SVD Based Noise Filter on the Samples in Toeplitz Form
The matrix M is called a Toeplitz matrix and they have a certain type of decomposition (but X should be the ACF and not the signal itself), so the resulting U and V would be a Vandermonde matrix which are very similar to Discrete Fourier transform.
So the whole process is similar to find spectrum of your signal and putting zero in bins which their power are below a threshold.