewed Bhattacharyya distances between the probability densities of an exponential family amount to skewed Jensen divergences induced by the cumulant function between their corresponding natural parameters, and that in limit cases the sided Kullback–Leibler divergences amount to reverse-sided Bregman divergences. In this work, we first show that the α -divergences between non-normalized densities of an exponential family amount to scaled α -skewed Jensen divergences induced by the partition function. We then show how comparative convexity with respect to a pair of quasi-arithmetical means allows both convex functions and their arguments to be deformed, thereby defining dually flat spaces with corresponding divergences when ordinary convexity is preserved. [ABSTRACT FROM AUTHOR]

The Gini index (GI), GI II, and GI III are proven to be effective sparsity measures in the fields of machine condition monitoring and fault diagnosis, and they can be reformulated as the ratio of different quasi-arithmetic means (RQAM). Under this framework, generalized Gini indices (GGIs) have been developed for sparse quantification by applying nonlinear weights to GI, and another generalized form of GI, referred to here as power function-based Gini indices I (PFGI1s), has been introduced by using power function as the generator of quasi-arithmetic means. The GGIs with different weight parameters exhibit reliable sparse quantization capability for repetitive transient features, while their repetitive transient discriminability is lower than kurtosis and negentropy under noise contamination. PFGI1 achieves enhanced repetitive transient discriminability with increasing power exponent, showing the advantage of the generalization approach. In this paper, based on RQAM, a single-parameter generalization method for generating PFGI1s is introduced into GI II and GI III from the perspective of the quasi-arithmetic mean generator, which leads to the power function-based Gini indices II and III (PFGI2s and PFGI3s) constructed from GI II and GI III, respectively. Mathematical derivation proves that PFGI2s and PFGI3s satisfy at least five of six typical attributes of sparsity measures and are two new families of sparsity measures. Simulation analysis shows that, similar to PFGI1s, PFGI2s and PFGI3s can monotonically estimate the sparsity of the data sequence and can simultaneously achieve strong random transient resistibility and high repetitive transient discriminability compared with traditional sparsity measures. The experimental results of bearing run-to-failure demonstrate that PFGI1s, PFGI2s, and PFGI3s with appropriate power exponents can effectively quantify the repetitive transient features caused by bearing faults and can accurately characterize the bearing degradation status compared with the state-of-the-art sparsity measures. [ABSTRACT FROM AUTHOR]

The main purpose of this manuscript is to present some new estimations of the Jensen gap in a discrete sense along with their applications. The proposed estimations for the Jensen gap are provided with the help of the notion of 6-convex functions. Some numerical experiments are performed to determine the significance and correctness of the intended estimates. Several outcomes of the main results are discussed for the Hölder inequality and the power and quasi-arithmetic means. Furthermore, some applications are presented in information theory, which provide some bounds for the divergences, Bhattacharyya coefficient, Shannon entropy, and Zipf–Mandelbrot entropy. [ABSTRACT FROM AUTHOR]

Abstract There are a lot of papers dealing with applications of the so-called cyclic refinement of the discrete Jensen’s inequality. A significant generalization of the cyclic refinement, based on combinatorial considerations, has recently been discovered by the author. In the present paper we give the integral versions of these results. On the one hand, a new method to refine the integral Jensen’s inequality is developed. On the other hand, the result contains some recent refinements of the integral Jensen’s inequality as elementary cases. Finally some applications to the Fejér inequality (especially the Hermite–Hadamard inequality), quasi-arithmetic means, and f-divergences are presented.

