Subset Of Space Research Articles

A Four-Grade Astrocytoma Classification System with Optimal Feature Selection Using Ant Colony Optimization Algorithm

An efficient Four-Grade Astrocytoma brain tumor classification is proposed in this research work. The work highlights the optimal feature selection using a hybrid combination of Ant Colony Optimization (ACO) algorithm and machine learning techniques, leading to better classification accuracy. A diverse set of features including spatial and frequency domain features will provide better discrimination of patterns. ACO provides a robust and efficient mechanism for exploring the vast space of possible feature subsets. A total of 135 features are extracted including Tamura features, Gabor Wavelet Features, Coiflet Wavelet Coefficients, Speeded Up Robust Features, and GLCM features. Ant Colony Optimization is used to find out the optimum set of features. The selected feature set is used for training a Decision Tree classifier, where the Astrocytoma MRI images are categorized into four grades of astrocyoma namely, pilocytic astrocytoma, diffuse astrocytoma, anaplastic astrocytoma and glioblastoma multiforne. Classification is also performed using KNN classifier, SVM classifier and Ensemble Bagged Tree Classifier. Performance metrics like accuracy, sensitivity, specificity, precision and FScore are utilized to evaluate the proposed system’s performance. It is found out that the decision tree classifier outperforms the other classifiers. Thus this integrated approach of ACO with machine learning classifiers results in improved Astrocytoma classification performance aiding clinicians in making accurate treatment decisions.

On the general fractal dimensions of hyperspace of compact sets

Consider a separable metric space (X,d), and let (K(X),d˜) denote the space of non-empty compact subsets of X equipped with the Hausdorff metric. This paper aims to introduce and investigate the concepts of two general fractal dimensions and general dimensions within the framework of (K(X),d˜). In particular, we explore a relationship between the general fractal dimensions of a set Z of a self-similar sequence space and their counterparts in the space of compact subsets K(Z).

Metric Approximation of Set-Valued Functions of Bounded Variation by Integral Operators

AbstractWe introduce an adaptation of integral approximation operators to set-valued functions (SVFs, multifunctions), mapping a compact interval [a, b] into the space of compact non-empty subsets of $${\mathbb {R}}^d$$ R d . All operators are adapted by replacing the Riemann integral for real-valued functions by the weighted metric integral for SVFs of bounded variation with compact graphs. For such a SVF F, we obtain pointwise error estimates for sequences of integral operators at points of continuity, leading to convergence at such points to F. At points of discontinuity of F, we derive estimates, which yield the convergence to a certain set described in terms of the metric selections of F. To obtain these estimates we refine and extend known results on approximation of real-valued functions by integral operators. Our analysis uses recently defined one-sided local quasi-moduli at points of discontinuity and several notions of local Lipschitz property at points of continuity. We also provide a global approach for error bounds. A multifunction F is represented by the set of all its metric selections, while its approximation (its image under the operator) is represented by the set of images of these metric selections under the operator. A bound on the Hausdorff distance between these two sets of single-valued functions in $$L^1$$ L 1 provides our global estimates. The theory is applied to concrete operators: the Bernstein–Durrmeyer operator and the Kantorovich operator.

Deep Curious Feature Selection: A Recurrent, Intrinsic-Reward Reinforcement Learning Approach to Feature Selection

Feature selection (FS) is an important step in the process of building machine learning-based models. The goal of the feature selection step is to find a small subset of features that will provide good prediction results by removing noisy, irrelevant, or repetitive features. Commonly used Wrapper methods use a machine learning model as a black box and its performance as the goal function for evaluating different features' subsets and selecting the best one. To avoid examining all possible subsets (an NP-hard problem), search algorithms are used to find the subsets to be examined, in a heuristic way. As exhaustive search methods are computationally complicated, most methods use simple and greedy search methods that yield only locally optimal results and are not sensitive to possible features interactions, which means that a feature may be chosen at the expense of two others that are more informative together. We analyze the problem of searching the features subset space with reference to two dimensions, namely, memory of past selected features subset and future selected features. We propose a new wrapper feature selection method based on the deep artificial curiosity framework, which implements intrinsic reward reinforcement learning, with long short-term memory unit (LSTM). This novel algorithm integrates these two elements of memory and future step. We show that our method, called the Deep Curious Feature Selection (DCFS) algorithm, deals with feature interactions, and provides a feature subset which improves the accuracy of learning models on artificial and real datasets.

On the operator of center of distances between the spaces of closed subsets of the real line

We study properties of an operator $S$ which assigns to compact subsets of $[0,1]$ their centers of distances. We consider its continuity points and its upper semicontinuity points as well as orbits and fixed points of this operator. We also compute centers of distances of some classic sets. Using properties of operator $S$ we show that the family of achievement sets is of the first category in the space of compact subsets of $[0,1]$.

