R.G. Jeroslow. On defining sets of vertices of the hypercube by linear inequalities. Discrete Mathematics, 11, 1975: 119--124.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....the unit. Obviously, a threshold gate has threshold number 1. Calculating the threshold number of a computational unit is a natural way to measure how much more powerful the unit is than a threshold gate. Investigations of the threshold number of Boolean functions can be traced back to Jeroslow [8]. He has shown that every Boolean function on n variables has threshold number at most 2 , and further that for each k, where 1 k 2 , there is a Boolean function having threshold number k, with the parity function attaining the maximum value. Maass and Schmitt [7] have shown that every ....

R. G. Jeroslow, On defining sets of vertices of the hypercube by linear inequalities, Discrete Mathematics 11 (1975) 119--124.

....a hyperplane to separate off a set of points all having the same classification (either all are positive points or all are negative points) These points can then be removed from consideration and the procedure iterated until no points remain. This procedure is similar in nature to one of Jeroslow [15], but at each stage in his procedure, only positive examples may be chopped off (not positive or negative) We give one example for illustration. Example: Suppose the data set D consists of all points of f0; 1g , labelled according to their parity, so the classification is 1 precisely when ....

....representation of the data. That is, since fewer hyperplanes might be used, the decision list could be smaller. Indeed, Jeroslow s method requires 2 iterations in the parity based Example given above, since at each stage it can only chop off one positive point. Note that Jeroslow s method [15] (described above) requires 2 iterations in this Example, since at each stage it can only chop off one positive point. The chopping procedure described above suggests that the use of threshold decision lists is fairly natural, if one is to take an iterative approach to data classification. ....

R.G. Jeroslow. On defining sets of vertices of the hypercube by linear inequalities. Discrete Mathematics, 11, 1975: 119--124.

....(either all are true points or all are false points) These points are then removed from consideration and the procedure is iterated until no points remain. For simplicity, we assume that at each stage, no data point lies on the hyperplane. This procedure is similar in nature to one of Jeroslow [14], but at each stage in his procedure, only positive examples may be chopped off (not positive or negative) We give one example for illustration. Example: Suppose the function f is the parity function, so that the true points are precisely those with an odd number of ones. We first find a ....

....n times, and at stage i in the procedure we chop off all data points having precisely (i 1) ones, by using the hyperplane y 1 y 2 y n = i 1 2, for example. These hyperplanes are in fact all parallel, but this is not necessary. 19 Note that, by contrast, Jeroslow s method [14] (described above) requires 2 iterations in this example, since at each stage it can only chop off one positive point. We may regard the chopping procedure as deriving a representation of the function by a threshold decision list. If, at stage i of the procedure, the hyperplane with ....

R.G. Jeroslow. On defining sets of vertices of the hypercube by linear inequalities. Discrete Mathematics, 11, 1975: 119--124.

....all are positive examples or all are negative examples) These points are then deleted from the database and the procedure is iterated until no points remain. For simplicity, we assume that at each stage, no data point lies on the hyperplane. This procedure is similar in nature to one of Jeroslow [4], but at each stage in his procedure, only positive examples may be chopped off (not positive or negative) We give one example for illustration. Example Suppose that the data points are all points of f0; 1g n and that the label accompanying x 2 f0; 1g n is 0 if x contains an even number ....

....and at stage i in the procedure we chop off all data points having precisely (i Gamma 1) ones, by using the hyperplane y 1 y 2 Delta Delta Delta y n = i Gamma 1=2, for example. These hyperplanes are parallel, but this is not necessary, or possible, in general. ut Jeroslow s method [4] (described above) requires 2 n Gamma1 iterations to explain the data given in this Example, since at each stage it can only chop off one positive point. 3 Decision lists The basic idea of a decision list was introduced by Rivest [5] We extend the definition slightly as follows. Let K be ....

R.G. Jeroslow. On defining sets of vertices of the hypercube by linear inequalities. Discrete Mathematics, 11 (1975): 119--124.

.... ffx fi; ff; fi) 2 IR n 1 ; is equivalent to a 0 1 inequality fl 1 x 1 Delta Delta Delta fl n x n ffi with integer coefficients fl 1 ; fl n ; ffi such that jfl i j 2 Gamman (n 1) n 1) 2 and jffij 2 Gamma(n 1) n 1) n 3) 2 1=2: By a result of Jeroslow [Jer75] at most 2 n Gamma1 linear 0 1 inequalities are needed to define a 0 1 set S f0; 1g n . For any k in the range 1 k 2 n Gamma1 , there exists S f0; 1g n , such that at least k linear 0 1 inequalities are needed to define S. Lipkin [Lip87] showed that in order to define a set S f0; ....

R. G. Jeroslow. On defining sets of vertices of the hypercube by linear inequalities. Discrete Mathematics, 11:119 -- 124, 1975.

....threshold number 1, a threshold gate has threshold number 1. Calculating the threshold number of a computational unit is a natural way to measure how much more powerful the unit is than a threshold gate. Investigations of the threshold number of Boolean functions can be traced back to Jeroslow [4] and Chv atal and Hammer [1] Jeroslow [4] has shown that every Boolean function on n variables has threshold number at most 2 n Gamma1 , and further that for each k, where 1 k 2 n Gamma1 , there is a Boolean function having threshold number k, with the parity function attaining the maximum ....

....threshold number 1. Calculating the threshold number of a computational unit is a natural way to measure how much more powerful the unit is than a threshold gate. Investigations of the threshold number of Boolean functions can be traced back to Jeroslow [4] and Chv atal and Hammer [1] Jeroslow [4] has shown that every Boolean function on n variables has threshold number at most 2 n Gamma1 , and further that for each k, where 1 k 2 n Gamma1 , there is a Boolean function having threshold number k, with the parity function attaining the maximum value. Chv atal and Hammer [1] proved that ....

R. G. Jeroslow, On defining sets of vertices of the hypercube by linear inequalities, Discrete Mathematics 11 (1975) 119--124.

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R.G. Jeroslow. On defining sets of vertices of the hypercube by linear inequalities. Discrete Mathematics, 11, 1975: 119--124.

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R.G. Jeroslow. On defining sets of vertices of the hypercube by linear inequalities. Discrete Mathematics, 11, 1975: 119--124.

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