A unified view of graph regularity via matrix decompositions

We give a unified proof of algorithmic weak and Szemerédi regularity lemmas for several well-studied classes sparse graphs, which only were previously known. These include core-dense low threshold rank (a version of) upper regular graphs. More precisely, we define cut pseudorandom prove our these then show that pseudorandomness captures all the above graph as special cases. The core approach is an abstracted matrix decomposition, can be computed by simple algorithm Charikar. Using work Oveis Gharan Trevisan, it also implies new PTASes MAX-CUT, MAX-BISECTION, MIN-BISECTION significantly expanded class input (It NP Hard to get graphs in general.)