Above are papers related to the following paper: [Paper_ID] Paper #285675. 	[Title] Parsimonious Least Norm Approximation [Abstract] A theoretically justifiable fast finite successive linear approximation algorithm is proposed for obtaining a parsimonious solution to a corrupted linear system Ax = b + p, where the corruption p is due to noise or error in measurement. The proposed linear-programming-based algorithm finds a solution x by parametrically minimizing the number of nonzero elements in x and the error k Ax b p k 1 . Numerical tests on a signal-processing-based example indicate that the proposed method is comparable to a method that parametrically minimizes the 1-norm of the solution x and the error k Ax b p k 1 , and that both methods are superior, by orders of magnitude, to solutions obtained by least squares as well by combinatorially choosing an optimal solution with a specific number of nonzero elements. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #25184. 	[Title] Feature Selection via Mathematical Programming [Abstract] The problem of discriminating between two finite point sets in n-dimensional feature space by a separating plane that utilizes as few of the features as possible, is formulated as a mathematical program with a parametric objective function and linear constraints. The step function that appears in the objective function can be approximated by a sigmoid or by a concave exponential on the nonnegative real line, or it can be treated exactly by considering the equivalent linear program with equilibrium constraints (LPEC). Computational tests of these three approaches on publicly available real-world databases have been carried out and compared with an adaptation of the optimal brain damage (OBD) method for reducing neural network complexity. One feature selection algorithm via concave minimization (FSV) reduced cross-validation error on a cancer prognosis database by 35.4% while reducing problem features from 32 to 4. Feature selection is an important problem in machine learning [18, 15, 16, 17, 33]. In its basic form the problem consists of eliminating as many of the features in a given problem as possible, while still carrying out a preassigned task with acceptable accuracy. Having a minimal number of features often leads to better generalization and simpler models that can be more easily interpreted. In the present work, our task is to discriminate between two given sets in an n-dimensional feature space by using as few of the given features as possible. We shall formulate this problem as a mathematical program with a parametric objective function that will attempt to achieve this task by generating a separating plane in a feature space of as small a dimension as possible while minimizing the average distance of misclassified points to the plane. One of the computational experiments that we carried out on our feature selection procedure showed its effectiveness, not only in minimizing the number of features selected, but also in quickly recognizing and removing spurious random features that were introduced. Thus, on the Wisconsin Prognosis Breast Cancer WPBC database [36] with a feature space of 32 dimensions and 6 random features added, one of our algorithms FSV (11) immediately removed the 6 random features as well as 28 of the original features resulting in a separating plane in a 4-dimensional reduced feature space. By using tenfold cross-validation [35], separation error in the 4-dimensional space was reduced 35.4% from the corresponding error in the original problem space. (See Section 3 for details.) We note that mathematical programming approaches to the feature selection problem have been recently proposed in [4, 22]. Even though the approach of [4] is based on an LPEC formulation, both the LPEC and its method of solution are different from the ones used here. The polyhedral concave minimization approach of [22] is principally involved with theoretical considerations of one specific algorithm and no cross-validatory results are given. Other effective computational applications of mathematical programming to neural networks are given in [30, 26]. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #385572. 	[Title] Individual and Collective Prognostic Prediction [Abstract] The prediction of survival time or recurrence time is an important learning problem in medical domains. The Recurrence Surface Approximation (RSA) method is a natural, effective method for predicting recurrence times using censored input data. This paper introduces the Survival Curve RSA (SC-RSA), an extension to the RSA approach which produces accurate predicted rates of recurrence, while maintaining accuracy on individual predicted recurrence times. The method is applied to the problem of breast cancer recurrence using two different datasets. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #25181. 	[Title] Bilinear Separation of Two Sets in n-Space [Abstract] The NP-complete problem of determining whether two disjoint point sets in the n-dimensional real space R n can be separated by two planes is cast as a bilinear program, that is minimizing the scalar product of two linear functions on a polyhedral set. The bilinear program, which has a vertex solution, is processed by an iterative linear programming algorithm that terminates in a finite number of steps at a point satisfying a necessary optimality condition or at a global minimum. Encouraging computational experience on a number of test problems is reported. