Use of negative correlation coefficient in pearson correlation algorithm for recommender systems
One thing you can do is to separate the contributions of:


a) who have a positive correlation with you
b) who have a negative correlation with you


Then you can:


Predict the rating using only users in a) $\to$ those will have positive correlation, this positive weights. Call this $\hat{r}_a$
Predict the rating using only users in b) $\to$ in this case, consider the weights as positive (even in they are negative correlation). Call this $\hat{r}_b$
The final predicted rating is $\hat{r} = \frac{\hat{r}_a - \hat{r}_b}{2}$