Topic

# Closed set

About: Closed set is a research topic. Over the lifetime, 3626 publications have been published within this topic receiving 45340 citations. The topic is also known as: closed subset.

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01 Jan 1960TL;DR: In this paper, the Stone-Czech Compactification is used to define a topological space, and a list of symbols for topological spaces is presented, including cardinal of closed sets in Beta-x, homomorphisms and continuous mapping.

Abstract: Contents: Functions of a Topological Space.- Ideals and Z-Filters.- Completely Regular Spaces.- Fixed Ideals. Compact Spaces.- Ordered Residue Class Rings.- The Stone-Czech Compactification.- Characterization of Maximal Ideals.- Realcompact Spaces.- Cardinals of Closed Sets in Beta-x.- Homomorphisms and Continuous Mappings.- Embedding in Products of Real Lines.- Discrete Spaces. Nonmeasurable Cardinals.- Hyper-Real Residue Class Fields.- Prime Ideals.- Uni- form Spaces.- Dimension.- Notes.- Bibliography.- List of Symbols.- Index.

1,259 citations

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1,245 citations

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01 Jan 2002TL;DR: CHARM is an efficient algorithm for mining all frequent closed itemsets that enumerates closed sets using a dual itemset-tidset search tree, using an efficient hybrid search that skips many levels, and uses a technique called diffsets to reduce the memory footprint of intermediate computations.

Abstract: The set of frequent closed itemsets uniquely determines the exact frequency of all itemsets, yet it can be orders of magnitude smaller than the set of all frequent itemsets. In this paper we present CHARM, an efficient algorithm for mining all frequent closed itemsets. It enumerates closed sets using a dual itemset-tidset search tree, using an efficient hybrid search that skips many levels. It also uses a technique called diffsets to reduce the memory footprint of intermediate computations. Finally it uses a fast hash-based approach to remove any “non-closed” sets found during computation. An extensive experimental evaluation on a number of real and synthetic databases shows that CHARM significantly outperforms previous methods. It is also linearly scalable in the number of transactions.

1,068 citations

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01 Jun 2016TL;DR: The proposed OpenMax model significantly outperforms open set recognition accuracy of basic deep networks as well as deep networks with thresholding of SoftMax probabilities, and it is proved that the OpenMax concept provides bounded open space risk, thereby formally providing anopen set recognition solution.

Abstract: Deep networks have produced significant gains for various visual recognition problems, leading to high impact academic and commercial applications. Recent work in deep networks highlighted that it is easy to generate images that humans would never classify as a particular object class, yet networks classify such images high confidence as that given class – deep network are easily fooled with images humans do not consider meaningful. The closed set nature of deep networks forces them to choose from one of the known classes leading to such artifacts. Recognition in the real world is open set, i.e. the recognition system should reject unknown/unseen classes at test time. We present a methodology to adapt deep networks for open set recognition, by introducing a new model layer, OpenMax, which estimates the probability of an input being from an unknown class. A key element of estimating the unknown probability is adapting Meta-Recognition concepts to the activation patterns in the penultimate layer of the network. Open-Max allows rejection of "fooling" and unrelated open set images presented to the system, OpenMax greatly reduces the number of obvious errors made by a deep network. We prove that the OpenMax concept provides bounded open space risk, thereby formally providing an open set recognition solution. We evaluate the resulting open set deep networks using pre-trained networks from the Caffe Model-zoo on ImageNet 2012 validation data, and thousands of fooling and open set images. The proposed OpenMax model significantly outperforms open set recognition accuracy of basic deep networks as well as deep networks with thresholding of SoftMax probabilities.

1,034 citations

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TL;DR: This paper explores the nature of open set recognition and formalizes its definition as a constrained minimization problem, and introduces a novel “1-vs-set machine,” which sculpts a decision space from the marginal distances of a 1-class or binary SVM with a linear kernel.

Abstract: To date, almost all experimental evaluations of machine learning-based recognition algorithms in computer vision have taken the form of “closed set” recognition, whereby all testing classes are known at training time. A more realistic scenario for vision applications is “open set” recognition, where incomplete knowledge of the world is present at training time, and unknown classes can be submitted to an algorithm during testing. This paper explores the nature of open set recognition and formalizes its definition as a constrained minimization problem. The open set recognition problem is not well addressed by existing algorithms because it requires strong generalization. As a step toward a solution, we introduce a novel “1-vs-set machine,” which sculpts a decision space from the marginal distances of a 1-class or binary SVM with a linear kernel. This methodology applies to several different applications in computer vision where open set recognition is a challenging problem, including object recognition and face verification. We consider both in this work, with large scale cross-dataset experiments performed over the Caltech 256 and ImageNet sets, as well as face matching experiments performed over the Labeled Faces in the Wild set. The experiments highlight the effectiveness of machines adapted for open set evaluation compared to existing 1-class and binary SVMs for the same tasks.

1,029 citations