Showing papers on "Equivalence class published in 2020"
TL;DR: This study introduces a classified nested equivalence class (CNEC)-based approach to calculate the information-entropy-based significance for feature selection using rough set theory and shows the use of CNECs is shown to significantly enhance three representative entropy-based feature selection algorithms that use roughSet theory.
34 citations
TL;DR: This paper analyzes the variations of equivalence classes, decision classes, conditional probability, internal grade and external grade in dynamic data sets while objects vary sequentially or simultaneously over time and proposes the updating mechanisms for two types of Dq-DTRS models from incremental perspective in dynamic decision information systems with the sequential and batch variations of objects.
Abstract: Double-quantitative decision-theoretic rough sets (Dq-DTRS) provide more comprehensive description methods for rough approximations of concepts, which lay foundations for the development of attribute reduction and rule extraction of rough sets. Existing researches on concept approximations of Dq-DTRS pay more attention to the equivalence class of each object in approximating a concept, and calculate concept approximations from the whole data set in a batch. This makes the calculation of approximations time consuming in dynamic data sets. In this paper, we first analyze the variations of equivalence classes, decision classes, conditional probability, internal grade and external grade in dynamic data sets while objects vary sequentially or simultaneously over time. Then we propose the updating mechanisms for the concept approximations of two types of Dq-DTRS models from incremental perspective in dynamic decision information systems with the sequential and batch variations of objects. Meanwhile, we design incremental sequential insertion, sequential deletion, batch insertion, batch deletion algorithms for two Dq-DTRS models. Finally, we present experimental comparisons showing the feasibility and efficiency of the proposed incremental approaches in calculating approximations and the stability of the incremental updating algorithms from the perspective of the runtime under different inserting and deleting ratios and parameter values.
29 citations
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TL;DR: This work compares multi-layer Graph Neural Networks with a simplified alternative that is called Graph-Augmented Multi-Layer Perceptrons (GA-MLPs), which first augments node features with certain multi-hop operators on the graph and then applies an MLP in a node-wise fashion.
Abstract: From the perspective of expressive power, this work compares multi-layer Graph Neural Networks (GNNs) with a simplified alternative that we call Graph-Augmented Multi-Layer Perceptrons (GA-MLPs), which first augments node features with certain multi-hop operators on the graph and then applies an MLP in a node-wise fashion. From the perspective of graph isomorphism testing, we show both theoretically and numerically that GA-MLPs with suitable operators can distinguish almost all non-isomorphic graphs, just like the Weifeiler-Lehman (WL) test. However, by viewing them as node-level functions and examining the equivalence classes they induce on rooted graphs, we prove a separation in expressive power between GA-MLPs and GNNs that grows exponentially in depth. In particular, unlike GNNs, GA-MLPs are unable to count the number of attributed walks. We also demonstrate via community detection experiments that GA-MLPs can be limited by their choice of operator family, as compared to GNNs with higher flexibility in learning.
22 citations
TL;DR: In this article, it was shown that the stable endomorphism rings of rigid objects in a suitable Frobenius category have only finitely many basic algebras in their derived equivalence class and these are precisely the contraction algebra of objects obtained by iterated mutation.
Abstract: We prove that the stable endomorphism rings of rigid objects in a suitable Frobenius category have only finitely many basic algebras in their derived equivalence class and that these are precisely the stable endomorphism rings of objects obtained by iterated mutation. The main application is to the Homological Minimal Model Programme. For a 3-fold flopping contraction
$$f :X \rightarrow {\mathrm{Spec}\;}\,R$$
, where X has only Gorenstein terminal singularities, there is an associated finite dimensional algebra
$$A_{{\text {con}}}$$
known as the contraction algebra. As a corollary of our main result, there are only finitely many basic algebras in the derived equivalence class of
$$A_{\text {con}}$$
and these are precisely the contraction algebras of maps obtained by a sequence of iterated flops from f. This provides evidence towards a key conjecture in the area.