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #27174. 	[Title] Mathematical Programming in Neural Networks [Abstract] This paper highlights the role of mathematical programming, particularly linear programming, in training neural networks. A neural network description is given in terms of separating planes in the input space that suggests the use of linear programming for determining these planes. A more standard description in terms of a mean square error in the output space is also given, which leads to the use of unconstrained minimization techniques for training a neural network. The linear programming approach is demonstrated by a brief description of a system for breast cancer diagnosis that has been in use for the last four years at a major medical facility. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #307336. 	[Title] A Support Vector Machine Approach to Decision Trees [Abstract] Key ideas from statistical learning theory and support vector machines are generalized to decision trees. A support vector machine is used for each decision in the tree. The "optimal" decision tree is characterized, and both a primal and dual space formulation for constructing the tree are proposed. The result is a method for generating logically simple decision trees with multivariate linear or nonlinear decisions. The preliminary results indicate that the method produces simple trees that generalize well with respect to other decision tree algorithms and single support vector machines. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #3243. 	[Title] A System for Induction of Oblique Decision Trees [Abstract] This article describes a new system for induction of oblique decision trees. This system, OC1, combines deterministic hill-climbing with two forms of randomization to find a good oblique split (in the form of a hyperplane) at each node of a decision tree. Oblique decision tree methods are tuned especially for domains in which the attributes are numeric, although they can be adapted to symbolic or mixed symbolic/numeric attributes. We present extensive empirical studies, using both real and artificial data, that analyze OC1's ability to construct oblique trees that are smaller and more accurate than their axis-parallel counterparts. We also examine the benefits of randomization for the construction of oblique decision trees. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #708945. 	[Title] Massive Data Discrimination via Linear Support Vector Machines [Abstract] A linear support vector machine formulation is used to generate a fast, finitely-terminating linear-programming algorithm for discriminating between two massive sets in n-dimensional space, where the number of points can be orders of magnitude larger than n. The algorithm creates a succession of sufficiently small linear programs that separate chunks of the data at a time. The key idea is that a small number of support vectors, corresponding to linear programming constraints with positive dual variables, are carried over between the successive small linear programs, each of which containing a chunk of the data. We prove that this procedure is monotonic and terminates in a finite number of steps at an exact solution that leads to a globally optimal separating plane for the entire dataset. Numerical results on fully dense publicly available datasets, numbering 20,000 to 1 million points in 32-dimensional space, confirm the theoretical results and demonstrate the ability to handle very large problems. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #174418. 	[Title] Improving the Accuracy and Speed of Support Vector Machines [Abstract] Support Vector Learning Machines (SVM) are finding application in pattern recognition, regression estimation, and operator inversion for ill-posed problems. Against this very general backdrop, any methods for improving the generalization performance, or for improving the speed in test phase, of SVMs are of increasing interest. In this paper we combine two such techniques on a pattern recognition problem. The method for improving generalization performance (the "virtual support vector" method) does so by incorporating known invariances of the problem. This method achieves a drop in the error rate on 10,000 NIST test digit images of 1.4% to 1.0%. The method for improving the speed (the "reduced set" method) does so by approximating the support vector decision surface. We apply this method to achieve a factor of fifty speedup in test phase over the virtual support vector machine. The combined approach yields a machine which is both 22 times faster than the original machine, and which has better generalization performance, achieving 1.1% error. The virtual support vector method is applicable to any SVM problem with known invariances. The reduced set method is applicable to any support vector machine. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #285687. 	