For a family (A x) x ∈ (0 , 1) of integral quasi-arithmetic means satisfying certain measurability-type assumptions we search for an integral mean K such that K ( (A x (P)) x ∈ (0 , 1) ) = K (P) for every compactly supported probability Borel measure P . Also some results concerning the uniqueness of invariant means will be given. [ABSTRACT FROM AUTHOR]

We extend some approach to the family of symmetric means (i.e. symmetric functions M : ⋃ n = 1 ∞ I n → I with min ≤ M ≤ max ; I is an interval). Namely, it is known that every symmetric mean can be written in a form M (v 1 , ⋯ , v n) : = F (f (v 1) + ⋯ + f (v n)) , where f : I → G and F : G → I (G is a commutative semigroup). For G = R k or G = R k × Z ( k ∈ N ) and continuous functions f and F we obtain two series of families (depending on k). It can be treated as a measure of complexity in a family of means (this idea is inspired by theory of regular languages and algorithmics). As a result we characterize the celebrated families of quasi-arithmetic means ( G = R × Z ) and Bajraktarević means ( G = R 2 under some additional assumptions). Moreover, we establish certain estimations of complexity for several other classical families. [ABSTRACT FROM AUTHOR]

There are a lot of papers dealing with applications of the so-called cyclic refinement of the discrete Jensen's inequality. A significant generalization of the cyclic refinement, based on combinatorial considerations, has recently been discovered by the author. In the present paper we give the integral versions of these results. On the one hand, a new method to refine the integral Jensen's inequality is developed. On the other hand, the result contains some recent refinements of the integral Jensen's inequality as elementary cases. Finally some applications to the Fejér inequality (especially the Hermite–Hadamard inequality), quasi-arithmetic means, and f-divergences are presented. [ABSTRACT FROM AUTHOR]

In this article, we give sharp two-sided bounds for the generalized Jensen functional Jn(f,g,h;p,x). Assuming convexity/concavity of the generating function h, we give exact bounds for the generalized quasi-arithmetic mean An(h;p,x). In particular, exact bounds are determined for the generalized power means in terms from the class of Stolarsky means. As a consequence, some sharp converses of the famous Hölder’s inequality are obtained.

This paper illustrates the new structure of artificial neuron based on root-power mean (RPM) aggregation for quaternionic-valued signals and also presented an efficient learning process of neural networks with quaternionic-valued RPM neurons. The main aim of this neuron is to present the potential capability of a nonlinear aggregation operation on the quaternionic-valued signals in neuron cell. A wide spectrum of aggregation ability of RPM in between minima and maxima has a beautiful property of changing its degree of compensation in the natural way which emulates the various existing neuron models as its special cases. Further, the quaternionic resilient propagation algorithm (ℍ-RPROP) with error-dependent weight backtracking step significantly accelerates the training speed and exhibits better approximation accuracy. The wide spectra of benchmark problem are considered to evaluate the performance of proposed quaternionic RPM neuron with ℍ-RPROP learning algorithm. [ABSTRACT FROM AUTHOR]

In this short communication, we will show that a quasi-arithmetic mean induced by an additive generator of a strict t-norm (strict t-conorm, respectively) can be represented by the composition of a unary operator (called the tau function) and the strict t-norm (strict t-conorm, respectively), both induced by the same generator function as the quasi arithmetic mean. Here, we will also state a connection between the idempotency of a transformed strict t-norm (strict t-conorm, respectively) and the tau function. [ABSTRACT FROM AUTHOR]

For a sequence of continuous, monotone functions f1,…,fn:I→R (I is an interval) we define the mapping M:In→In as a Cartesian product of quasi-arithmetic means generated by fj-s. It is known that, for every initial vector, the iteration sequence of this mapping tends to the diagonal of In. We will prove that whenever all fj-s are C2 with nowhere vanishing first derivative, then this convergence is quadratic. Furthermore, the limit VarMk+1(v)(VarMk(v))2 will be calculated in a nondegenerated case. [ABSTRACT FROM AUTHOR]

In the 1960s Cargo and Shisha introduced a metric in a family of quasi-arithmetic means defined on a common interval as the maximal possible difference between these means taken over all admissible vectors with corresponding weights. During the years 2013-2016 we proved that, having two quasi-arithmetic means, we can majorize the distance between them in terms of the Arrow-Pratt index. In this paper we are going to prove that this operator can also be used to establish certain lower bounds of this distance. [ABSTRACT FROM AUTHOR]

