Machine learning algorithm that uses the Pearson or Spearman correlation?
I do not think this exists. I suppose an algorithm could use pearson coefficients as starting coefficients, but honestly it seems like a waste of computational resources. Here are some reasons that occur to me as to why it is a bad idea:


Pearson and Spearman correlations become decreasingly meaningful as
the number of dimensions increase. I commonly work with millions of dimensions...Spearman correlations for individual features? 
In sparse matrices, these coefficients mean next to nothing as there will be only a very slight correlations between a feature and the target. Usually, it is a multi dimensional relationship that we are trying to find (lots of caveats placed here :P)
Pearson and Spearman correlations assume certain parameters which are not usually true in ML applications ie homoscedasticity, linearity, normality, etc.


For the above and many other reasons imho it doesn't serve any purpose to use these anywhere in ML algorithms.