Fundamental limits of singular value based signal detection from randomly compressed signal-plus-noise matrices

The singular value spectrum of a data matrix is commonly used to detect high-dimensional signals. However, as the size of this data matrix grows, taking its SVD becomes intractable. We consider projecting the data matrix into a lower dimensional space and using the resulting singular value spectrum for signal detection. We derive the almost sure limit of the top singular values of the resulting projected matrix both when using a Gaussian and unitary projection matrix. We highlight our prediction accuracy and discuss the benefits and drawbacks of each projection matrix using numerical simulations.