Measure functions for frames

This paper addresses the natural question: “How should frames be compared?” We answer this question by quantifying the overcompleteness of all frames with the same index set. We introduce the concept of a frame measure function: a function which maps each frame to a continuous function. The comparison of these functions induces an equivalence and partial order that allows for a meaningful comparison of frames indexed by the same set. We define the ultrafilter measure function, an explicit frame measure function that we show is contained both algebraically and topologically inside all frame measure functions. We explore additional properties of frame measure functions, showing that they are additive on a large class of supersets—those that come from so called non-expansive frames. We apply our results to the Gabor setting, computing the frame measure function of Gabor frames and establishing a new result about supersets of Gabor frames.     

Decomposition of conflict as a distribution on hypotheses in the framework on belief functions

In this paper, we address the problem of identifying the potential sources of conflict between information sources in the framework of belief function theory. To this aim, we propose a decomposition of the global measure of conflict as a function defined over the power set of the discernment frame. This decomposition, which associates a part of the conflict to some hypotheses, allows identifying the origin of conflict, which is hence considered as “local” to some hypotheses. This is more informative than usual global measures of conflict or disagreement between sources. Having shown the unicity of this decomposition, we illustrate its use on two examples. The first one is a toy example where the fact that conflict is mainly brought by one hypothesis allows identifying its origin. The second example is a real application, namely robot localization, where we show that focusing the conflict measure on the “favored” hypothesis (the one that would be decided) helps us to robustify the fusion process.     

Generalized information theory for hints

This paper develops a new uncertainty measure for the theory of hints that complies with the established semantics of statistical information theory and further satisfies all classical requirements for such a measure imposed in the literature. The proposed functional decomposes into conversant uncertainty measures and therefore discloses a new interpretation of the latters as well. By abstracting to equivalence classes of hints we transport the new measure to mass functions in Dempster-Shafer theory and analyse its relationship with the aggregate uncertainty, which currently is the only known functional for the Dempster-Shafer theory of evidence that satisfies the same set of properties. Moreover, the perspective of hints reveals that the standard independence notion in Dempster-Shafer theory called non-interactivity corresponds to an amalgamation of probabilistic independence and qualitative independence between frames of discernment. All results in this paper are developed for arbitrary families of compatible frames generalizing the very specialized multi-variate systems that are usually studied in information theory.     

Evidential reasoning in large partially ordered sets

The Dempster-Shafer theory of belief functions has proved to be a powerful formalism for uncertain reasoning. However, belief functions on a finite frame of discernment Ω are usually defined in the power set 2^Ω, resulting in exponential complexity of the operations involved in this framework, such as combination rules. When Ω is linearly ordered, a usual trick is to work only with intervals, which drastically reduces the complexity of calculations. In this paper, we show that this trick can be extrapolated to frames endowed with an arbitrary lattice structure, not necessarily a linear order. This principle makes it possible to apply the Dempster-Shafer framework to very large frames such as the power set, the set of partitions, or the set of preorders of a finite set. Applications to multi-label classification, ensemble clustering and preference aggregation are demonstrated.     

A proof for the positive definiteness of the Jaccard index matrix

In this paper we provide a proof for the positive definiteness of the Jaccard index matrix used as a weighting matrix in the Euclidean distance between belief functions defined in Jousselme et al. [13]. The idea of this proof relies on the decomposition of the matrix into an infinite sum of positive semidefinite matrices. The proof is valid for any size of the frame of discernment but we provide an illustration for a frame of three elements. The Jaccard index matrix being positive definite guaranties that the associated Euclidean distance is a full metric and thus that a null distance between two belief functions implies that these belief functions are strictly identical.     

An uncertainty principle for finite frames

We consider the notion of uncertainty for finite frames. Using a difference operator inspired by the Gauss-Hermite differential equation we obtain a time-frequency measure for finite frames. We then find the minimizer of the measure over all equal norm Parseval frames, dependent on the dimension of the space and the number of elements in the frame. Next we show that given a frame one can find the dual frame that minimizes this time-frequency measure, generalizing some work of Daubechies, Landau and Landau to the finite case and extending some recent work on Sobolev duals for finite frames.     

Propagating belief functions in qualitative Markov trees

This article is concerned with the computational aspects of combining evidence within the theory of belief functions. It shows that by taking advantage of logical or categorical relations among the questions we consider, we can sometimes avoid the computational complexity associated with brute-force application of Dempster's rule.The mathematical setting for this article is the lattice of partitions of a fixed overall frame of discernment. Different questions are represented by different partitions of this frame, and the categorical relations among these questions are represented by relations of qualitative conditional independence or dependence among the partitions. Qualitative conditional independence is a categorical rather than a probabilistic concept, but it is analogous to conditional independence for random variables.We show that efficient implementation of Dempster's rule is possible if the questions or partitions for which we have evidence are arranged in a qualitative Markov tree—a tree in which separations indicate relations of qualitative conditional independence. In this case, Dempster's rule can be implemented by propagating belief functions through the tree.     

