  We give a complete classification of differential $\mathbb{Z}$-graded
homotopy categories of matrix factorizations of isolated singularities up to
quasi-equivalence. This answers a question of Bernhard Keller and Evgeny
Shinder.
  More generally, we show that a quasi-equivalence between the dg singularity
category of a Gorenstein isolated singularity $R$ and the dg singularity
category of a complete local Noetherian $\mathbb{C}$-algebra $S$ of different
Krull dimension can always be realized by Kn\"orrer's periodicity -- in
particular, the existence of such an equivalence implies that $R$ and $S$ are
hypersurface singularities. This uses and is complemented by a recent
categorical version of the Mather--Yau theorem for hypersurfaces of the same
Krull dimension due to Hua & Keller, which completes the classification
mentioned above.
