A Proof of Soft Medoid breakdown point

We start with a discussion of some preliminaries for the proofs. In A.2, we build upon Lopuhaä and Rousseeuw [39]'s work to prove Lemma 1. In A.3, we prove Theorem 1. A.1 Preliminaries The adversary can replace m arbitrary points. For a concise notation we simply write that we replace the first m values, but the points come with an arbitrary order beforehand. In case (a), just some values during the derivation change, but the results are essentially the same. For case (b), the number of samples is no longer n.