Sometimes, I get a white square in which it should say "Deus Ex GOTY" or something like that. This is before the game goes fullscreen. If you get this, just quit and try again. I don't know why this happens, but quit and open again worked after the second or third try. I hope this is what you mean by white screen.

Did you try software rendering? This is the slowest because the GPU is not used, but on current machines, there's not much of a difference for such an old game. You can choose this on first start or if the game crashes or if you go to Settings->Display->Change Renderer.

Please tell me how the game performs with different renderers on your machine. Most renderers make the scene too dark. Just use the software renderer and go to Training. The very first hall will look like it should. Try OpenGL or Direct3D to see what I mean by dark.

I tried it just last week, but deinstalled it again because I have to wait to get the cool Matrix Mod, which I cannot find anywhere in the internet. I have it on a backup disk somewhere.
This one.

Edit: I found the mod here
See my other post on how to install it.