Next we show that ˜f is upper semicontinuous at x 0. Suppose that y 0 i → x 0, g 0 i ∈ ˜fc(y 0 i
) for each i ∈ N, and g 0is an accumulation point of (g 0 i
)i∈N. Then there is a i0 ∈ N such that y 0 i ∈ Uδ(x 0) for all i ≥ i0. From Uδ(x 0) = Uδ(γ(x)) = γ (Uδ(x))

```
follows that there are vector representations yi ∈ Uδ(x) with γ(yi) = y
                                                                          0
                                                                          i
                                                                            for

```

each i ≥ i0. From continuity of γ
−1follows that yi → x. By construction of ∂ ˜f follows that g 0 i ∈ ∂ ˜f (y 0 i
) = ∂ ˜f (γ (yi)) = γ
∂ ˜f (yi)

for each i ≥ i0. Hence, there are vector representations gi ∈ ∂ ˜f(yi) with γ(gi) =
g 0 i for each i ≥ i0. Since ˜f is upper semicontinuous at x, we find that g ∈ ∂ ˜f(x).

Again by construction of ∂ ˜f follows that g 0 = γ(g) ∈ γ
∂ ˜f(x)
= ∂ ˜f (γ(x)) = ∂ ˜f(x 0).

This proves upper semicontinuity of ∂ ˜f at all vector representations projecting to X = π(x).

Finally, we prove that ˜f satisfies the subderivative property at x 0. Suppose that y 0, y ∈ X with y 0 = γ(y). By Γ-invariance of ˜f, we have ˜f(y 0) = ˜f(y).

Since ˜f is generalized differentiable at x, we find a g ∈ ∂ ˜f(y) such that
˜f(y 0) = ˜f(y) = ˜f(x) + hg, y − xi + o(x, y, g)
with o(x, y, g) tending faster to zero than ky − xk. Let g 0 = γ(g). Exploiting Γ-invariance of ˜f as well as isometry and linearity of γ yields
˜f(y 0) = ˜f(γ(x)) + hγ(g), γ(y − x)i + o(x, y, g)
= ˜f(x 0) + hg 0, y 0 − x 0i + o(x, y, g).

We define o 0(x 0, y 0, g 0) = o ◦ γ
−1(x 0, y 0, g 0) = o(x, y, g) showing that o 0tends faster to zero than *norm*y 0 − x. This proves the subderivative property of ˜f at all vector representations projecting to X = π(x).

Putting all results together yields that ˜f is generalized differentiable at γ(x)
for all γ ∈ Γ. ut

## B.2 Lloyd-Max Necessary Conditions For Optimality

Due to the comparable nice analytical properties of Riemannian orbifolds, the proofs for the nearest neighbor and centroid condition of optimal graph quantizers are similar to their respective counterparts in vector quantization.

Theorem 3 (Nearest Neighbor Condition). Suppose that C is a fixed codebook. Any graph quantizer Q : XΓ → C *with* Q(X) = arg min Y ∈C
d (*X, Y* )
for all X ∈ XΓ *, where ties are resolved according to some rule, has minimal* expected distortion.