On the construction of frames for spaces of distributions

We introduce a new method for constructing frames for general distribution spaces and employ it to the construction of frames for Triebel-Lizorkin and Besov spaces on the sphere. Conceptually, our scheme allows the freedom to prescribe the nature, form or some properties of the constructed frame elements. For instance, our frame elements on the sphere consist of smooth functions supported on small shrinking caps.     

On the Evaluation of Uncertain Courses of Action

We consider the problem of decision making under uncertainty. The fuzzy measure is introduced as a general way of representing available information about the uncertainty. It is noted that generally in uncertain environments the problem of comparing alternative courses of action is difficult because of the multiplicity of possible outcomes for any action. One approach is to convert this multiplicity of possible of outcomes associated with an alternative into a single value using a valuation function. We describe various ways of providing a valuation function when the uncertainty is represented using a fuzzy measure. We then specialize these valuation functions to the cases of probabilistic and possibilistic uncertainty.     

Sparse matrices in frame theory

Frame theory is closely intertwined with signal processing through a canon of methodologies for the analysis of signals using (redundant) linear measurements. The canonical dual frame associated with a frame provides a means for reconstruction by a least squares approach, but other dual frames yield alternative reconstruction procedures. The novel paradigm of sparsity has recently entered the area of frame theory in various ways. Of those different sparsity perspectives, we will focus on the situations where frames and (not necessarily canonical) dual frames can be written as sparse matrices. The objective for this approach is to ensure not only low-complexity computations, but also high compressibility. We will discuss both existence results and explicit constructions.     

Active classification using belief functions and information gain maximization

Obtaining reliable estimates of the parameters of a probabilistic classification model is often a challenging problem because the amount of available training data is limited. In this paper, we present a classification approach based on belief functions that makes the uncertainty resulting from limited amounts of training data explicit and thereby improves classification performance. In addition, we model classification as an active information acquisition problem where features are sequentially selected by maximizing the expected information gain with respect to the current belief distribution, thus reducing uncertainty as quickly as possible. For this, we consider different measures of uncertainty for belief functions and provide efficient algorithms for computing them. As a result, only a small subset of features need to be extracted without negatively impacting the recognition rate. We evaluate our approach on an object recognition task where we compare different evidential and Bayesian methods for obtaining likelihoods from training data and we investigate the influence of different uncertainty measures on the feature selection process.     

Recovery of signals from unordered partial frame coefficients

In this paper, we study the feasibility and stability of recovering signals in finite-dimensional spaces from unordered partial frame coefficients. We prove that with an almost self-located robust frame, any signal except from a Lebesgue measure zero subset can be recovered from its unordered partial frame coefficients. However, the recovery is not necessarily stable with almost self-located robust frames. We propose a new class of frames, namely self-located robust frames, that ensures stable recovery for any input signal with unordered partial frame coefficients. In particular, the recovery is exact whenever the received unordered partial frame coefficients are noise-free. We also present some characterizations and constructions for (almost) self-located robust frames. Based on these characterizations and construction algorithms, we prove that any randomly generated frame is almost surely self-located robust. Moreover, frames generated with cube roots of different prime numbers are also self-located robust.     

Measures of uncertainty for imprecise probabilities: An axiomatic approach

The aim of this paper is to formalize, within a broad range of theories of imprecise probabilities, the notion of a total, aggregate measure of uncertainty and its various disaggregations into measures of nonspecificity and conflict. As a framework for facilitating this aim, we introduce a system of well-justified axiomatic requirements for such measures. It is shown that these requirements can be equivalently defined for belief functions and credal sets. Some important consequences are then derived within this framework, which clarify the role of various uncertainty measures proposed in the literature. Moreover, some well-defined new open problems for future research also emerge from the introduced framework.     

Necessary conditions and sufficient conditions of irregular shearlet frames

In this paper, necessary conditions and sufficient conditions for the irregular shearlet systems to be frames are studied. We show that the irregular shearlet systems to possess upper frame bounds, the space‐scale‐shear parameters must be relatively separated. We prove that if the irregular shearlet systems possess the lower frame bound and the space‐scale‐shear parameters satisfy certain condition, then the lower shearlet density is strictly positive. We apply these results to systems consisting only of dilations, obtaining some new results relating density to the frame properties of these systems. We prove that for a feasible class of shearlet generators introduced by P. Kittipoom et al., each relatively separated sequence with sufficiently hight density will generate a frame. Explicit frame bounds are given. We also study the stability of shearlet frames and show that a frame generated by certain shearlet function remains a frame when the space‐scale‐shear parameters and the generating function undergo small perturbations. Explicit stability bounds are given. Using pseudo‐spline functions of type I and II, we construct a family of irregular shearlet frames consisting of compactly supported shearlets to illustrate our results.     

Belief functions on real numbers

We generalize the TBM (transferable belief model) to the case where the frame of discernment is the extended set of real numbers , under the assumptions that ‘masses’ can only be given to intervals. Masses become densities, belief functions, plausibility functions and commonality functions become integrals of these densities and pignistic probabilities become pignistic densities. The mathematics of belief functions become essentially the mathematics of probability density functions on .     