18 citations
TL;DR: It is shown that the sequence of dimensions of the linear spaces generated by a given rank-metric code together with itself under several applications of a field automorphism are invariants for the whole equivalence class of the code, which give rise to easily computable criteria to check if two codes are inequivalent.
15 citations
05 Jan 2020
TL;DR: A natural class of range query problems is defined, and it is proved that all problems within this class have the same time complexity (up to polylogarithmic factors); the equivalence is very general, and even applies to online algorithms.
Abstract: We define a natural class of range query problems, and prove that all problems within this class have the same time complexity (up to polylogarithmic factors). The equivalence is very general, and even applies to online algorithms. This allows us to obtain new improved algorithms for all of the problems in the class. We then focus on the special case of the problems when the queries are offline and the number of queries is linear. We show that our range query problems are runtime-equivalent (up to polylogarithmic factors) to counting for each edge e in an m-edge graph the number of triangles through e. This natural triangle problem can be solved using the best known triangle counting algorithm, running in O(m2ω/(ω+1)) ⩽ O(m1.41) time. Moreover, if ω = 2, the O(m2ω/(ω+1)) running time is known to be tight (within mo(1) factors) under the 3SUM Hypothesis. In this case, our equivalence settles the complexity of the range query problems. Our problems constitute the first equivalence class with this peculiar running time bound. To better understand the complexity of these problems, we also provide a deeper insight into the family of triangle problems, in particular showing black-box reductions between triangle listing and per-edge triangle detection and counting. As a byproduct of our reductions, we obtain a simple triangle listing algorithm matching the state-of-the-art for all regimes of the number of triangles. We also give some not necessarily tight, but still surprising reductions from variants of matrix products, such as the (min, max)-product.
15 citations
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TL;DR: This paper focuses on the problem of robust estimation of tree-structured Ising models, and proves that this problem is unidentifiable, however, this unidentifiability is limited to a small equivalence class of trees formed by leaf nodes exchanging positions with their neighbors.
Abstract: We consider the task of learning Ising models when the signs of different random variables are flipped independently with possibly unequal, unknown probabilities. In this paper, we focus on the problem of robust estimation of tree-structured Ising models. Without any additional assumption of side information, this is an open problem. We first prove that this problem is unidentifiable, however, this unidentifiability is limited to a small equivalence class of trees formed by leaf nodes exchanging positions with their neighbors. Next, we propose an algorithm to solve the above problem with logarithmic sample complexity in the number of nodes and polynomial run-time complexity. Lastly, we empirically demonstrate that, as expected, existing algorithms are not inherently robust in the proposed setting whereas our algorithm correctly recovers the underlying equivalence class.
10 citations
17 Feb 2020
TL;DR: The classical concept of reducibility in hypergroups is extended to the fuzzy case and new fundamental relations are defined on a crisp hypergroup endowed with a fuzzy set, that lead to the concept of fuzzy reduced hypergroup.
Abstract: The fuzzyfication of hypercompositional structures has developed in several directions. In this note we follow one direction and extend the classical concept of reducibility in hypergroups to the fuzzy case. In particular we define and study the fuzzy reduced hypergroups. New fundamental relations are defined on a crisp hypergroup endowed with a fuzzy set, that lead to the concept of fuzzy reduced hypergroup. This is a hypergroup in which the equivalence class of any element, with respect to a determined fuzzy set, is a singleton. The most well known fuzzy set considered on a hypergroup is the grade fuzzy set, used for the study of the fuzzy grade of a hypergroup. Based on this, in the second part of the paper, we study the fuzzy reducibility of some particular classes of crisp hypergroups with respect to the grade fuzzy set.
8 citations
TL;DR: A nested equivalence class (NEC) approach is introduced to calculate dependency, which not only simplifies dependency calculation but also reduces the universe correspondingly, in most cases.
