### abstract ###
We consider the problem of reconstructing a low rank  matrix from a small subset of its entries
In this paper,  we describe the implementation of an efficient algorithm proposed in  CITATION ,  based on singular value decomposition followed by local manifold optimization,  for solving the low-rank matrix completion problem
It has been shown that if the number of revealed entries is large enough,  the output of singular value decomposition gives a good estimate for the original matrix,  so that local optimization reconstructs the correct matrix with high probability
We present numerical results which show that  this algorithm can reconstruct the low rank matrix exactly  from a very small subset of its entries
We further study the robustness of the algorithm with respect to noise, and its performance on actual collaborative filtering datasets
### introduction ###
In this paper we consider the problem of reconstructing an  SYMBOL  low rank matrix  SYMBOL  from a small set of observed entries
This problem is of considerable practical interest and has many applications
One example is collaborative filtering, where users submit rankings for small subsets of, say, movies, and the goal is to infer the preference of unrated movies for a recommendation system  CITATION
It is believed that the movie-rating matrix is approximately low-rank, since only a few factors contribute to a user’s preferences
Other examples of matrix completion include the problem of inferring 3-dimensional structure from motion  CITATION  and triangulation from incomplete data of distances between wireless sensors, also known as the sensor localization problem  CITATION ,  CITATION
