  In this paper we provide a new efficient algorithm for approximately
computing the profile maximum likelihood (PML) distribution, a prominent
quantity in symmetric property estimation. We provide an algorithm which
matches the previous best known efficient algorithms for computing approximate
PML distributions and improves when the number of distinct observed frequencies
in the given instance is small. We achieve this result by exploiting new
sparsity structure in approximate PML distributions and providing a new matrix
rounding algorithm, of independent interest. Leveraging this result, we obtain
the first provable computationally efficient implementation of PseudoPML, a
general framework for estimating a broad class of symmetric properties.
Additionally, we obtain efficient PML-based estimators for distributions with
small profile entropy, a natural instance-based complexity measure. Further, we
provide a simpler and more practical PseudoPML implementation that matches the
best-known theoretical guarantees of such an estimator and evaluate this method
empirically.