The Hurewicz property and the Vietoris hyperspace

In this note, we characterize when the Vietoris space of compact subsets of a given space has the Hurewicz property in terms of a selection principle on the given space itself using k-covers and the notion of groupability introduced by Kočinac and Scheepers. We comment that the same technique establishes another equivalent condition to a space being Hurewicz in each of its finite powers. We end with some characterizations involving spaces of continuous functions and answer a question posed by Kočinac.

The Fermat–Steiner Problem in the Space of Compact Subsets of the Euclidean Plane

The Fermat–Steiner problem is the problem of finding all points of a metric space Y such that the sum of the distances from them to points of a certain fixed finite subset A of the space Y is minimal. In this paper, we examine the Fermat–Steiner problem in the case where Y is the space of compact subsets of the Euclidean plane endowed with the Hausdorff metric, and points of A are finite pairwise disjoint compact sets.

Efficient cross-validation traversals in feature subset selection

Sparse and robust classification models have the potential for revealing common predictive patterns that not only allow for categorizing objects into classes but also for generating mechanistic hypotheses. Identifying a small and informative subset of features is their main ingredient. However, the exponential search space of feature subsets and the heuristic nature of selection algorithms limit the coverage of these analyses, even for low-dimensional datasets. We present methods for reducing the computational complexity of feature selection criteria allowing for higher efficiency and coverage of screenings. We achieve this by reducing the preparation costs of high-dimensional subsets {mathscr {O}}({n}m^2) to those of one-dimensional ones {mathscr {O}}(m^2). Our methods are based on a tight interaction between a parallelizable cross-validation traversal strategy and distance-based classification algorithms and can be used with any product distance or kernel. We evaluate the traversal strategy exemplarily in exhaustive feature subset selection experiments (perfect coverage). Its runtime, fitness landscape, and predictive performance are analyzed on publicly available datasets. Even in low-dimensional settings, we achieve approximately a 15-fold increase in exhaustively generating distance matrices for feature combinations bringing a new level of evaluations into reach.

Iterated Minkowski sums, horoballs and north-south dynamics

Given a finite generating set A for a group \Gamma , we study the map W \mapsto WA as a topological dynamical system – a continuous self-map of the compact metrizable space of subsets of \Gamma . If the set A generates \Gamma as a semigroup and contains the identity, there are precisely two fixed points, one of which is attracting. This supports the initial impression that the dynamics of this map is rather trivial. Indeed, at least when \Gamma= \mathbb{Z}^d and A \subseteq \mathbb{Z}^d is a finite positively generating set containing the identity, the natural invertible extension of the map W \mapsto W+A is always topologically conjugate to the unique “north-south” dynamics on the Cantor set. In contrast to this, we show that various natural “geometric” properties of the finitely generated group (\Gamma,A) can be recovered from the dynamics of this map, in particular, the growth type and amenability of \Gamma . When \Gamma = \mathbb{Z}^d , we show that the volume of the convex hull of the generating set A is also an invariant of topological conjugacy. Our study introduces, utilizes and develops a certain convexity structure on subsets of the group \Gamma , related to a new concept which we call the sheltered hull of a set. We also relate this study to the structure of horoballs in finitely generated groups, focusing on the abelian case.

Toward the group completion of the Burau representation

Following Boardman-Vogt, McDuff, Segal, and others, we construct a monoidal topological groupoid or space of finite subsets of the plane, and interpret the Burau representation of knot theory as a topological quantum field theory defined on it. Its determinant or writhe is an invertible braided monoidal TQFT which group completes to define a Hopkins-Mahowald model for integral homology as an E 2 E_2 Thom spectrum. We use these ideas to construct an infinite cyclic (Alexander) cover for the space of finite subsets of C \mathbb {C} , and we argue that the TQFT defined by Burau is closely related to the SU(2)-valued Wess-Zumino-Witten model for string theory on R + 3 \mathbb {R}^3_+ .

The Epistemology of Nondeterminism

This paper proposes new semantics for propositional dynamic logic (PDL), replacing the standard relational semantics. Under these new semantics, program execution is represented as fundamentally deterministic (i.e., functional), while nondeterminism emerges as an epistemic relationship between the agent and the system: intuitively, the nondeterministic outcomes of a given process are precisely those that cannot be ruled out in advance. We formalize these notions using topology and the framework of dynamic topological logic (DTL) (Kremer and Mints in Ann Pure Appl Logic 131:133–158, 2005). We show that DTL can be used to interpret the language of PDL in a manner that captures the intuition above, and moreover that continuous functions in this setting correspond exactly to deterministic processes. We also prove that certain axiomatizations of PDL remain sound and complete with respect to the corresponding classes of dynamic topological models. Finally, we extend the framework to incorporate knowledge using the machinery of subset space logic (Dabrowski et al. in Ann Pure Appl Logic 78:73–110, 1996), and show that the topological interpretation of public announcements as given in Bjorndahl (in: van Ditmarsch and Sandu (eds) Jaakko Hintikka on knowledge and game theoretical semantics, outstanding contributions to Logic, vol 12, Springer, 2018) coincides exactly with a natural interpretation of test programs.

