Breaking the Small Cluster Barrier of Graph Clustering Supplementary Material

In this supplementary material, we present proof details. 1 Notation and Conventions We use the following notation and conventions throughout the supplement. For a real n× n matrix M , we use the unadorned norm ‖M‖ to denote its spectral norm. The notation ‖M‖F refers to the Frobenius norm, ‖M‖1 is ∑ i,j |M(i, j)| and ‖M‖∞ is maxij |M(i, j)|. We will also study operators on the space of matrices. To distinguish them from the matrices studied in this work, we will simply call these objects “operators”, and will denote them using a calligraphic font, e.g. P. The norm ‖P‖ of an operator is defined as ‖P‖ = sup M :‖M‖F =1 ‖PM‖F , where the supremum is over matrices M . For a fixed, real n× n matrix M , we define the matrix linear subspace T (M) as follows: T (M) := {YM +MX : X,Y ∈ Rn×n} . In words, this subspace is the set of matrices spanned by matrices each row of which is in the row space of M , and matrices each column of which is in the column space of M . For any given subspace of matrices S ⊆ Rn×n, we let PS denote the orthogonal projection onto S with respect to the the inner product 〈X,Y 〉 = ∑n i,j=1X(i, j)Y (i, j) = trX Y . This means that for any matrix M , PSM = argminX∈S ‖M −X‖F . For a matrix M , we let Γ(M) denote the set of matrices supported on a subset of the support of M . Note that for any matrix X, (PΓ(X)M)(i, j) = { M(i, j) X(i, j) 6= 0 0 otherwise . It is a well known fact that PT (X) is given as follows: PT (X)M = PC(X)M +MPR(X) − PC(X)MPR(X) ,