F.W. Smith, Pattern classifier design by linear programming, IEEE Trans. Computers, C-17, 4, April 1968, 367-372.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... a recent study we found that, for determining linear separability, linear programming methods far outperform the adaptive methods in terms of definiteness and correctness of decisions and time efficiency [1] To handle both separable and nonseparable cases, we use a formulation proposed by Smith [10] that minimizes an error function: minimize a t t subject to Z t w t b t 0 where a, b are arbitrary constant vectors (both chosen to be 1) w is the weight vector, t is an error vector, and Z is a matrix where each column z is defined on an input vector x (augmented by adding one ....

F.W. Smith, Pattern classifier design by linear programming, IEEE Trans. Computers, C-17, 4, April 1968, 367-372.

....constraining the location and orientation of the optimal separating hyperplane. With a properly defined objective function, a separating hyperplane can be obtained by solving a linear programming problem. Several alternative formulations have been proposed in the past ( 3] 5] 10] 15] [17]) employing different objective functions. An early survey of these methods is given in [7] Here we mention a few representative formulations. In a very simple formulation described in [15] the objective function is trivial, so that it is simply a test of linear separability by finding a ....

....separable data, it is impossible for all components of the error vector to be negative at any given iteration. This formulation gives only a test for linear separability but does not lead to any useful solution if the data are not linearly separable. Another formulation suggested by Smith ( 6] [17]) minimizes an error function: minimize a t t subject to Z t w t b t 0 (9) where Z is the augmented data matrix as before, a is a positive vector of weights, b is a positive margin vector chosen arbitrarily (e.g. b = 1) and t and w are the error and weight vectors which are also ....

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Smith, F.W., Pattern Classifier Design by Linear Programming, IEEE Transactions on Computers, C-17, 4, April 1968, 367-372.

.... 1 2C m X i=1 ff 2 i subject to the constraint : m X i=1 ff i y i = 0; which makes clear how the trade off parameter C in their formulation is related to the kernel parameter Delta, namely C = 1 Delta 2 : Note that this approach to handling non separability goes back to Smith [26], with Bennett and Mangasarian [6] giving essentially the same formulation as Cortes and Vapnik [10] but with a different optimisation of the function class. The expression also shows how moving to the soft margin ensures separability of the data, since both primal and dual problems are feasible. ....

F. W. Smith. Pattern classifier design by linear programming. IEEE Transactions on Computers, C-17:367--372, 1968.

....(1.2) that is a null solution that does not provide any error minimizing separation. For this purpose, we utilize the linear program introduced recently in [8] and which has the following desirable features not all of which are possessed by any other previous linear programming formulation [11, 27, 28, 45, 20, 19]: i) A strict separating plane (that is neither set lies on the separating plane) for linearly separable sets A and B (ii) An error minimizing plane is obtained when the sets A and B are linearly inseparable. iii) No extraneous constraints are used to exclude the null solution for linearly ....

F. W. Smith. Pattern classifier design by linear programming. IEEE Transactions on Computers, C-17:367--372, 1968.

.... T z k fi fi fi fi fi fi fi GammaAw efl e y; Bw Gamma efl e z; y 0; z 0 9 = 7) Robustness here refers to the fact that the useless null vector (w = 0) is naturally excluded as a solution of (7) which is not the case in other linear programming formulations of this problem [41, 66, 26, 25]. Note that because of the constraints of the problem, the variables y and z will satisfy the following conditions: y minf0; GammaAw efl eg and z minf0; Bw Gamma efl eg: Hence minimizing e T m e T z k will force the satisfaction in some best sense of (6) and equivalently (5) ....

F. W. Smith. Pattern classifier design by linear programming. IEEE Transactions on Computers, C-17:367--372, 1968.

....configurations (a) and (b) of Figure 1 are equivalent as can easily be seen if the roles of A and B are interchanged. Bilinear separation is a natural extension of linear separation which, for a long time, has been known to be equivalent to the polynomial time solution of a single linear program [9, 13, 26, 5]. Linear separation is also equivalent to separation by Rosenblatt s perceptron or linear threshold unit (LTU) 24, 25, 11] see Figure 2) However most problems are not linearly separable. For example the simple Minsky Papert exclusive or classical problem [20] is not linearly separable, but is ....

....When the convex hulls of A and B are also disjoint then A and B are said to be linearly separable. By using the duality theory of linear programming, this can be shown to be equivalent to the existence of a plane wx = fl strictly separating A from B (see Figure 2(a) which is equivalent to [12, 13, 26, 5] Gamma Aw efl e 0; Bw Gamma efl e 0; for some w 2 R n ; fl 2 R: 5) Based on the above definition of linear separability, we now define bilinear separability as follows: Definition 3.1 (Bilinear separability definition (See Figure 5) The sets A and B are bilinearly separable if and ....

F. W. Smith. Pattern classifier design by linear programming. IEEE Transactions on Computers, C-17:367--372, 1968.

....as follows. Given two disjoint finite points sets A and B in the n dimensional real space R n , construct a discriminant function f , from R n into the real line R, such that f(A) 0 and f(B) 0. When the convex hulls of the two point sets A and B do not intersect, a single linear program [6,7,9,2,3] can be used to obtain a linear discriminant function of the following type (1:1) f(x) cx fl where c is in R n and fl is in R. Unfortunately in many real life problems the convex hulls of the sets A and B intersect and one must resort to a more complex discriminant function, such as a ....

....separation of these sample sets, but also correctly classified each of 45 new points subsequently obtained. It is worthwhile to point out here, that for the linearly inseparable case (i.e. the case of intersecting convex hulls) solution of any of the single linear programs proposed in [6] [9] or [2,3] may not provide any useful information. We will demonstrate this by means of small examples and by citing computational experience with the medical diagnosis problem. A brief word about the notation employed. For vectors x and y in the n dimensional real space R n , xy will denote ....

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F.W. Smith, Pattern classifier design by linear programming, IEEE Transaction on Computers, C-17, 4, (1968), pp. 367-372.

....and Qualcomm, Inc. 1 Introduction The problem of designing a statistical classifier to minimize the probability of misclassification or a more general risk measure has been a topic of continuing interest since the 1950s. Much of the early, classical work focused on linear classifiers [40] 14] [46] and parametric classifiers, e.g. 9] More recently, with the increase in power of serial and parallel computing resources, a number of more complex classifier structures have been proposed, along with associated learning algorithms to design them. The most prominent research has focused on ....

F.W. Smith. Pattern classifier design by linear programming. IEEE Trans. on Comp., C-17:367-- 372, 1968.

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F. W. Smith, Pattern Classifier Design by Linear Programming, IEEE Transactions on Computers C-17, 4, 1968, pp. 367-372.

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