Basically I want to convert similarity measures into weights which are used as predictors.  The similarities will be on [0,1], and I will restrict the weights to also be on [0,1]. I'd like a paramteric function that does this mapping which I'll likely optimize using gradient descent.  The requirements are that 0 maps to 0, 1 maps to 1 and it be strictly increasing.  A simple derivative is also appreciated.  Thanks in advance