8 citations
TL;DR: In this article, the authors considered six possible criticisms of using yield to study equivalence classes and each criticism was supported; instead, each disclosed a nonyield factor that could play a critical role in the measurement of class formation but has not yet been explored experimentally.
Abstract: “Yield,” the percentage of participants in a group who form a set of equivalence classes, has been used very broadly to identify the effect of different training protocols on class formation and expansion, identify variables that enhance the immediate emergence of these classes, and characterize the differential relatedness of class members. In addition, yield is now being used to document the formation of educationally relevant equivalence classes. To further understand the value of using yield, we considered six possible criticisms of its use to study equivalence classes. Upon analysis, each criticism was supported; instead, each disclosed a nonyield factor that could play a critical role in the measurement of class formation but has not yet been explored experimentally. Finally, yield cannot be replaced with trial-based measures of responding or vice versa; rather, both types of measures are needed to obtain a comprehensive understanding of equivalence class formation.
8 citations
TL;DR: In this article, the norm in classical Sobolev spaces can be expressed as a difference quotient, which can be used to generalize the space to the fractional smoothness case.
Abstract: The norm in classical Sobolev spaces can be expressed as a difference quotient. This expression can be used to generalize the space to the fractional smoothness case. Because the difference quotien...
TL;DR: This paper analyzes the structure and generation of p-augmented matrix, and proposes an intelligent acquisition algorithm of information equivalence class based on matrix reasoning, and analyzes two types of information fusion, namely information redundancy fusion and information supplement fusion.
Abstract: The development of information technology brings the challenge of data redundancy and data shortage to information fusion. Based on the dynamic boundary characteristics of p-set, this paper analyzes the structure and generation of p-augmented matrix, and then analyzes the dynamic generation of information equivalence class, and then proposes an intelligent acquisition algorithm of information equivalence class based on matrix reasoning. In addition, this paper analyzes two types of information fusion, namely information redundancy fusion and information supplement fusion. Then, the relationship among redundant information fusion, supplementary information fusion, and information equivalence classes is analyzed. Finally, this paper presents the application of intelligent acquisition of information equivalence class in information retrieval.
TL;DR: In this paper, a pertinent equivalence relation defined on the spaces of almost periodic functions in Bohr, Stepanov, Weyl and Besicovitch's sense is defined and the condition of almost periodicity of a function in any of these generalized spaces can be interpreted in the way that the closure of its set of translates coincides with its corresponding equivalence class.
Abstract: Our paper is focused on spaces of generalized almost periodic functions which, as in classical Fourier analysis, are associated with a Fourier series with real frequencies. In fact, based on a pertinent equivalence relation defined on the spaces of almost periodic functions in Bohr, Stepanov, Weyl and Besicovitch’s sense, we refine the Bochner-type property by showing that the condition of almost periodicity of a function in any of these generalized spaces can be interpreted in the way that, with respect to the topology of each space, the closure of its set of translates coincides with its corresponding equivalence class.
01 Jan 2020
TL;DR: In this paper, the spectral representation and classification of Boolean functions have been studied and found to be useful in logic design and testing, and a single efficient recursive classification algorithm has been presented to determine all equivalence classes for small n, and to determine if two functions fall in the same equivalence class for larger n.
Abstract: The spectral representation and classification of Boolean functions have been previously studied and found to be useful in logic design and testing. Spectral techniques also have potential application for reversible and quantum circuits. This paper considers the partitioning of Boolean functions into linear, affine and spectral equivalence classes. A single efficient recursive classification algorithm is presented. It can be used to determine all equivalence classes for small n, and to determine if two functions fall in the same equivalence class for larger n. For two functions in the same equivalence class, the algorithm identifies a sequence of translations to map one to the other.
01 Jul 2020
TL;DR: The model checking algorithm for another epistemic operator $\mathcal{C}$ is designed and developed, and examples of privacy protocols illustrate the usefulness of the model and method.
