**Obsessively** (can be consisdered negative)
[Dictionary.com][1]:
> the domination of one's thoughts or feelings by a persistent idea, image, desire, etc.

**Single-mindedly** (positive)
[Dictionary.com][2]
> having or showing a single aim or purpose

**Faithfully** (positive)
[Dictionary.com][3]
> steady in allegiance or affection; loyal; constant

**Unswervingly** (positive)
[The Free Dictionary][4]
> 1. Not veering or turning aside  
> 2.  Constant; steady


  [1]: http://dictionary.reference.com/browse/obsession
  [2]: http://dictionary.reference.com/browse/single-minded
  [3]: http://dictionary.reference.com/browse/faithfully?s=t
  [4]: http://www.thefreedictionary.com/unswervingly