In Nagoya most of the homeless I saw had jobs, but didn't make enough to really afford rent. They were always really nice, and seemed to try to go the extra mile to stay out of the way and not make a mess.
I'd bet that less of the homeless in Japan are on the streets due to mental problems, as compared to the US. It seems to be much more based on economics than mental health.
It's easier here to keep someone locked in a hospital. I don't think anyone has ever done a homeless census, so there's no way of knowing for sure. The only other comment I'll make is that it sure is weird to see someone grimy from head to foot, but be sleeping with his shoes stacked neatly to one side. Some customs run very deep.