Incremental Computation of Pseudo-Inverse of Laplacian

Abstract

A divide-and-conquer based approach for computing the Moore-Penrose pseudo-inverse of the combinatorial Laplacian matrix (L + ) of a simple, undirected graph is proposed. The nature of the underlying sub-problems is studied in detail by means of an elegant interplay between L + and the effective resistance distance (Ω). Closed forms are provided for a novel two-stage process that helps compute the pseudo-inverse incrementally. Analogous scalar forms are obtained for the converse case, that of structural regress, which entails the breaking up of a graph into disjoint components through successive edge deletions. The scalar forms in both cases, show absolute element-wise independence at all stages, thus suggesting potential parallelizability. Analytical and experimental results are presented for dynamic (time-evolving) graphs as well as large graphs in general (representing real-world networks). An order of magnitude reduction in computational time is achieved for dynamic graphs; while in the general case, our approach performs better in practice than the standard methods, even though the worst case theoretical complexities may remain the same: an important contribution with consequences to the study of online social networks.

Cascading failures in power grids

“…Using Theorem 1 and Corollary 1, in Section 5 we provide upper bounds on the mutual edge flow change ratios (M e,e ). We note that a similar result to Theorem 1 was independently proved in a very recent technical report [40].…”

Pseudo-inverses of difference matrices and their application to sparse signal approximation

“…The following theorem generalizes the ideas of [20,2] for pseudo-inverses of circulant matrices of rank N − 1, as well as of [22,17] for pseudo-inverses of the Laplacian. According to the usual definition, we say that L + is the Moore-Penrose inverse of L if…”

“…This formula has already been proposed for finding generalized inverses of circulant matrices of size N ×N with rank N −1 in [20] and for discrete Laplace operators considered e.g. in [22,17]. We generalize this formula to arbitrary symmetric N × N difference matrices of rank N − 1 possessing the eigenvector 1 = (1, .…”

Global Registration of Multiple Point Clouds Using Semidefinite Programming

“…Applied to (14), this means that we can compute L † in O(|E(Γ)|) time (up to logarithmic factors). Note that, even if L is dense, it is still possible to speed up the inversion (say, compared to a direct Gaussian elimination) using the formula [33,50]: The main computational blocks in GRET-SDP are identical to that in GRET-SPEC, plus the SDP computation. The SDP solution can be computed in polynomial time using interior-point programming [67].…”