  Finite unit norm tight frames provide Parseval-like decompositions of vectors
in terms of redundant components of equal weight. They are known to be
exceptionally robust against additive noise and erasures, and as such, have
great potential as encoding schemes. Unfortunately, up to this point, these
frames have proven notoriously difficult to construct. Indeed, though the set
of all unit norm tight frames, modulo rotations, is known to contain manifolds
of nontrivial dimension, we have but a small finite number of known
constructions of such frames. In this paper, we present a new iterative
algorithm---gradient descent of the frame potential---for increasing the degree
of tightness of any finite unit norm frame. The algorithm itself is trivial to
implement, and it preserves certain group structures present in the initial
frame. In the special case where the number of frame elements is relatively
prime to the dimension of the underlying space, we show that this algorithm
converges to a unit norm tight frame at a linear rate, provided the initial
unit norm frame is already sufficiently close to being tight. By slightly
modifying this approach, we get a similar, but weaker, result in the
non-relatively-prime case, providing an explicit answer to the Paulsen problem:
"How close is a frame which is almost tight and almost unit norm to some unit
norm tight frame?"
