It seems that many of the motifs that have become her trademarks have been embedded in this tradition. The first Kusama pumpkin to see was with her grandfather. When she went to pick it up, she started talking to her. It was the size of the head of a guy. She decorated the pumpkin and received an award for it, her first 11-year-old. Eighty years later, her biggest silver pumpkin sculptures were sold for $500,000. After the assault on Pearl Harbor, when Kusama was 13, she was forced to work in a factory making parachute fabrics. She painted intricate flowers over and over throughout the evening. In a notice of her first show, the local paper reported her producing 70 aquarelles a day 

Watching the stills of Kusama’s early life in Lenz’s documentary – her hair cut straight across her forehead, photographed between flowers – is a sharp and moving contrast to the footage of the artist at work in her studio. The same slightly bulbous eyes look out from under the red wig as she connects her dots with a magic marker, chewing her lip like a girl. “To me,” Lenz says, “Kusama’s childhood trauma was instrumental in her work, not only because of her difficult family, but also because of her society and the nightmare of the Second World War.”

Lenz came to understand these stresses more profoundly because, when making a film, she married from a Japanese family and heard the story of her husband’s grandfather, killed by the Hiroshima bomb, and her mother-and father-in-law, who had an arranged marriage. “That gave me a better understanding of her childhood,” she says. “The standards of the time for a young woman, an arranged marriage, children. Kusama confidently took the decision to leave Japan and go to New York though it was a fairly surprising thing to do.”

The second chapter of Kusama’s journey started when she first visited Georgia O’Keeffe’s work in a bookshop in Matsumoto, her hometown. She found O’Keeffe’s address in New Mexico, and wrote to her for advice on how to make her way into the art world of New York, sending some of her own intricate aquariums of abstract vegetal shapes and bursting seed pods. O’Keeffe answered, puzzled at first why anyone, let alone a young woman in rural Japan, would want to do such a thing, but the interest has grown over a number of years into a kind of mentorship. “The artist has a hard time making a living in this country,” O’Keeffe answered. “You’re just going to have to find the best way you can.”

Kusama arrived in New York in 1958, at the age of 27, with a few hundred dollars sewn in the linen of her dresses, along with 60 kimonos of silk and some sketches. Her intention was to survive the sale of one or the other. In her own account, she initially subsisted on food scraps, including fish heads scavenged from the fishmonger’s trash, which she boiled for soup. She’s been following her job around the area. “One day,” she recalls in her autobiography, “I brought a canvas more than 40 blocks in the streets of Manhattan to present it for consideration at the Whitney Annual. My painting was not chosen, and I had to take it back 40 blocks. The wind was blowing hard that day, and more than once it looked as though the canvas was floating up into the air, taking me with it. When I got home, I was so tired that I slept like a dead man for two days.”

Her breakthrough pieces, the Infinity Net paintings, originated from an earlier collection of aquariums called the Pacific Ocean, which she had made in response to the tracery of waves on the surface of the sea when she first flew from Tokyo. The nets she painted were made of a repetitive singular impasto gesture in small circles, like interlocking scales; the longest canvases were 30ft in length. One of these canvases sold for $7.1m in 2014, a record for a living female artist. The first ones she sold for $75 to fellow artists Frank Stella and Donald Judd in 1962.

Judd and Kusama have been living in the same building on 19th Street in Manhattan for a while. “She’d sit around my apartment and talk, or I’d go down there and talk,” Judd said in a 1988 interview. “She must have worked through the night, as far as I could tell. Most of the paintings were done in one take. I don’t understand how she’d be able to do that, but she’d start in the corner and then go over.” One of the startling aspects about seeing Lenz’s film is the way Kusama appeared to be written out of the history of pop art. There was a period in the 1960s when she shared nearly equal billing – and popularity – with the likes of Andy Warhol and Claes Oldenburg. Part of this eclipse seems to have been by design – Kusama has long said that the Waspish men around her appropriated her original ideas and moved away as their own.

In 1963, she began to create chairs and other objects decorated, fungi-like, with white painted phallic shapes made of stuffed fabric; her piece of resistance was a rowing boat, complete with oars, which she and Judd rescued from a junkyard. It was presented in a box-like room, the walls, the ceiling and the floor of which were papered with 999 silk-screen images of the phallic ship. She saw this as her own private aversion therapy.

“I started making penises to heal my feelings of disgust with sex,” she wrote later. “My fear was the hide-in-the-closet-trembling sort. I was told that sex was filthy, shameful, something to hide. Complicating matters even more was all the talk of ‘healthy families’ and ‘arranged marriages’ and the utter resistance to romantic love… I also experienced a sex act when I was a toddler, and the terror that came through my eye blew up inside me.” There is a grim irony in this act of therapy in that Oldenburg seems to have embraced her soft sculpture technique and Warhol’s repeated wallpaper prints. She was desperate at the way the men around her sought fame for her theories.

Lenz’s film aims to reveal this appropriation. “Every single Q&A I get a question about how true the allegations that these white male artists stole her ideas were,” Lenz says. “Obviously, I’ve reviewed all the dates, and they’re all working out like she said. People who had degrees in art history always questioned this, though; it was as if they didn’t want to change their mind. They know what they know, I suppose.” Kusama saw something like her ideal man in Joseph Cornell, the reclusive genius of the outer world of art, the founder of surreal boxes of found objects, and a man who had always lived with his mother in the 1950s. Cornell became fascinated with Kusama, giving her a dozen poems a day, never hanging from a phone call, so he was there when she picked it up. This was her only known romantic relationship, though, “he didn’t like sex, and I didn’t like sex, so we weren’t having sex.” He wasn’t a very easy guy.