206660013
An Improved Intermediate Value Theorem and Rough Fix-Point Theorem of Roughly Continuous Discrete Functions
The incompleteness and insufficiency in the investigation of rough continuity in Pawlak rough function model are pointed out. Proposing the e-d definition of rough continuity of a discrete function, the concept of Pawlak rough continuity is deduced in the form of a proposition. A series of operating properties of roughly continuous functions are discussed such as maximization, minimization, complementarity, etc.. By an opposite example, we point out that the sufficient condition in the intermediate value theorem Pawlak proposed is not valid. An improved intermediate value theorem of roughly continuous discrete functions on closed intervals is then proposed. The concept of connectivity function closely connected with the rough continuity is introduced, by which the improved intermediate value theorem is proved strictly. Moreover, the concept of rough fix-points of discrete functions is introduced. The rough fix-point theorem of roughly continuous functions is proposed and studied, and some new results are achieved.
452800
Rough Derivatives in Rough Function Model and Their Application
On the theory and application of rough derivatives in rough function model, the notion of generalized rough functions is defined to improve the definition of higher order rough derivatives Pawlak proposed, and analyze the functional features of roughly derived functions and higher order roughly derived functions. The rough extremum definition of discrete functions is given, Fermat theorem and Rolle theorem of roughly smooth discrete functions are proposed and proved, which complete and perfect the theoretical foundation of rough derivatives application in rough function model. The concepts that are rough monotone and rough convexity of discrete functions are defined. By comparing with the derivative application of real continuous function, a series of theorems are put forward which are the relation theorem of rough derivatives and rough monotone, the two sufficient conditions of rough extrema, the two relation theorems of rough derivatives and rough convexity. Moreover, some new results are achieved such as the sufficient condition of rough smoothing of discrete functions, etc.
21906808
The Theory and Application of Rough Integration in Rough Function Model
The definition of the rough integral Pawlak proposed is improved. New notions are proposed, which are respectively the rough integral on a constant interval and the roughly integral upper limit function, and so on. By comparing with the definite integral of real functions, the properties of rough integrals are analyzed. Giving the concept of mean value of discrete functions, the mean value method for rough integration is derived. At the same time, the intermediate value theorem of rough integration is proposed and its geometric significance is analyzed, which provides a dependable theoretical tool for rough integral operation, etc.. Conclusions including the existence theorem of rough primitives and the fundamental formula of rough calculus are proposed. By the representative of discrete functions, the method of computing a primitive is given, by which basic formulas for rough integration in common use are derived, and the method of rough direct integration is obtained. There is also the method of rough integration by parts for rough integrals which is like that of definite integrals. Thus the recurrence formula for rough integration is deduced, in which the integrand of the rough integral is in the shape of the product of a rough power function and a rough exponential function. It is pointed out that integration by substitution is not applicable for rough integral operation.
19552464
Attribute reduction and optimal decision rules acquisition for continuous valued information systems
For continuous valued information systems, the attribute values of objects for the same attribute represent not only their ordinal relationship but also their relative distances. Therefore, the classical rough set model is not suitable for deducing attribute reductions and optimal decision rules for continuous valued information systems. Though some discretization methods are proposed to transform the continuous valued information systems into discrete ones, but those methods are too categorical and may lead to loss of information in some cases. To solve such information loss problem, we propose a tolerance rough set model in this paper. With a given level, the proposed model can divide a universe into some maximal tolerance classes. Also two types of lower and upper approximations are defined accordingly. Then the reductions of the maximal tolerance class and optimal decision rules based on the proposed attribute descriptors are defined, and the approximate discernibility function for the maximal tolerance class is constructed and used to compute all the corresponding optimal decision rules via using Boolean reasoning techniques. Finally, the general reductions and consistent reductions for continuous valued information systems are discussed.
