  We show how to use equidimensional algebraic correspondences between complex
algebraic varieties to construct pull-backs and transforms of certain classes
of geometric currents. Using this construction we produce explicit formulas at
the level of complexes for a regulator map from the Higher Chow groups of
smooth complex quasi-projective algebraic varieties to Deligne-Beilinson
cohomology with integral coefficients. A distinct aspect of our approach is the
use of Suslin's complex of equidimensional cycles over $ \Delta^n $ to compute
Bloch's higher Chow groups. We calculate explicit examples involving the
M\"{a}hler measure of Laurent polynomials.
