  Sketched gradient algorithms have been recently introduced for efficiently
solving the large-scale constrained Least-squares regressions. In this paper we
provide novel convergence analysis for the basic method {\it Gradient
Projection Classical Sketch} (GPCS) to reveal the fast linear convergence rate
of GPCS towards a vicinity of the solution thanks to the intrinsic
low-dimensional geometric structure of the solution prompted by constraint set.
Similar to our analysis we observe computational and sketch size trade-offs in
numerical experiments. Hence we justify that the combination of gradient
methods and the sketching technique is a way of designing efficient algorithms
which can actively exploit the low-dimensional structure to accelerate
computation in large scale data regression and signal processing applications.