5078035
Order-Based Decision Rules Acquisition in Continuous-Valued Decision Information Systems
In a continuous-valued information system, attribute values of objects represent not only the relative distances, but also the ordinal relation of objects. By considering the relative distances of objects, some discretization approaches and other approaches were proposed in previous literature, to discuss the attribute reduces of the system. This paper considers ordinal relations of objects, and utilizes dominance relation rough set model to discuss decision rules acquisition problems. Firstly, concepts of order-based decision rules and generalized order-based decision rules are introduced. Then the concept of optimal order-based decision rule is defined. In order to compute optimal order-based decision rules, the concept of reduct of the object is proposed. Lastly, discernibility functions of objects are constructed, and used to compute optimal order-based decision rules by Boolean reasoning techniques.
6225988
A global optimal algorithm for class-dependent discretization of continuous data
This paper presents a new method to convert continuous variables into discrete variables for inductive machine learning. The method can be applied to pattern classification problems in machine learning and data mining. The discretization process is formulated as an optimization problem. We first use the normalized mutual information that measures the interdependence between the class labels and the variable to be discretized as the objective function, and then use fractional programming (iterative dynamic programming) to find its optimum. Unlike the majority of class-dependent discretization methods in the literature which only find the local optimum of the objective functions, the proposed method, OCDD, or Optimal Class-Dependent Discretization, finds the global optimum. The experimental results demonstrate that this algorithm is very effective in classification when coupled with popular learning systems such as C4.5 decision trees and Naive-Bayes classifier. It can be used to discretize continuous variables for many existing inductive learning systems.
14397325
Constraint-Sensitive Privacy Management for Personalized Web-Based Systems
This research aims at reconciling web personalization with privacy constraints imposed by legal restrictions and by users' privacy preferences. We propose a software product line architecture approach, where our privacyenabling user modeling architecture can dynamically select personalization methods that satisfy current privacy constraints to provide personalization services. A feasibility study is being carried out with the support of an existing user modeling server and a software architecture based development environment.
21082976
Real-time machine learning in embedded software and hardware platforms
This paper describes on-going research work into real-time machine learning using embedded software and reconfigurable hardware. The main focus of the work is to develop real-time incremental learning methods particularly targeted at demonstration in mobile robot environments. Three main areas are described. The first represents reactive robot navigation knowledge using a novel frequency table technique whose memory requirement is known a priori. The second area investigates a Genetic Algorithm (GA) method that combines planning and reactive approaches to allow navigation to proceed even in the face of time constraints. In the third area we are developing novel hardware-based machine learning systems suitable for implementation in reconfigurable platforms.
15223911
Boosting with incomplete information
In real-world machine learning problems, it is very common that part of the input feature vector is incomplete: either not available, missing, or corrupted. In this paper, we present a boosting approach that integrates features with incomplete information and those with complete information to form a strong classifier. By introducing hidden variables to model missing information, we form loss functions that combine fully labeled data with partially labeled data to effectively learn normalized and unnormalized models. The primal problems of the proposed optimization problems with these loss functions are provided to show their close relationship and the motivations behind them. We use auxiliary functions to bound the change of the loss functions and derive explicit parameter update rules for the learning algorithms. We demonstrate encouraging results on two real-world problems --- visual object recognition in computer vision and named entity recognition in natural language processing --- to show the effectiveness of the proposed boosting approach.
6932570
3D Surface Matching and Recognition Using Conformal Geometry
3D surface matching is a fundamental issue in computer vision with many applications such as shape registration, 3D object recognition and classification. However, surface matching with noise, occlusion and clutter is a challenging problem. In this paper, we analyze a family of conformal geometric maps including harmonic maps, conformal maps and least squares conformal maps with regards to 3D surface matching. As a result, we propose a novel and computationally efficient surface matching framework that uses least squares conformal maps. According to conformal geometry theory, each 3D surface with disk topology can be mapped to a 2D domain through a global optimization and the resulting map is a diffeomorphism, i.e., one-to-one and onto. This allows us to simplify the 3D surface-matching problem to a 2D image-matching problem, by comparing the resulting 2D conformal geometric maps, which are stable, insensitive to resolution changes and robust to occlusion and noise. Therefore, highly accurate and efficient 3D surface matching algorithms can be achieved by using conformal geometric maps. Finally, the performance of conformal geometric maps is evaluated and analyzed comprehensively in 3D surface matching with occlusion, noise and resolution variation. We also provide a series of experiments on real 3D face data that achieve high recognition rates.