  A fast non-convex low-rank matrix decomposition method for potential field
data separation is proposed. The singular value decomposition of the large size
trajectory matrix, which is also a block Hankel matrix, is obtained using a
fast randomized singular value decomposition algorithm in which fast block
Hankel matrix-vector multiplications are implemented with minimal memory
storage. This fast block Hankel matrix randomized singular value decomposition
algorithm is integrated into the \texttt{Altproj} algorithm, which is a
standard non-convex method for solving the robust principal component analysis
optimization problem. The improved algorithm avoids the construction of the
trajectory matrix. Hence, gravity and magnetic data matrices of large size can
be computed. Moreover, it is more efficient than the traditional low-rank
matrix decomposition method, which is based on the use of an inexact augmented
Lagrange multiplier algorithm. The presented algorithm is also robust and,
hence, algorithm-dependent parameters are easily determined. The improved and
traditional algorithms are contrasted for the separation of synthetic gravity
and magnetic data matrices of different sizes. The presented results
demonstrate that the improved algorithm is not only computationally more
efficient but it is also more accurate. Moreover, it is possible to solve far
larger problems. As an example, for the adopted computational environment,
matrices of sizes larger than $205 \times 205$ generate "out of memory"
exceptions with the traditional method, but a matrix of size $2001\times 2001$
can be calculated in $1062.29$s with the new algorithm. Finally, the improved
method is applied to separate real gravity and magnetic data in the Tongling
area, Anhui province, China. Areas which may exhibit mineralizations are
inferred based on the separated anomalies.
