Getting a "non-refundable" deposit back is hard since it is, well, non-refundable. The purpose of a deposit is so that they hold the rental for you. If they really didn't lose anything because they had back-up applications, they might opt to be nice and return it to you. However, they are under no obligation to do so.
If the deposit is nonrefundable and you decide to cancel the deal, then you can expect go get you deposit buck. It is nonrefundable. The nonrefundable deposit has the purpose to secure that you wont cancel the deal.
That will be tough! You have to get the management company on your side and explain why you have to cansel, so that they help you to convince the owner to give it back. You may also put all your reasons in writing and give them a letter that they can forward to the owner. I assume the reasons are very compelling like a job offer has been cancelled or something else beyond your control that they may relate to. Otherwise, non-refundable deposit does not have to be refunded. You may also try that that they are in business of renting apartments and not collecting deposits and that they should be able to find another tenant quickly in this market.