It is known that the Power Means tend to the maximum of their arguments when the exponents tend to $${+\infty}$$ . We give certain necessary and sufficient conditions for a 1-parameter family of quasi-arithmetic means generated by functions satisfying certain smoothness conditions to have an analogous property. Our results are deeply connected with operators introduced by Mikusiński and Páles in the late 1940s and late 1980s, respectively. The main result is a generalization of the author's earlier results. [ABSTRACT FROM AUTHOR]

The aim of this paper is to characterize the distributivity equations between quasi-arithmetic means and some binary operators, such as, nullnorms, semi-nullnorms and semi-t-operators. It is shown that the distributivity equations between nullnorms (semi-nullnorms) and quasi-arithmetic means degenerate into the distributivity equations between t-norms or t-conorms (semi-t-norms or semi-t-conorms) and quasi-arithmetic means. But for semi-t-operators, the distributive equations have new solutions. These new results extend the previous ones about the distributivity between t-norms (t-conorms) and quasi-arithmetic means. [ABSTRACT FROM AUTHOR]

Highlights: [•] Bayes error expressed using the total variation distance. [•] Chernoff bound interpreted as the minimization of a geometric weighted mean. [•] Chernoff bound construction generalized using quasi-arithmetic weighted means. [Copyright &y& Elsevier]

In this paper we give converses of the operator version for the Jensen inequality and the scalar Lah-Ribarič inequality. We also apply those results to quasi-arithmetic means and power means for selfadjoint operators. [ABSTRACT FROM AUTHOR]

Abstract: In many practical applications, the need arises to aggregate data of varying dimension. Following from the self-identity property, some recent studies have looked at the stability of aggregation operators in terms of their behavior as the dimensionality is increased from to n. We use the penalty-based representation of aggregation functions in order to investigate the conditions for weighting vectors associated with some important weighted families, extending on the results already established for quasi-arithmetic means. In particular, we obtain results for quasi-medians and functions that involve a reordering of the inputs such as the OWA and order statistics. [Copyright &y& Elsevier]

This paper describes an artificial neuron structure and an efficient learning procedure in the complex domain. This artificial neuron aims at incorporating an improved aggregation operation on the complex-valued signals. The aggregation operation is based on the idea underlying the weighted root-power mean of input signals. This aggregation operation allows modeling the degree of compensation in a natural manner and includes various aggregation operations as its special cases. The complex resilient propagation algorithm (\BBC-RPROP) with error-dependent weight backtracking step accelerates the training speed significantly and provides better approximation accuracy. Finally, performance evaluation of the proposed complex root-power mean neuron with the \BBC-RPROP learning algorithm on various typical examples is given to understand the motivation. [ABSTRACT FROM AUTHOR]

Abstract: This article discusses Lipschitz properties of generated aggregation functions. Such generated functions include triangular norms and conorms, quasi-arithmetic means, uninorms, nullnorms and continuous generated functions with a neutral element. The Lipschitz property guarantees stability of aggregation operations with respect to input inaccuracies, and is important for applications. We provide verifiable sufficient conditions to determine when a generated aggregation function holds the k-Lipschitz property, and calculate the Lipschitz constants of power means. We also establish sufficient conditions which guarantee that a generated aggregation function is not Lipschitz. We found the only 1-Lipschitz generated function with a neutral element . [Copyright &y& Elsevier]

Abstract: This article studies a large class of averaging aggregation functions based on minimizing a distance from the vector of inputs, or equivalently, minimizing a penalty imposed for deviations of individual inputs from the aggregated value. We provide a systematization of various types of penalty based aggregation functions, and show how many special cases arise as the result. We show how new aggregation functions can be constructed either analytically or numerically and provide many examples. We establish connection with the maximum likelihood principle, and present tools for averaging experimental noisy data with distinct noise distributions. [Copyright &y& Elsevier]

