Given his talent, ambition and track record - not to mention his looks and his A-list wife - it's no surprise that Benjamin Millepied has been dubbed dance's biggest crossover star since Mikhail Baryshnikov.

Born in Bordeaux to a former ballerina, the 36-year-old trained in a Lyon conservatory from the age of 13 and in 1993, while still in his teens, moved to New York to study at New York City Ballet.

By 2001, he was the company's principal dancer, but it was his involvement in Darren Aronofsky's Oscar-winning 2010 film Black Swan that propelled him out of the arts pages and into the gossip columns. As well as choreographing the film's dance scenes and appearing in them, he began a relationship with the film's star, Natalie Portman.

But Millepied has crossed over in other ways too. In 2011, he quit New York City Ballet. A year later he quit New York as well, marrying Portman and moving to Los Angeles, where he founded LA Dance Project, the enterprise which brings him to the Edinburgh International Festival this week. The company will perform works by William Forsythe and Merce Cunningham as well as Millepied's own piece, Moving Parts.

While notionally a dance company, LA Dance Project is actually a collaborative arts co-operative whose other members include composer Nico Muhly (who has worked with Philip Glass, Bjork and Brooklyn art rockers Grizzly Bear among others) and producer Charles Fabius. The two other members of the controlling quintet are art consultant Matthieu Humery, formerly of Christie's, and filmmaker Dimitri Chamblas, who also works as a choreographer.

"Knowing that you were pretty much set for failure trying to build a dance company from scratch in America, since no-one's been able to do it for some time, I took the time to think about how I wanted the company to be structured," Millepied explains.

"The collective was just a way for me to say it's important to surround yourself with people who are specialists in their own right. I wanted to have the ability to be able to count on other people to help me gather talent and think about the company.''

He compared the approach to that of George Balanchine, who co-founded New York City Ballet in the late 1940s with wealthy, Harvard-educated Lincoln Kirstein, a writer and art collector-turned-impressario.

''Sergei Diaghilev [the founder of Ballets Russes] too was very, very cultured and involved in all aspects of the arts,'' adds Millepied. ''So I thought that it would be a very positive thing to make these people responsible."

While Diaghilev was a one-off, Millepied thinks he has his 21st-century version of Kirstein in producer Charles Fabius. The Dutch-born producer is a former programme director of the Paris Grand Opera and is currently a consultant for New York's Guggenheim Museum.

In other words he's plugged into virtually every mover, shaker and (importantly) benefactor in the world of American arts. Working with him, says Millepied, has been key to the success of LA Dance Project since its inaugural performances last year.

Despite the difficulties of starting a dance company - especially in Los Angeles, whose "provincial quality" is allied to a mistrust of incomers with new ideas, says Millepied - LA Dance Project has so far resisted the temptation to sweeten its repertoire with easy work. Accordingly, the rarely performed Merce Cunningham piece Millepied is bringing to Edinburgh this week is difficult, even by Cunningham's standards.

Called Winterbranch, it dates from 1964 and, in its original staging, began with Cunningham crawling over a floor holding a torch. It explores what Cunningham called "facts in dancing" - those facts in this context being falling down and getting up again - and has a score by La Monte Young. It consists of two sounds which the composer once described as ashtrays scraping against a mirror and wooden blocks rubbed against a Chinese gong.

The original lighting concept was by painter Robert Rauschenberg, a regular Cunningham collaborator, and was intended to mimic car headlights ­sweeping across the dancers. Rauschenberg also designed the costumes.

"That piece is a shock for a lot of people," says Millepied. "Some people cannot sit through it." He chuckles devilishly at the idea. "I find it so modern and so emotional in a way that his works aren't necessarily. It is tortuous.

"It's not the Merce you're used to … But it's a great piece and we wanted to make a statement that some of that modern American dance repertoire is going to stay alive and be performed by our company."

As well as his own piece, Moving Parts, which has a score by Nico Muhly and costumes by designers Kate and Laura Mulleavy of fashion label Rodarte, Millepied has included William Forsythe's Quintett in his Edinburgh programme. Created in 1993 as an elegy of sorts for the choreographer's wife, then terminally-ill, it's one of Forsythe's most celebrated works and uses as its soundtrack Gavin Bryar's 1972 piece Jesus' Blood Never Failed Me Yet. "It has a very liberating quality, very much a celebration-of-life quality," says Millepied. "So it's a good counterpoint to Winterbranch, in my view."

Away from the rarefied atmosphere of the Edinburgh International Festival, Millepied has made it his mission to take dance beyond the proscenium arch, to use it to colonise other spaces and by doing so win new audiences.

The company has already performed at the Design Museum in London and at the Museum of Contemporary Art in Los Angeles, and this autumn he's planning a performance at the iconic Union Station in Los Angeles. "We're doing an opera," he explains. "Dancers will be navigating through the station during rush hour with singers and an orchestra. So that's the sort of thing I'm very interested in." He has something similar planned for the Pompidou Centre in Paris.

It was that same yearning for artistic expansion that saw Millepied say yes to film-maker Darren Aronofsky's request that he choreograph Black Swan when filming began in 2009. This was despite the fact that its gothic plot is hardly the best advertisement for dance: it turns on a ballerina whose obsession with her role in Swan Lake brings her close to mental breakdown.

"It was a no-brainer," he says of the decision. "All stories are allowed to be told in film and in art, and the idea that the dance world wasn't portrayed well … well, not so long ago [Bolshoi Ballet artistic director] Sergei Filin had acid thrown in his face, so it is an obsessional art form. But Black Swan brought a lot of people to dance for the first time and brought a lot of momentum and interest to dance that we're still feeling right now. And it's a movie, for Christ's sake. It was a thriller, a rollercoaster."

Next year, Benjamin Millepied embarks on yet another crossover when he leaves Los Angeles to take up the role of director of dance at Paris Opera Ballet. He will retain an interest in LA Dance Project and continue to act in an advisory capacity, but with the world's oldest and most prestigious national ballet company under his command it's fair to say transatlantic crossings will be kept to a minimum.

The Paris job sees him return to his native country, of course, but also to a European style of arts funding which is very different from the American approach he's used to, in which rich patrons are wooed over cocktails.

Broadly he's optimistic about the state of dance in France - "one of the remarkable things about the country and the culture is the fact that you have young people in the theatres still," he says - but aware, too, that money is tight everywhere and arts budgets are no exception.

"When things are hard you can get away with more and you can do more," he says. "People have more of a thirst and a need for a change. So I actually think it's a good time to take it on."

Perhaps he should make his first task a production of the ballet Diaghalev and Balanchine mounted in Paris in 1929: The Prodigal Son.

LA Dance Project performs Quintett, Winterbranch and Moving Parts at the Edinburgh Playhouse, August 24-26