Ainslie, a British sailor who has won the Finn class gold medal twice in a row and also captured the Laser class in 2000, finished second to Hoegh-Christensen in both the races contested off the southern coast of England.
There will be 11 races contested in all with the first-place finisher receiving one point, the second-place finisher receiving two points and on down the line through 23 contestants in the competition.
The worst score during the first 10 races can be dropped, but the 11th and final race must count and the points in that one are doubled.
Hoegh-Christensen had two points after two races, Ainslie had four and Ivan Gaspic of Croatia had six after finishing third in both races.
The Star class regatta also got under way with two races and Brazilians Robert Scheidt and Bruno Prada took the lead with five points. They won the day's second race.
Irish sailors Peter O'Leary and David Burrows were second with eight points, followed by boats from Poland and Norway tied for third. Xavier Rohart and Pierre-Alexis Ponsot of France won the first race, but finished 13th in the second.