The purpose of this research is to compare the risks and benefits of two different procedures used to help patients with mitral valve regurgitation (also known as MR). MR is a condition where the valve does not close fully when it is supposed to, and some blood can then leak back into the left atrium instead of circulating to the rest of the body. The treatment options this study will compare are: (1) transcatheter edge-to-edge repair (abbreviated as TEER; which is a catheter procedure for repairing the mitral valve that doesn't require surgery to open up the heart) and (2) mitral valve repair surgery, which is an open-heart surgical procedure. There are no new or "experimental" procedures being tested in this study: both treatment options are well-established treatments and are regularly performed in patients who have MR.