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I have a problem similar to this question:

How do I use negative examples (in addition to positive ones) for training a multiclass softmax classifier (or a neural net with softmax output)?

where I have a multi-class classifier that is now trained only on positive examples. But, there are also cases where I don't know the correct class for a sample, I just know what label it is not.So, I'm thinking about how this information can be used to improve the current classifier. So, as suggested in the great answer of the above post, we should change the loss function to also maximize the sum of probabilities of other labels when we know that a label is not correct.

But, I wonder if there is first any other ready to use implementation for this case as the answer suggests that this new term in the loss can be problematic due to numerical issues.

Second, would it help if I change my problem to some sort of regression-form where if I know the label for a sample, the output of the classifier for that label will be +1 (learned from a positive example), if I know it's definitely not (learned from a negative example) it will be -1, and will be 0 in all other cases. Like when we use only positive examples, the labels will be 0 and 1 only (1 only for the correct class), and now they can take an additional value of -1 if they are definitely incorrect. Would this make sense and solvable?

Any feedback will be highly appreciated. I thought considering negative examples in multi-class case would be more common.

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