The innovation about soft point in this study is, soft point's soft matrix form which were not described before is defined for each set of parameters. The matrix representation of soft points is useful for storing all soft points that can be obtained in all different parameters. The proposed soft matrix provides every soft point that changes with each parameter that takes place in a soft set is proved and showed that it enables detailed examination in application of soft set theory. General Terms 2010 AMS Classification: 03G25,20D05 Keywords Soft set, soft point, soft matrix, soft matrix form of soft point.

Rough sets and fuzzy sets have been proved to be powerful mathematical tools to deal with uncertainty, it soon raises a natural question of whether it is possible to connect rough sets and fuzzy sets. The existing generalizations of fuzzy rough sets are all based on special fuzzy relations (fuzzy similarity relations, -similarity relations), it is advantageous to generalize the fuzzy rough sets by means of arbitrary fuzzy relations and present a general framework for the study of fuzzy rough sets by using both constructive and axiomatic approaches. In this paper, from the viewpoint of constructive approach, we first propose some definitions of upper and lower approximation operators of fuzzy sets by means of arbitrary fuzzy relations and study the relations among them, the connections between special fuzzy relations and upper and lower approximation operators of fuzzy sets are also examined. In axiomatic approach, we characterize different classes of generalized upper and lower approximation operators of fuzzy sets by different sets of axioms. The lattice and topological structures of fuzzy rough sets are also proposed. In order to demonstrate that our proposed generalization of fuzzy rough sets have wider range of applications than the existing fuzzy rough sets, a special lower approximation operator is applied to a fuzzy reasoning system, which coincides with the Mamdani algorithm.

Pattern recognition and machine learning techniques have been increasingly adopted in adversarial settings such as spam, intrusion, and malware detection, although their security against well-crafted attacks that aim to evade detection by manipulating data at test time has not yet been thoroughly assessed. While previous work has been mainly focused on devising adversary-aware classification algorithms to counter evasion attempts, only few authors have considered the impact of using reduced feature sets on classifier security against the same attacks. An interesting, preliminary result is that classifier security to evasion may be even worsened by the application of feature selection. In this paper, we provide a more detailed investigation of this aspect, shedding some light on the security properties of feature selection against evasion attacks. Inspired by previous work on adversary-aware classifiers, we propose a novel adversary-aware feature selection model that can improve classifier security against evasion attacks, by incorporating specific assumptions on the adversary's data manipulation strategy. We focus on an efficient, wrapper-based implementation of our approach, and experimentally validate its soundness on different application examples, including spam and malware detection.

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The generalization error bounds found by current error models using the number of effective parameters of a classifier and the number of training samples are usually very loose. These bounds are intended for the entire input space. However, support vector machine (SVM), radial basis function neural network (RBFNN), and multilayer perceptron neural network (MLPNN) are local learning machines for solving problems and treat unseen samples near the training samples to be more important. In this paper, we propose a localized generalization error model which bounds from above the generalization error within a neighborhood of the training samples using stochastic sensitivity measure. It is then used to develop an architecture selection technique for a classifier with maximal coverage of unseen samples by specifying a generalization error threshold. Experiments using 17 University of California at Irvine (UCI) data sets show that, in comparison with cross validation (CV), sequential learning, and two other ad hoc methods, our technique consistently yields the best testing classification accuracy with fewer hidden neurons and less training time.