Inspired by high-dimensional data analysis and multi-spectral imaging, we aim to reconstruct a finite dimensional vector from a set of magnitudes of its subspace components. First, we develop closed formulas for signal reconstruction. Second, we use semi-definite programming and random subspaces to reduce the number of required subspace components. We also address the optimal choice of the subspace dimensions. Motivated by applications in physics, we have some suggestions for the reconstruction of a finite-dimensional signal from the absolute values of its Fourier coefficients.