Abstract: Computation Tree Logic of Knowledge (CTLK) can specify many requirements of privacy or security of multi-agent systems (MAS). In our previous paper, we defined Knowledge-oriented Petri Nets (KPN) to model MAS. A KPN is a special Petri net in which some places are used to represent knowledge. Consequently, KPN can formally represent not only the interaction/collaboration process of multiple agents but also their epistemic evolutions. These epistemic evolutions are closely related to privacy and security of MAS. Then we defined the similar reachability graphs of KPN and constructed the equivalence classes of knowledge for each agent. We considered an epistemic operator $\mathcal{K}$ and designed its model checking algorithm based on similar reachability graph and equivalence class. In this paper, we design the model checking algorithm for another epistemic operator $\mathcal{C}$. Additionally, we develop a related tool. Examples of privacy protocols illustrate the usefulness of our model and method.
24 Oct 2020
TL;DR: In this paper, the configuration space of adaptive modular robots with a triangular structure based on extended binary trees is described, and a reconfiguration is performed by populating the binary tree indices of a desired target configuration in an ascending manner, moving modules along the surface of the robot.
Abstract: In this paper, we present a novel description for the configuration space of adaptive modular robots with a triangular structure based on extended binary trees. In general, binary trees can serve as a representation of kinematic trees with a maximum of two immediate descendants per element. Kinematic loops are incorporated in the tree structure by an ingenious extension of the binary tree indices. The introduction of equivalence classes then allows a unique mathematical description of specific configurations of the robot system. Subsequently, we show how the extended binary tree can serve as a systematic tool for reconfiguration planning, allowing to solve the self-reconfiguration problem for modular robots with a triangular structure, which has as yet no general solution. Reconfiguration is performed by populating the binary tree indices of a desired target configuration in an ascending manner, moving modules along the surface of the robot. We demonstrate the planning algorithm on a simple example and conclude by outlining a way to translate the individual reconfiguration steps to specific module movement commands.
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TL;DR: The proposed method, GENNI, allows us to efficiently identify parameters that are functionally equivalent and then visualise the subspace of the resulting equivalence class, to better explore questions surrounding identifiability.
Abstract: We propose an efficient algorithm to visualise symmetries in neural networks. Typically, models are defined with respect to a parameter space, where non-equal parameters can produce the same input-output map. Our proposed method, GENNI, allows us to efficiently identify parameters that are functionally equivalent and then visualise the subspace of the resulting equivalence class. By doing so, we are now able to better explore questions surrounding identifiability, with applications to optimisation and generalizability, for commonly used or newly developed neural network architectures.
21 Jan 2020
TL;DR: A virtual link is a generalization of a classical link that is defined as an equivalence class of certain diagrams, called virtual link diagrams, further generalized to a twisted link as mentioned in this paper.
Abstract: A virtual link is a generalization of a classical link that is defined as an equivalence class of certain diagrams, called virtual link diagrams. It is further generalized to a twisted link. Twisted links are in one-to-one correspondence with stable equivalence classes of links in oriented thickenings of (possibly non-orientable) closed surfaces. By definition, equivalent virtual links are also equivalent as twisted links. In this paper, we discuss when two virtual links are equivalent as twisted links, and give a necessary and sufficient condition for this to be the case.
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TL;DR: In this article, the authors give a formal meaning to the symbol $dX_t$ and to Ito stochastic differential equations at an instant in time by interpreting it as an equivalence class on a subspace of continuous semimartingales.
Abstract: We give a formal meaning to the symbol $dX_t$ and to Ito stochastic differential equations at an instant in time. We interpret $dX_t$ as an equivalence class on a subspace of continuous semimartingales. We define a vector space structure on this set of equivalence classes and deduce key properties consistent with classical Ito integration theory. In particular, we link our notion of a differential with Ito integration via a Stochastic Fundamental Theorem of Calculus.
19 Aug 2020
TL;DR: After surveying the published literature, and searching for similar commercial products, the authors did not find a comparable technology, to assess the contributions made by Caliper, at the time of writing, and so it is claimed that Caliper is the only product of its kind today.
