Are there algorithms to invert a magnitude spectrum?
Apart from the stochastic literature, there are a set of deterministic algorithms that claim exact(?) recovery of a signal form its partial Fourier description, collectively referred to as:


  signal reconstruction from DTFT phase or magnitude alone


They are generally iterative in nature and exact convergence is hard to achieve in practice. Monson Hayes [et al] published relevant IEEE papers to propose several algorithms that reconstruct a real signal from its DTFT phase function alone, which is based on the phase sufficiency claim. The magnitude is a little more diffcult as I know. Have a look at those papers for your benefit.