Viability and invariance of systems on metric spaces

We consider a generalized control system on a metric space and investigate necessary and sufficient conditions for viability and invariance of proper subsets, describing state constraints. Viability means that for every initial condition in the set of constraints we can find trajectories of control system starting at this condition and satisfying state constraints forever. Invariance means that every trajectory of control system starting in the set of constraints never violates them. As examples of application we consider controlled continuity equations on the metric space of Borel probability measures having compact support, endowed with the Wasserstein distance, and controlled morphological systems on the space of nonempty compacts subsets of the Euclidean space endowed with the Hausdorff metric. We also provide sufficient conditions for the existence and uniqueness of contingent solutions to the Hamilton–Jacobi–Bellman equation on proper metric spaces.

Stability and measurability of the modified lower dimension

The lower dimension dim L \dim _L is the dual concept of the Assouad dimension. As it fails to be monotonic, Fraser and Yu [Adv. Math. 329 (2018), pp. 273–328] introduced the modified lower dimension d i m {ML} dim_\textit {{ML}} by making the lower dimension monotonic with the simple formula d i m {ML} X = sup { dim L ⁡ E : E ⊂ X } dim_\textit {{ML}}X=\sup \{\dim _L E: E\subset X\} . As our first result we prove that the modified lower dimension is finitely stable in any metric space, answering a question of Fraser and Yu. We prove a new, simple characterization for the modified lower dimension. For a metric space X X let K ( X ) \mathcal {K}(X) denote the metric space of the non-empty compact subsets of X X endowed with the Hausdorff metric. As an application of our characterization, we show that the map d i m {ML} : K ( X ) → [ 0 , ∞ ] dim_\textit {{ML}}\colon \mathcal {K}(X)\to [0,\infty ] is Borel measurable. More precisely, it is of Baire class 2 2 , but in general not of Baire class 1 1 . This answers another question of Fraser and Yu. Finally, we prove that the modified lower dimension is not Borel measurable defined on the closed sets of ℓ 1 \ell ^1 endowed with the Effros Borel structure.

On non-surjective ε-isometric embeddings between Hausdorff metric spaces of compact convex subsets

Let X be a real Banach space, (cc(X),H) be the metric space of all non-empty compact convex subsets of X equipped with the Hausdorff metric H, f:(cc(X),H)→(cc(Y),H) be an ε-isometric embedding with ε≥0, σA(x⁎)=supa∈A⁡〈x⁎,a〉, ∀A∈(cc(X),H) and x⁎∈B(X⁎), and σf(A)(y⁎)=supy∈f(A)⁡〈y⁎,y〉, ∀A∈(cc(X),H) and y⁎∈B(Y⁎). We show that, for instance,(1)if X is Gâteaux smooth, then for any ϕ∈(σcc(X)−σcc(X)‾)⁎ there exists ψ∈(span‾[σf(cc(X))−σf(cc(X))])⁎ with ‖ψ‖=‖ϕ‖≡γ such that|〈ψ,σf(A)−σf(B)〉−〈ϕ,σA−σB〉|≤6γε,∀A,B∈cc(X);(2)if X is Gâteaux smooth and ε=0, then there exists a unique surjective bounded linear operator F:span‾[σf(cc(X))−σf(cc(X))]→σcc(X)−σcc(X)‾ with ‖F‖=1 such thatF(σf(A)−σf({0}))=σA,∀A∈cc(X);(3)if X is Gâteaux smooth, then there exists a unique surjective linear isometric embedding g:σcc(X)−σcc(X)‾→span‾[σf(cc(X))−σf(cc(X))] such that‖g(σA)−(σf(A)−σf({0}))‖≤6ε,∀A∈cc(X), if and only if for any μ∈span‾[σf(cc(X))−σf(cc(X))] with ‖μ‖=1,liminf|t|→∞dist(tμ,σf(cc(X))−σf(cc(X)))/|t|<1/2;(4)if X and Y are Banach spaces of the same finite dimension, ε=0, and f(A+B)=f(A)+f(B) for any pair A,B∈cc(X), then there is a surjective linear isometric embedding f‾:X→Y such thatf(A)={f‾(a):a∈A},∀A∈cc(X), where σcc(X)−σcc(X)≡{σA−σB:A,B∈cc(X)}, σf(cc(X))−σf(cc(X))≡{σf(A)−σf(B):A,B∈cc(X)}, and dist(tμ,σf(cc(X))−σf(cc(X)))≡inf⁡{‖tμ−v‖:v∈σf(cc(X))−σf(cc(X))}.