Abstract: Empirical studies of Program Performance, are limited by the choice and the resulting bias, from the input samples used in the experiment. Estimation and Prediction based on static analysis, are more universal, superior and widely accepted. However the higher language artifacts such as Procedures, Loops, Conditionals and Recursion which ease program development can be an hindrance to quality analysis and performance study, both in terms of time and effort spent and in some extreme cases making it impractical. However, we could transform the program, eliminate the constraints imposed by these program structures and greatly ease the process of quality analysis and performance study. This process may also reduce the errors in the estimation, and help deliver timely results, when there is still an opportunity to use them in a later analysis phase. We propose transformations prior to estimation, such as Procedure Call Expansion, Loop Unrolling and Control Predication collectively referred to as Program Shape Flattening here with the structural hindrances themselves referred to as the Program Shape. The outcome of this transformation, is sequential code that is easy to work with. Specifically, for parallel performance estimations, we now have code that is free from Control Dependencies. We use the concept of Equivalence Classes to group statements based on their Data Dependence behavior. Statements that belong to an Equivalence Class are mutually dependent directly or transitively. On the other hand statements that belong to separate Equivalence Classes are dependence free and can be run in parallel without compromising on the program correctness. With this arrangement of program statements we claim that the program run time is now equal to the run time of the class that runs the longest. While this scheme of grouping program instructions, can be viewed as a method of parallel conversion, we use this method here specifically for parallel performance estimation and prediction. After surveying the published literature, and searching for similar commercial products, we did not find a comparable technology, to assess the contributions made by Caliper, at the time of writing, and so we claim that Caliper is the only product of its kind today.
TL;DR: In this paper, a partial order on neighborhood equivalence classes of maximally spread essential multibranched surfaces embedded in a 3-manifold is introduced, and it is shown that if a maximal spread essential multi-branched surface is atoroidal and acylindrical, then its equivalence class is minimal with respect to the partial order.
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TL;DR: It is proved that the submonoid generated by the generators of the free nil-$2 group on $m$ generators is isomorphic to the quotient of thefree monoid $\{ 1, \ldots , m\}^{*}$ by the $2$-binomial equivalence.
Abstract: Two finite words $u$ and $v$ are $k$-binomially equivalent if, for each word $x$ of length at most $k$, $x$ appears the same number of times as a subsequence (i.e., as a scattered subword) of both $u$ and $v$. This notion generalizes abelian equivalence. In this paper, we study the equivalence classes induced by the $k$-binomial equivalence with a special focus on the cardinalities of the classes. We provide an algorithm generating the $2$-binomial equivalence class of a word. For $k \geq 2$ and alphabet of $3$ or more symbols, the language made of lexicographically least elements of every $k$-binomial equivalence class and the language of singletons, i.e., the words whose $k$-binomial equivalence class is restricted to a single element, are shown to be non context-free. As a consequence of our discussions, we also prove that the submonoid generated by the generators of the free nil-$2$ group on $m$ generators is isomorphic to the quotient of the free monoid $\{ 1, \ldots , m\}^{*}$ by the $2$-binomial equivalence.
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TL;DR: This work proposes to test, and when possible establish, an equivalence between two different artificial neural networks by attempting to construct a data-driven transformation between them, using manifold-learning techniques and employing diffusion maps with a Mahalanobis-like metric.
Abstract: We propose to test, and when possible establish, an equivalence between two different artificial neural networks by attempting to construct a data-driven transformation between them, using manifold-learning techniques. In particular, we employ diffusion maps with a Mahalanobis-like metric. If the construction succeeds, the two networks can be thought of as belonging to the same equivalence class.
We first discuss transformation functions between only the outputs of the two networks; we then also consider transformations that take into account outputs (activations) of a number of internal neurons from each network. In general, Whitney's theorem dictates the number of measurements from one of the networks required to reconstruct each and every feature of the second network. The construction of the transformation function relies on a consistent, intrinsic representation of the network input space.
