  We study the factorization of soft and collinear singularities in
dimensionally-regularized fixed-angle scattering amplitudes in massless gauge
theories. Our factorization is based on replacing the hard massless partons by
light-like Wilson lines, and defining gauge-invariant jet and soft functions in
dimensional regularization. In this scheme the factorized amplitude admits a
powerful symmetry: it is invariant under rescaling of individual Wilson-line
velocities. This symmetry is broken by cusp singularities in both the soft and
the eikonal jet functions. We show that the cancellation of these cusp
anomalies in any multi-leg amplitude imposes all-order constraints on the
kinematic dependence of the corresponding soft anomalous dimension, relating it
to the cusp anomalous dimension. For amplitudes with two or three hard partons
the solution is unique: the constraints fully determine the kinematic
dependence of the soft function. For amplitudes with four or more hard partons
we present a minimal solution where the soft anomalous dimension is a sum over
colour dipoles, multiplied by the cusp anomalous dimension. In this case
additional contributions to the soft anomalous dimension at three loops or
beyond are not excluded, but they are constrained to be functions of conformal
cross ratios of kinematic variables.
