10.5 Continuous Extensions, Part II скачать в хорошем качестве

10.5 Continuous Extensions, Part II

We define the multilinear extension, and prove several important properties when it is derived from a monotone or submodular function. In particular: The gradient of F is nonnegative when f is monotone. The elements of the Hessian are nonnegative (with zeros on the diagonal). These imply that: F is non-decreasing along any direction d \geq 0 F is concave along any direction d \geq 0 F is convex along any direction d = e_i - e_j. We also show that by iid sampling, we can obtain a good estimate of F(x) with high probability.