  We generalize the notions of $n$-cluster tilting subcategories and
$\tau$-selfinjective algebras into $n$-precluster tilting subcategories and
$\tau_n$-selfinjective algebras, where we show that a subcategory naturally
associated to $n$-precluster tilting subcategories has a higher
Auslander--Reiten theory. Furthermore, we give a bijection between
$n$-precluster tilting subcategories and $n$-minimal Auslander--Gorenstein
algebras, which is a higher dimensional analog of Auslander--Solberg
correspondence (Auslander--Solberg, 1993) as well as a Gorenstein analog of
$n$-Auslander correspondence (Iyama, 2007). The Auslander--Reiten theory
associated to an $n$-precluster tilting subcategory is used to classify the
$n$-minimal Auslander--Gorenstein algebras into four disjoint classes. Our
method is based on relative homological algebra due to Auslander--Solberg.
