Lecture : Local Spectral Methods ( 3 of 4 )

Warning: these notes are still very rough. They provide more details on what we discussed in class, but there may still be some errors, incomplete/imprecise statements, etc. in them. Last time, we considered local spectral methods that involve short random walks started at a small set of localized seed nodes. Several things are worth noting about this. • The basic idea is that these random walks tend to get trapped in good conductance clusters, if there is a good conductance cluster around the seed node. A similar statement holds for approximate localized random walks, e.g., the ACL push procedure—meaning, in particular, that one can implement them " quickly, " e.g., with the push algorithm, without even touching all of the nodes in G. • The exact statement of the theorems that can be proven about how these procedures can be used to find good locally-biased clusters is quite technically complicated—since, e.g., one could step outside of the initial set of nodes if one starts near the boundary—certainly the statement is much more complicated than that for the vanilla global spectral method. • The global spectral method is on the one hand a fairly straightforward algorithm (compute an eigenvector or some other related vector and then perform a sweep cut with it) and on the other hand a fairly straightforward objective (optimize the Rayleigh quotient variance subject to a few reasonable constraints). • Global spectral methods often do very well in practice. • Local spectral methods often do very well in practice. • A natural question is: what objective do local spectral methods optimize—exactly, not ap-proximately? Or, relatedly, can one construct an objective that is quickly-solvable and that also comes with similar locally-biased Cheeger-like guarantees? To this end, today we will present a local spectral ansatz that will have several appealing properties: • It can be computed fairly quickly, as a PPR. • It comes with locally-biased Cheeger-like guarantees. • It has the same form as several of the semi-supervised objectives we discussed. • Its solution touches all the nodes of the input graph (and thus it is not as quick to compute as the push procedure which does not).