Discrete Moving Frames on Lattice Varieties and Lattice-Based Multispaces

In this paper, we develop the theory of the discrete moving frame in two different ways. In the first half of the paper, we consider a discrete moving frame defined on a lattice variety and the equivalence classes of global syzygies that result from the first fundamental group of the variety. In the second half, we consider the continuum limit of discrete moving frames as a local lattice coalesces to a point. To achieve a well-defined limit of discrete frames, we construct multispace, a generalisation of the jet bundle that also generalises Olver’s one-dimensional construction. Using interpolation to provide coordinates, we prove that it is a manifold containing the usual jet bundle as a submanifold. We show that continuity of a multispace moving frame ensures that the discrete moving frame converges to a continuous one as lattices coalesce. The smooth frame is, at the same time, the restriction of the multispace frame to the embedded jet bundle. We prove further that the discrete invariants and syzygies approximate their smooth counterparts. In effect, a frame on multispace allows smooth frames and their discretisations to be studied simultaneously. In our last chapter we discuss two important applications, one to the discrete variational calculus, and the second to discrete integrable systems. Finally, in an appendix, we discuss a more general result concerning equicontinuous families of discretisations of moving frames, which are consistent with a smooth frame.     

Frames and Coorbit Theory on Homogeneous Spaces with a Special Guidance on the Sphere

The topic of this article is a generalization of the theory of coorbit spaces and related frame constructions to Banach spaces of functions or distributions over domains and manifolds. As a special case one obtains modulation spaces and Gabor frames on spheres. Group theoretical considerations allow first to introduce generalized wavelet transforms. These are then used to define coorbit spaces on homogeneous spaces, which consist of functions having their generalized wavelet transform in some weighted L_p space. We also describe natural ways of discretizing those wavelet transforms, or equivalently to obtain atomic decompositions and Banach frames for the corresponding coorbit spaces. Based on these facts we treat aspects of nonlinear approximation and show how the new theory can be applied to the Gabor transform on spheres. For the S¹ we exhibit concrete examples of admissible Gabor atoms which are very closely related to uncertainty minimizing states.     

基于TOPSIS的模糊群体多属性决策方法

针对模糊群体多属性决策问题,给出一种基于理想点法(TOPSIS)的多属性决策方法.方法先用三角模糊数的形式表示专家评价值的模糊性和不确定性,而后考虑了专家在不同评价属性中的重要程度和意见的相似度,并将专家意见进行集结得到专家群体关于方案集的模糊决策矩阵,最后定义了三角模糊数形式的正负理想方案,通过计算各方案与正负理想方案的距离以及各方案与理想点的相对接近度,最终确定最优方案.通过实例分析说明了该方法的可行性和有效性.     

Theory of evidence — A survey of its mathematical foundations,applications and computational aspects

The mathematical theory of evidence has been introduced by Glenn Shafer in 1976 as a new approach to the representation of uncertainty. This theory can be represented under several distinct but more or less equivalent forms. Probabilistic interpretations of evidence theory have their roots in Arthur Dempster's multivalued mappings of probability spaces. This leads to random set and more generally to random filter models of evidence. In this probabilistic view evidence is seen as more or less probable arguments for certain hypotheses and they can be used to support those hypotheses to certain degrees. These degrees of support are in fact the reliabilities with which the hypotheses can be derived from the evidence. Alternatively, the mathematical theory of evidence can be founded axiomatically on the notion of belief functions or on the allocation of belief masses to subsets of a frame of discernment. These approaches aim to present evidence theory as an extension of probability theory. Evidence theory has been used to represent uncertainty in expert systems, especially in the domain of diagnostics. It can be applied to decision analysis and it gives a new perspective for statistical analysis. Among its further applications are image processing, project planning and scheduling and risk analysis. The computational problems of evidence theory are well understood and even though the problem is complex, efficient methods are available.Research partly supported by Grants No. 21-30186.90 and 21-32660.91 of the Swiss National Foundation for Scientific Research.     

三角模糊偏好下冲突型多属性群决策方法研究

针对三角模糊偏好下冲突型群决策问题,本文提出一种新的决策方法。在冲突消解阶段,用三角模糊数表示决策专家偏好,定义两三角模糊数型偏好矢量间的相似度,通过计算专家对各个方案的偏好矢量与各方案的群偏好矢量间的相似度,以此为基础定义专家的冲突测度。给出阈值和协商机制调控专家的冲突测度,直到所有的专家的冲突测度都小于给定阈值,进入决策阶段。在决策阶段,利用三角模糊数的期望函数确定属性权重,计算各个方案群偏好矢量与理想方案偏好矢量之间的加权相似度,由加权相似度大小排列决策,选出最优方案。最后给出案例应用,利用Matlab画出各方案的冲突测度图,数值结果表明本文方法的可行性及有效性。