I think people would be a lot more compassionate towards homeless people generally if the violent and destructive subset of homeless people were put in prison where they belong. With the awful ones out of the way, the peaceful sympathetic homeless people would become the public face of homelessness and the general public would be much more willing to to address their problems constructively (e.g. provide housing to them.)

But instead the justice system is set up to give effective impunity to the worst sort of homeless people; they're back on the street days after being arrested (if they are even arrested in the first place.) They cause incredible damage and commotion, so they hog all the public attention and give all homeless people a very bad name through association.