We illustrate our algorithm by matching neural network pairs trained to learn (a) observations of scalar functions; (b) observations of two-dimensional vector fields; and (c) representations of images of a moving three-dimensional object (a rotating horse). The construction of such equivalence classes across different network instantiations clearly relates to transfer learning. We also expect that it will be valuable in establishing equivalence between different Machine Learning-based models of the same phenomenon observed through different instruments and by different research groups.
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TL;DR: This work proposes directed graphical models based on Hurdle conditional distributions parametrized in terms of polynomials in parent variables and their 0/1 indicators of being zero or nonzero, and shows that, under a natural and weak assumption, the exact directed acyclic graph of the authors' zero-inflated models can be identified.
Abstract: Modern RNA sequencing technologies provide gene expression measurements from single cells that promise refined insights on regulatory relationships among genes. Directed graphical models are well-suited to explore such (cause-effect) relationships. However, statistical analyses of single cell data are complicated by the fact that the data often show zero-inflated expression patterns. To address this challenge, we propose directed graphical models that are based on Hurdle conditional distributions parametrized in terms of polynomials in parent variables and their 0/1 indicators of being zero or nonzero. While directed graphs for Gaussian models are only identifiable up to an equivalence class in general, we show that, under a natural and weak assumption, the exact directed acyclic graph of our zero-inflated models can be identified. We propose methods for graph recovery, apply our model to real single-cell RNA-seq data on T helper cells, and show simulated experiments that validate the identifiability and graph estimation methods in practice.
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TL;DR: This paper shows that the task computability of 2-process affine models is decidable and presents a complete hierarchy of the five equivalence classes of 1-process Affine models, defined for a system of two processes.
Abstract: An affine model of computation is defined as a subset of iterated immediate-snapshot runs, capturing a wide variety of shared-memory systems, such as wait-freedom, t-resilience, k-concurrency, and fair shared-memory adversaries. The question of whether a given task is solvable in a given affine model is, in general, undecidable. In this paper, we focus on affine models defined for a system of two processes. We show that the task computability of 2-process affine models is decidable and presents a complete hierarchy of the five equivalence classes of 2-process affine models.
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TL;DR: Motivated by the principle of functoriality in category theory, a new method is proposed that allows to tie role and positional analysis together and illustrated on two well-studied data sets in network science.
Abstract: A key concern in network analysis is the study of social positions and roles of actors in a network The notion of "position" refers to an equivalence class of nodes that have similar ties to other nodes, whereas a "role" is an equivalence class of compound relations that connect the same pairs of nodes An open question in network science is whether it is possible to simultaneously perform role and positional analysis Motivated by the principle of functoriality in category theory we propose a new method that allows to tie role and positional analysis together We illustrate our methods on two well-studied data sets in network science
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17 Apr 2020
TL;DR: In this article, a privacy protection method for track data release based on track segmentation is proposed, which comprises the following steps: applying an equivalence class division algorithm based on tracking segmentation filling to an original track data set accumulated by a location-based service application provider; applying a clustering group construction algorithm based in trajectory segmentation clustering to each equivalence classification; determining the starting time of a clustining group and performing dividing to obtain candidate clustering groups; traversing each candidate clustaging group, determining a trajectory set, and constructing trajectory segments outside a clustered
Abstract: The invention discloses a privacy protection method for track data release based on track segmentation. The privacy protection method comprises the following steps: applying an equivalence class division algorithm based on track segmentation filling to an original track data set accumulated by a location-based service application provider; applying a clustering group construction algorithm based on trajectory segmentation clustering to each equivalence class; determining the starting time of a clustering group and performing dividing to obtain candidate clustering groups; traversing each candidate clustering group, determining a trajectory set, and constructing trajectory segments outside a clustering group time interval; inserting the tracks which are not added into the clustering group into the clustering group with the same time interval and the nearest spatial position; and performing spatial disturbance on each position point on each track in each clustering group, and changing each clustering group into an anonymous track set as a track data set which can be directly published. According to the method, an equivalence class division algorithm is adopted, the number of deletedspace-time points is reduced, it is guaranteed that the equivalence class contains enough tracks, and the availability of the to-be-published data is improved.