[Title] Mathematical Programming in Data Mining [Abstract] Mathematical programming approaches to three fundamental problems will be described: feature selection, clustering and robust representation. The feature selection problem considered is that of discriminating between two sets while recognizing irrelevant and redundant features and suppressing them. This creates a lean model that often generalizes better to new unseen data. Computational results on real data confirm improved generalization of leaner models. Clustering is exemplified by the unsupervised learning of patterns and clusters that may exist in a given database and is a useful tool for knowledge discovery in databases (KDD). A mathematical programming formulation of this problem is proposed that is theoretically justifiable and computationally implementable in a finite number of steps. A resulting k-Median Algorithm is utilized to discover very useful survival curves for breast cancer patients from a medical database. Robust representation is concerned with minimizing trained model degradation when applied to new problems. A novel approach is proposed that purposely tolerates a small error in the training process in order to avoid overfitting data that may contain errors. Examples of applications of these concepts are given. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #3229. 	[Title] Book Review  Introduction to the Theory of Neural Computation Reviewed by: 2 [Abstract] Neural computation, also called connectionism, parallel distributed processing, neural network modeling or brain-style computation, has grown rapidly in the last decade. Despite this explosion, and ultimately because of impressive applications, there has been a dire need for a concise introduction from a theoretical perspective, analyzing the strengths and weaknesses of connectionist approaches and establishing links to other disciplines, such as statistics or control theory. The Introduction to the Theory of Neural Computation by Hertz, Krogh and Palmer (subsequently referred to as HKP) is written from the perspective of physics, the home discipline of the authors. The book fulfills its mission as an introduction for neural network novices, provided that they have some background in calculus, linear algebra, and statistics. It covers a number of models that are often viewed as disjoint. Critical analyses and fruitful comparisons between these models Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #3231. 	[Title] Irrelevant Features and the Subset Selection Problem [Abstract] We address the problem of finding a subset of features that allows a supervised induction algorithm to induce small high-accuracy concepts. We examine notions of relevance and irrelevance, and show that the definitions used in the machine learning literature do not adequately partition the features into useful categories of relevance. We present definitions for irrelevance and for two degrees of relevance. These definitions improve our understanding of the behavior of previous subset selection algorithms, and help define the subset of features that should be sought. The features selected should depend not only on the features and the target concept, but also on the induction algorithm. We describe a method for feature subset selection using cross-validation that is applicable to any induction algorithm, and discuss experiments conducted with ID3 and C4.5 on artificial and real datasets. Please summarise all the given information.

Paper #285675 has the title `Parsimonious Least Norm Approximation  
` and the abstract `A theoretically justifiable fast finite successive linear approximation algorithm is proposed for obtaining a parsimonious solution to a corrupted linear system Ax = b + p, where the corruption p is due to noise or error in measurement. The proposed linear-programming-based algorithm finds a solution x by parametrically minimizing the number of nonzero elements in x and the error k Ax b p k 1 . Numerical tests on a signal-processing-based example indicate that the proposed method is comparable to a method that parametrically minimizes the 1-norm of the solution x and the error k Ax b p k 1 , and that both methods are superior, by orders of magnitude, to solutions obtained by least squares as well by combinatorially choosing an optimal solution with a specific number of nonzero elements. 
`.
Question: Which category should Paper #285675 be classified as ?
You can select one from ['Neural_Networks', 'Case_Based', 'Theory', 'Reinforcement_Learning', 'Probabilistic_Methods', 'Rule_Learning', 'Genetic_Algorithms'].


Paper #285675 has the title `Parsimonious Least Norm Approximation  
` and the abstract `A theoretically justifiable fast finite successive linear approximation algorithm is proposed for obtaining a parsimonious solution to a corrupted linear system Ax = b + p, where the corruption p is due to noise or error in measurement. The proposed linear-programming-based algorithm finds a solution x by parametrically minimizing the number of nonzero elements in x and the error k Ax b p k 1 . Numerical tests on a signal-processing-based example indicate that the proposed method is comparable to a method that parametrically minimizes the 1-norm of the solution x and the error k Ax b p k 1 , and that both methods are superior, by orders of magnitude, to solutions obtained by least squares as well by combinatorially choosing an optimal solution with a specific number of nonzero elements. 
`.
Question: Which category should Paper #285675 be classified as ?
You can select one from ['Neural_Networks', 'Case_Based', 'Theory', 'Reinforcement_Learning', 'Probabilistic_Methods', 'Rule_Learning', 'Genetic_Algorithms'].

This paper should be classified into: Neural_Networks