Compressing Interactive Communication under Product Distributions

We study the problem of compressing interactive communication to its information content I, defined as the amount of information that the participants learn about each other's inputs. We focus on the case when the participants' inputs are distributed independently and show how to compress the communication to O(I log
²
I) bits, with no dependence on the original communication cost. This result improves quadratically on previous work by Kol (STOC 2016) and essentially matches the well-known lower bound Ω(I).