Materials for “ Statistical - Computational Phase Transitions in Planted Models : The High - Dimensional Setting ”

We provide the proofs for the theorems in the main paper. 1 Proofs for Planted Clustering In this section, Theorems 1–6 refer to the theorems in the main paper. Equations are numbered continuously from the main paper. We let n1 := rK and n2 := n − rK be the numbers of nonisolated and isolated nodes, respectively. 1.1 Proof of Theorem 1 The proof relies on information theoretical arguments and the Fano’s inequaliy [4]. We use D (Ber(p)‖Ber(q)) to denote the KL divergence between two Bernoulli distributions with mean p and q. We first state an upper bound on D (Ber(p)‖Ber(q)), which is used later in the proof: D (Ber(p)‖Ber(q)) = p log p q + (1− p) log 1− p 1− q (a) ≤ pp− q q + (1− p) − p 1− q = (p− q)2 q(1− q) , (16) where (a) follows from the inequality log x ≤ x− 1,∀x ≥ 0. Let P(Y ∗,A) be the joint distribution of Y ∗ and A when Y ∗ is sampled from Y uniformly at random and A is generated according to the planted clustering model. Because the supremum is lower bounded by the average, we have inf Ŷ sup Y ∗∈Y P [ Ŷ 6= Y ∗ ] ≥ inf Ŷ P(Y ∗,A) [ Ŷ 6= Y ∗ ] . (17) Let H(X) be the entropy of a random variable X and I(X;Z) the mutual information between X and Z. By Fano’s inequality, we have for any Ŷ , P(Y ∗,A)(Ŷ 6= Y ∗) ≥ 1− I(Y ∗;A) + 1 log |Y| . (18) Simple counting gives that |Y| = ( n n1 ) n1! r!(K!)r . Note that ( n n1 ) ≥ ( n n1 ) n1 and √ n(ne ) n ≤ n! ≤ e √ n(ne ) n. It follows that |Y| ≥ (n/n1) √ n1(n1/e) n1 e √ r(r/e)rerKr/2(K/e)n1 ≥ ( n K )n1 1 e(r √ K)r .