Graph Powering and Spectral Robustness

Abstract

Spectral algorithms, such as principal component analysis and spectral clustering, typically require careful data transformations to be effective: upon observing a matrix A, one may look at the spectrum of ψ(A) for a properly chosen ψ. The issue is that the spectrum of A might be contaminated by non-informational top eigenvalues, e.g., due to scale' variations in the data, and the application of ψ aims to remove these.Designing a good functional ψ (and establishing what good means) is often challenging and model dependent. This paper proposes a simple and generic construction for sparse graphs,where A denotes the adjacency matrix and r is an integer (less than the graph diameter). This produces a graph connecting vertices from the original graph that are within distance r, and is referred to as graph powering. It is shown that graph powering regularizes the graph and decontaminates its spectrum in the following sense: (i) If the graph is drawn from the sparse Erdős-Rényi ensemble, which has no spectral gap, it is shown that graph powering produces a "maximal" spectral gap, with the latter justified by establishing an Alon-Boppana result for powered graphs; (ii) If the graph is drawn from the sparse SBM, graph powering is shown to achieve the fundamental limit for weak recovery (the KS threshold) similarly to [Mas14], settling an open problem therein. Further, graph powering is shown to be significantly more robust to tangles and cliques than previous spectral algorithms based on self-avoiding or nonbacktracking walk counts [Mas14, MNS14, BLM15, AS15]. This is illustrated on a geometric block model that is dense in cliques. *

Achieving the Bayes Error Rate in Synchronization and Block Models by SDP, Robustly

“…Work of [39,44,49] establishes that the necessary and sufficient condition of weak recovery is n(p−q) 2 p+q > 2. Subsequent work proves similar phase transitions and shows that various algorithms achieve weak recovery above the optimal threshold for the SBM with k ≥ 2 and possibly unbalanced clusters; see, e.g., [7,4,12,14,16,19,51,56]. As discussed later, our results also imply weak recovery guarantees with a sub-optimal constant.…”

“…Most related to us is a line of work that characterizes minimax optimal error rates for partial recovery. For the binary symmetric SBM, the work [62] establishes the aforementioned minimax lower bound (4). They also provide an exponential-time algorithm that achieves a matching upper bound (up to an o(1) factor in the exponent).…”

“…We remark that neither SDP requires knowing the parameters of the data generating processes (that is, τ 2 , α, , p and q in Models 1-3). 4 The SDP (7) was considered in [11,10] and [34] for studying the exact recovery threshold in Z2 and CBM, respectively, and the SDP (8) was considered in [33] for exact recovery under the binary symmetric SBM. These formulations can be further traced back to the work of [24] on SDP relaxation for MIN BISECTION.…”

“…In particular, whennI * > (1 + δ) log n for any positive constant δ, the bound (3) ensures that err( σ sdp , σ * ) < 1 n and hence err( σ sdp , σ * ) = 0. Moreover, the lower bound (4) shows that exact recovery is information-theoretically impossible whennI * < log n. In the literature, establishing such tight exact recovery thresholds often involves specialized and sophisticated arguments, and has been the milestones in the remarkable recent development on community detection (see Section 2 for a discussion of related work). We recover these results, for all three models, as a corollary of our main theorem by plugging in the corresponding expressions of I * andn.…”

A literature survey of matrix methods for data science

“…For example the diagonal entries of W 2 give the degrees of the nodes in the network. Note that powers of the graph Laplacian are considered in [1] for the purpose of spectral clustering and within graph convolutional networks for semi-supervised learning in [187]. The use of W 2 suggests to consider higher powers of the adjacency matrix for longer walks and one obtains (…”