She can run, but Kate Hudson cannot hide!

The actress has regularly been seen in the crowd at numerous New York Yankee games, cheering on her new beau Alex Rodriguez.

But when it comes to being seen out in public with Madonna's ex, Kate doesn't want anything to do with it.

Maybe that's why the second she spotted our photographer last night, Kate made a dash from her cab to the door of Alex's NY apartment.

Meanwhile, the shy baseball player who was inside the taxi with her, left the scene, only to return minutes later and run inside.

And, because we know you care, YES Kate DID stay the night.

This relationship/lovefest is definitely on...even if they are an odd celebrity match-up.

See more pictures of Kate Hudson in the gallery below...