The first two sources I find suggest the following

The idea that mental illness alone causes homelessness is naive and inaccurate, for two major reasons. First, the overwhelming majority of those living with mental illness are not homeless (and studies have failed to demonstrate a causal relationship between the two).

The ultimate causes of homelessness are upstream, i.e. a profound lack of affordable housing due in large part to neo-liberal government austerity policies that prevent or limit public funding for housing, gentrification that displaces working and poor families, and growing income disparities that make paying the rent beyond the means of millions of households.