What about the heat loss in compression/expansion? Fong says: "It became clear that what you wanted to do for maximum efficiency was keep the temperature as close to constant as possible in compression and expansion. It turned out nobody had figured out how to do that, and I read a Wikipedia article saying it was impossible to do it, and I said, ‘My god, that’s not true. You can just spray water in.’ And then I was like, ‘Wait. I could just spray water in.’ And thus the company and core idea was born."
So how does it work? : "Instead of wasting the heat, we collect it by spraying water into the air during the compression process. That keeps the temperature down, and it keeps the pressure down, so you have to put less energy in to compress the same amount of air. During expansion, spraying water sends heat back into the air, which keeps the pressure high, and increases the amount of energy you get back.” Science aside, the numbers don’t lie: LightSail’s process recovers 70% of the energy it puts out, pretty much doubling the efficiency of the standard compression method. "
Their website has more information about the technology they've developed, including some experimental calculations. This sounds like an innovative idea, although past experience has made me cynical about actual practicality or implementability of innovative ideas.