There's an essential flaw in the question, Brian. You can't really talk about "homeless people" or "poor people" in terms of being anything but homeless or poor, in the very same way that you can't talk about "black people" or "white people" in terms of being anything other than black or white.
There are many reasons why someone might be homeless, and it's problematic to talk about a mentally disadvantaged person in the same breath as a person who's living in a shelter with her two children because she lost her job and has no other safe place to go, or the woman whose drug habit has turned her to squatting and street prostitution, or the ex-felon who's just been released from prison with $25 and absolutely no where to go.
Let me put it another way. For most homeless people, homelessness is actually the least of their problems. This actually divides the two largest groups of homeless people - those for whom homelessness is the problem, and those for whom homelessness is a symptom of a deeper problem.
I can tell you this - until the "experts" - let alone the average person - really embraces this fact, it's all just talk and wasted money. Always be wary of anyone who says their organization aims to "end homelessness," because that's a sure sign that their organization doesn't understand homelessness. Or worse, that they do understand it, but are banking on the fact that you don't.
For those for whom homelessness is the actual problem, the solution is obvious and relatively inexpensive. But for the others (the majority), the real solution is complex and, frankly, very, very expensive. In lots of people's eyes, prohibitively expensive - which is why homeless organizations do much better when they group all homeless people together, then tout their successes with the easy cases when they make their public appeals for more money.
The homeless problem was first introduced in the US with the de-institutionalization of people from state mental hospitals, under your friend and mine, dear old Ronald Reagan.* The promise was that these people could be better treated in neighborhood-based adult homes. But this system was never adequately funded, and so the care and oversight of the patients was abysmal to say the least, and all of a sudden there was a huge "homeless problem."
I could go on and on - I won't, for everyone's sake. Suffice it to say that unresearched, armchair conversation about this problem probably does more harm than good.
* This is not a myth, it's true. There are tons and tons of people who worship Reagan and want to erase the statement that "Ronald Reagan caused the homeless problem as we know it," and who have undertaken a campaign to re-write the history on this point. Consequently, anyone and their brother can find an article via Google under "Debunking the myth of Reagan's Deinstitutionalization." If you want to do some actual research to test the veracity of my statement, you have to first look at what Reagan did as governor in California, leading up to his presidential bid. There's a very clear line of budget cuts starting in California and ending with HUD budget cuts from the White House that made tens of thousands of mentally ill patients in state institutions ineligible for residential care. As I said above, the promise was that these patients would be cared for in community-based facilities. This was, of course, a broken promise.