  Smooth convex minimization over the unit trace-norm ball is an important
optimization problem in machine learning, signal processing, statistics and
other fields, that underlies many tasks in which one wishes to recover a
low-rank matrix given certain measurements. While first-order methods for
convex optimization enjoy optimal convergence rates, they require in worst-case
to compute a full-rank SVD on each iteration, in order to compute the
projection onto the trace-norm ball. These full-rank SVD computations however
prohibit the application of such methods to large problems. A simple and
natural heuristic to reduce the computational cost is to approximate the
projection using only a low-rank SVD. This raises the question if, and under
what conditions, this simple heuristic can indeed result in provable
convergence to the optimal solution. In this paper we show that any optimal
solution is a center of a Euclid. ball inside-which the projected-gradient
mapping admits rank that is at most the multiplicity of the largest singular
value of the gradient vector. Moreover, the radius of the ball scales with the
spectral gap of this gradient vector. We show how this readily implies the
local convergence (i.e., from a "warm-start" initialization) of standard
first-order methods, using only low-rank SVD computations. We also quantify the
effect of "over-parameterization", i.e., using SVD computations with higher
rank, on the radius of this ball, showing it can increase dramatically with
moderately larger rank. We extend our results also to the setting of
optimization with trace-norm regularization and optimization over bounded-trace
positive semidefinite matrices. Our theoretical investigation is supported by
concrete empirical evidence that demonstrates the \textit{correct} convergence
of first-order methods with low-rank projections on real-world datasets.