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TL;DR: This paper proposes a lazy listing algorithm that, instead of listing all solutions, lists, in an efficient way, directly only the equivalence classes or one representative per class.
Abstract: When a problem has more than one solution, it is often important, depending on the underlying context, to list them all. Even when the listing can be done in polynomial delay, that is, spending no more than polynomial time to go from one solution to the next, this can be costly as the number of solutions themselves may be huge, including sometimes exponential. This paper addresses this problem by proposing what we called a \emph{lazy listing} algorithm. By this we mean that, instead of listing all solutions, we list, in an efficient way, directly only the equivalence classes or one representative per class. Besides the need to then provide an \emph{a priori} relation of equivalence between solutions, we place ourselves in the special context where the problem to be addressed, either after some preliminary polynomial time dynamic programming computation or directly from the start, leads to a decomposable tropical circuit.
TL;DR: In this article, the authors introduced coherent subspaces spanned by a finite number of coherent states, in a quantum system with Hilbert space that has odd prime dimension $d. The set of all coherent sub-spaces is partitioned into equivalence classes, with $d^2$ subspace in each class, and corresponding coherent projectors within an equivalence class have the closure under displacements property.
Abstract: Coherent subspaces spanned by a finite number of coherent states are introduced, in a quantum system with Hilbert space that has odd prime dimension $d$. The set of all coherent subspaces is partitioned into equivalence classes, with $d^2$ subspaces in each class.The corresponding coherent projectors within an equivalence class, have the `closure under displacements property' and also resolve the identity. Different equivalence classes provide different granularisation of the Hilbert space, and they form a partial order `coarser' (and `finer'). In the case of a two-dimensional coherent subspace spanned by two coherent states, the corresponding projector (of rank $2$) is different than the sum of the two projectors to the subspaces related to each of the two coherent states. We quantify this with `non-addditivity operators' which are a measure of quantum interference in phase space, and also of the non-commutativity of the projectors. Generalized $Q$ and $P$ functions of density matrices, which are based on coherent projectors in a given equivalence class, are introduced. Analogues of the Lorenz values and the Gini index (which are popular quantities in Mathematical Economics) are used here to quantify the inequality in the distribution of the $Q$ function of a quantum state, within the granular structure of the Hilbert space....
TL;DR: In this paper, the authors introduced coherent subspaces spanned by a finite number of coherent states, in a quantum system with Hilbert space that has odd prime dimension $d. The set of all coherent sub-spaces is partitioned into equivalence classes, with $d^2$ subspace in each class, and corresponding coherent projectors within an equivalence class have the closure under displacements property.
Abstract: Coherent subspaces spanned by a finite number of coherent states are introduced, in a quantum system with Hilbert space that has odd prime dimension $d$. The set of all coherent subspaces is partitioned into equivalence classes, with $d^2$ subspaces in each class.The corresponding coherent projectors within an equivalence class, have the `closure under displacements property' and also resolve the identity. Different equivalence classes provide different granularisation of the Hilbert space, and they form a partial order `coarser' (and `finer'). In the case of a two-dimensional coherent subspace spanned by two coherent states, the corresponding projector (of rank $2$) is different than the sum of the two projectors to the subspaces related to each of the two coherent states. We quantify this with `non-addditivity operators' which are a measure of quantum interference in phase space, and also of the non-commutativity of the projectors. Generalized $Q$ and $P$ functions of density matrices, which are based on coherent projectors in a given equivalence class, are introduced. Analogues of the Lorenz values and the Gini index (which are popular quantities in Mathematical Economics) are used here to quantify the inequality in the distribution of the $Q$ function of a quantum state, within the granular structure of the Hilbert space....