Sparsity measures that can quantify the sparsity of signals are often used as objective functions of signal processing and machine learning algorithms (e.g., sparse filtering, compressive sensing, blind deconvolution, and the fast Kurtogram, etc.). Classic sparsity measures include kurtosis, Gini index, negative entropy, the ratio of L p norm to L q norm, smoothness index, L1 norm, L0 norm, etc. To enrich the library of sparsity measures, this paper aims to generalize a classic Gini index to construct generalized Gini indices (GGIs). Firstly, inspired by the ratio of different quasi-arithmetic means, investigations on the GGIs showed that the GGIs satisfy all six properties of good sparsity measures. The GGIs only have one parameter a ⩾ 0. When a = 1 , the GGIs are reduced to the classic Gini index. Secondly, investigations reveal that: (1) the GGIs can monotonically quantify the sparsity changes of Bernoulli coefficients. Sparsity quantification curves given by the GGIs are complementary to those generated by our recently proposed Box-Cox sparsity measures (BCSMs). Especially, when the unique parameters of the GGIs and the BCSMs are equal to zero, these sparsity quantification curves of the GGIs and BCSMs are reduced to that of negative entropy; (2) the GGIs are convergent with an increase of signal lengths; (3) when the squared envelope of white Gaussian noise is quantified by the GGIs, theoretical values of the GGIs are derived as a / (a + 1) , which can be used as baselines for machine abnormality detection. At last, two bearing run-to-failure datasets are used to validate the effectiveness of the GGIs for machine condition monitoring and their results showed that the GGIs are effective in detecting incipient faults and indicating informative frequency bands. In future, the proposed GGIs can be used in any algorithms that need sparsity measures. [ABSTRACT FROM AUTHOR]

Abstract: This article examines the construction of aggregation functions from data by minimizing the least absolute deviation criterion. We formulate various instances of such problems as linear programming problems. We consider the cases in which the data are provided as intervals, and the outputs ordering needs to be preserved, and show that linear programming formulation is valid for such cases. This feature is very valuable in practice, since the standard simplex method can be used. [Copyright &y& Elsevier]

Abstract: A variant of Jensen’s operator inequality for convex functions, which is a generalization of Mercer’s result, is proved. Obtained result is used to prove a monotonicity property for Mercer’s power means for operators, and a comparison theorem for quasi-arithmetic means for operators. [Copyright &y& Elsevier]

Abstract: A variant of Jensen–Steffensen''s inequality is proved. Necessary and sufficient conditions for the equality in Jensen–Steffensen''s inequality are established. Several inequalities involving more than two monotonic functions and generalized quasi-arithmetic means with not only positive weights are proved. It is shown that such generalized quasi-arithmetic means have the same comparability properties as those with positive weights. [Copyright &y& Elsevier]

Motivated by problems on street’s lighting where an empirical illuminance index is being employed in the technical manuals, we present a functional equation model and we obtain a complete characterization of various classes of indices, extending the usual one. [ABSTRACT FROM AUTHOR]

We say that f is reciprocally convex if x↦f(x) is concave and x↦f(1/x) is convex on (0,+∞). Reciprocally convex functions generate a sequence of quasi-arithmetic means, with the first one between harmonic and arithmetic mean and others above the arithmetic mean. We present several examples related to the gamma function and we show that if f is a Stieltjes transform, then −f is reciprocally convex. An application in probability is also presented. [Copyright &y& Elsevier]

Standard ordering of aggregation operators in several distinguished classes is characterized. Lower and upper bounds of (finite) systems in those classes of aggregation operators are discussed. Several examples are given. [Copyright &y& Elsevier]

We will prove that every pair of quasi-arithmetic means satisfying certain smoothness assumption has the supremum and the infimum in this set. More precisely, if f and g are C 2 functions with nowhere vanishing first derivative then there exists a function h such that: (i) A [ f ] ≤ A [ h ] , (ii) A [ g ] ≤ A [ h ] , and (iii) for every continuous strictly monotone function s : I → R A [ f ] ≤ A [ s ] and A [ g ] ≤ A [ s ] implies A [ h ] ≤ A [ s ] (A [ f ] stands for a quasi-arithmetic mean). Moreover, h ∈ C 2 , h ′ ≠ 0 , and it is a solution of the differential equation h ″ h ′ = max ⁡ ( f ″ f ′ , g ″ g ′ ). We also provide some extension to a finite family of means and dual (reflected) result. [ABSTRACT FROM AUTHOR]