Let us consider a Riemannian manifold with bounded Ricci curvature $$\|Ric\|\leq n-1$$ and the noncollapsing lower volume bound $$Vol(B_1(p))>v>0$$. In their codimension four paper, Cheeger-Naber conjecture that the $L^2$ Riemann curvature is bounded, i.e., $$\fint_{B_1(p)}|Rm|^2 < C(n,v)$$. In the same paper, they give a proof of this conjecture for dimensional four manifolds by using Gauss-Bonnet theorem. In this talk, we will discuss a new proof of this conjecure for dimensional four manifolds. The proofs involve some estimates on the $W$-entropy. This is part of the joint work with Professor Aaron Naber of Northwestern University.