I want to get the first few eigenvectors of real symmetric matrices with missing values. Since it has missing values, I won't be able to use the common linear programming techniques, but stochastic gradient will work. Funk's SVD used in recommendation engines solves a more general problem of low-rank SVD approximation using gradient descent. So some flavor of Funk's SVD should be able to solve this. I will be using this in R so any R or C++ library that plays well with Rcpp will do. Are there any standard implementations or libraries that I can use? It doesn't make sense to roll my own for such a standard algorithm.
In my comments I already mentioned two threads (here and here) in SE that presented some interesting discussion points regarding FunkSVD. The only place I have found a decent FunkSVD implementation that could potentially work with R is LensKit's, Matrix Factorization CF module. Let me point out that this will not be straightforward as you will need to use a JAVA API to connect from R to Lenskit.
Given you problem maybe checking the R package recommenderlab might also be beneficial. Let me note that in many cases recommender systems essentially give out an informed nearest-neighbour suggestion so for your particular problem you might be able to re-express it as clustering/classification task rather than a manifold-learning one.