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We prove algorithmic weak and \Szemeredi{} regularity lemmas for severalclasses of sparse graphs in the literature, for which only weak regularitylemmas were previously known. These include core-dense graphs, low thresholdrank graphs, and (a version of) $L^p$ upper regular graphs. More precisely, wedefine \emph{cut pseudorandom graphs}, we prove our regularity lemmas for thesegraphs, and then we show that cut pseudorandomness captures all of the abovegraph classes as special cases.  The core of our approach is an abstracted matrix decomposition, roughlyfollowing Frieze and Kannan [Combinatorica '99] and \Lovasz{} and Szegedy[Geom.\ Func.\ Anal.\ '07], which can be computed by a simple algorithm byCharikar [AAC0 '00]. This gives rise to the class of cut pseudorandom graphs,and using work of Oveis Gharan and Trevisan [TOC '15], it also implies newPTASes for MAX-CUT, MAX-BISECTION, MIN-BISECTION for a significantly expandedclass of input graphs. (It is NP Hard to get PTASes for these graphs ingeneral.)

A Unified View of Graph Regularity via Matrix Decompositions