Renormalization group (RG) flows are commonly used in quantum field theory to describe the changes in degrees of freedom as we "flow" from very high to very low energies. In particular, there have been a number of proofs of the irreversibility of RG flows in various dimensions, i.e., the statement that the number of high-energy degrees of freedom is larger than the number of low-energy degrees of freedom. In this talk, I will discuss RG flows across dimensions, which one can think of as probing compact extra dimensions. We will do so using the powerful tool of holography, which allows us to recast this field theoretic problem as a geometric flow in anti-de Sitter space.