Lipschitz Clustering in Metric Spaces

In this paper, the Lipschitz clustering property of a metric space refers to the existence of Lipschitz retractions between its finite subset spaces. Obstructions to this property can be either topological or geometric features of the space. We prove that uniformly disconnected spaces have the Lipschitz clustering property, while for some connected spaces, the lack of sufficiently short connecting curves turns out to be an obstruction. This property is shown to be invariant under quasihomogeneous maps, but not under quasisymmetric ones.

On the properties of the mean orbital pseudo-metric

Given a topological dynamical system (X,T), we study properties of the mean orbital pseudo-metric E¯ defined byE¯(x,y)=limsupn→∞minσ∈Sn⁡1n∑k=0n−1d(Tk(x),Tσ(k)(y)), where x,y∈X and Sn is the permutation group of {0,1,…,n−1}. Let ωˆT(x) denote the set of measures quasi-generated by a point x∈X. We show that the map x↦ωˆT(x) is uniformly continuous if X is endowed with the pseudo-metric E¯ and the space of compact subsets of the set of invariant measures is considered with the Hausdorff distance. We also obtain a new characterisation of E¯-continuity, which connects it to other properties studied in the literature, like continuous pointwise ergodicity introduced by Downarowicz and Weiss. Finally, we apply our results to reprove some known results on E¯-continuous and mean equicontinuous systems.

Automated Feature Selection: A Reinforcement Learning Perspective

Feature selection is a critical step in machine learning that selects the most important features for a subsequent prediction task. Effective feature selection can help to reduce dimensionality, improve prediction accuracy, and increase result comprehensibility. It is traditionally challenging to find the optimal feature subset from the feature subset space as the space could be very large. While much effort has been made on feature selection, reinforcement learning can provide a new perspective towards a more globally-optimal searching strategy. In the preliminary work, we propose a multi-agent reinforcement learning framework for the feature selection problem. Specifically, we first reformulate feature selection with a reinforcement learning framework by regarding each feature as an agent. Besides, we obtain the state of the environment in three ways, i.e., statistic description, autoencoder, and graph convolutional network (GCN), in order to derive a fixed-length state representation as the input of reinforcement learning. In addition, we study how the coordination among feature agents can be improved by a more effective reward scheme. Also, we provide a GMM-based generative rectified sampling strategy to accelerate the convergence of multi-agent reinforcement learning. Our method searches the feature subset space more globally and can be easily adapted to real-time scenarios due to the nature of reinforcement learning. In the extended version, we further accelerate the framework from two aspects. From the sampling aspect, we show the indirect acceleration by proposing a rank-based softmax sampling strategy. From the exploration aspect, we show the direct acceleration by proposing an interactive reinforcement learning (IRL)-based exploration strategy. Extensive experimental results show the significant improvement of the proposed method over conventional approaches.

Feature space and label space selection based on Error-correcting output codes for partial label learning

Partial label learning (PLL) is a type of weakly supervised learning. This paper proposes a new heuristic algorithm for Partial Label learning through the combination of Feature space And Label space using Error-correcting output codes (PL-FALE for short). In the proposed framework, the feature space of different training sample subsets is exploited to generate diverse positive group and negative group pairs by the divide-and-conquer strategy. Then, the training sample label space is exploited to deal with the partial label overlap in every group pair. A set of experiments are conducted on nine controlled UCI datasets and five real-world datasets, and the experimental results show that PL-FALE can improve the classification performance by utilizing the information embedded in the feature space and label space in most cases. Also, the proposed algorithm achieves more stable performance than other ECOC-based PLL algorithms. The source code of PL-FALE is publicly available for non-commercial and research use at: https://github.com/xiaoziyang1/plfale1.

EXPSPACE-Completeness of the Logics K4 × S5 and S4 × S5 and the Logic of Subset Spaces

It is known that the satisfiability problems of the product logics K4 × S5 and S4 × S5 are NEXPTIME-hard and that the satisfiability problem of the logic SSL of subset spaces is PSPACE-hard. Furthermore, it is known that the satisfiability problems of these logics are in N2EXPTIME. We improve the lower and the upper bounds for the complexity of these problems by showing that all three problems are in ESPACE and are EXPSPACE-complete under logspace reduction.

Selection games and the Vietoris space

We explore the connections between selection games on Hausdorff spaces and their corresponding Vietoris space of compact subsets. These considerations offer a similar relationship as the well-known relationship between ω-covers of X and ordinary open covers of the finite powers of X. The primary utility of this method is to establish similar relationships with k-covers and the Vietoris space of compact subsets. Particularly, we show that some commonly studied selection principles are equivalent to a related hyperspace being Menger or Rothberger. We then apply these equivalences to correct a flawed argument in a previous paper which attempted to show that Hurewicz/Pawlikowski theorems are true for k-covers.