# Regression algorithm on [0,1] with lots of mislabeled data

I have a training set mapping some Likert-scale variables (integers between 1 and 7, rescaled to real numbers between 0 and 1) to predict a continuous variable between 0 and 1. The data set is reasonable large ($$10^4$$-$$10^5$$ rows) but very noisy (10-20% has bogus labeling).

When applying regression algorithms to this dataset, the regression algorithm predicts scores only in a subinterval of [0,1], e.g. scores below 0.1 and above 0.9 are not reached for any input value of the columns. This is expected given the bogus labels; but also inacceptable, as the prediction for non-bogus rows is severely impacted. Moreover, since 0.0 and 1.0 are the most common target values in the training set (it is a 'How much ... are you?' type prediction) it is not appreciated by the users that 0.0 and/or 1.0 would be unreachable. (Simply stretching the result to cover [0,1] greatly increases the error, so is probably suboptimal.)

What method could I use to detect and remove these mislabeled/bogus rows, or at least a significant amount of them? Since they are a minority, I would think something with majority voting should be the answer, but it's not clear how to implement such an approach, since $$7^{n_{cols}} > n_{rows}$$.

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