Supplementary material for Manifold structure in graph

Proof of Theorem 3. We have Z Lemma 4. Consider a polynomial kernel over a bounded region Z R The associated integral operator has finite rank. Lemma 5. Suppose f is analytic on Z with f ( x, y)= Figure 4: Kernel density ridge sets (red) as estimates of the underlying manifold (blue), for embed-dings of simulated graphs described in Section 2 and also shown in Figure 1.