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I have a 3 class sample labeled data set, which I have divided into 2 parts. I am using the first part to train the two-class perceptron classifier.

One approach is to train $\binom{3}{2}$ two-class classifiers on training data for two classes at a time, and then use voting to classify the test samples for a multi-category case. The obvious problem with the above approach is the presence of ambiguous regions like the ones shown in the image below:

Linear decision boundary for a 4-class problem

Pattern classification by Duda, Hart, Stork suggests training $c$ different linear discriminant functions, where $c$ is the number of unique classes, such that

$$g_{i}(x) = W_{i}^tX + w_{i0} \quad \quad \quad i = 1,..., c$$

and assigning $X$ to $\omega_{i}$ if $g_{i}(X) > g_{j}(X)$. The resulting classifier is called in the text book as a linear machine.

Following is an illustration from the book showing decision boundary produced by a linear machine for 3-class problem.

decision boundary produced by a linear machine for 3-class problem

My doubt is how is the training process of the $c$ linear discriminant functions $g_{i}(X)$ different from training of $\binom{c}{2}$ 2-class classifiers?

UPDATE: The one vs rest approach mentioned in DavidDLewis's answer is not the one I am referring to. With the one vs rest approach, there are infact even more ambiguous regions. I am referring to linear machine classifier using which there are no ambiguous regions. See illustration of 1 vs rest approach below:

1 vs rest

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  • $\begingroup$ I believe the 'linear machine' is a 'one vs. rest' approach, but the resultant classifiers are used differently than in your diagram. $\endgroup$ – shabbychef Sep 2 '11 at 20:11
  • $\begingroup$ It would be really helpful, if you could please elaborate on the usage differences you mentioned in your comment. $\endgroup$ – stressed_geek Sep 4 '11 at 9:54
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    $\begingroup$ imagine $k$ 1-v-rest classifiers as each describing regions where the $i$th class dominates the rest, _i.e._ $\left\{x|g_i(x) \le c_i\right\}$. This can result in the ambiguous region in your bottom figure. The linear machine approach trains the classifiers in the same way, but then defines the region where the $i$th class dominates as $\left\{x|g_i(x) \le g_j(x),\,\forall j\right\}$. $\endgroup$ – shabbychef Sep 4 '11 at 23:29
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Training c linear discriminant functions is a example of a "1-vs-all" or "1-against-the-rest" approach to building a multiclass classifier given a binary classifier learning algorithm. Training C(c,2) 2-class classifiers is an example of the "1-vs-1" approach. As c gets larger, the "1-vs-1" approach builds a lot more classifiers (but each from a smaller training set).

This paper compares these two approaches, among others, using SVMs as the binary classifier learner:

C.-W. Hsu and C.-J. Lin. A comparison of methods for multi-class support vector machines , IEEE Transactions on Neural Networks, 13(2002), 415-425.

Hsu and Lin found 1-vs-1 worked best with SVMs, but this would not necessarily hold with all binary classifier learners, or all data sets.

Personally, I prefer polytomous logistic regression.

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  • $\begingroup$ I am not referring to "1 vs rest" approach. Please see the update on my question post for clarification on the question. Thanks! $\endgroup$ – stressed_geek Sep 2 '11 at 17:37

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