1. Matrix Completion from a few entries
Reading Group
Matrix Completion from a few entries
Raghunandan H. Kesavan, Andrea Montanari, and Sewoong Oh
Presenter: Raghu Ranganathan
ECE / CMR
Tennessee Technological University
March 4th, 2011

2. Paper overview
Let be a matrix of rank , and we observe a uniformly random subset of its entries
An efficient algorithm called OPTSPACE is described that reconstructs from observed entries with RMSE
With probability larger than
Further, if , and is sufficiently unstructured, then OPTSPACE reconstructs it exactly with entries with probability larger than
Complexity of OPTSPACE is

3. 11/5/10

4. 11/5/10

5. 11/5/10

6. Matrix Completion problem

7. Assumptions: Incoherence property (M unstructured)

8. Algorithm: Naive approach

9. Naive approach fails
For , this approach performs poorly, because contains rows and columns with non-zero revealed entries (number of non zero entries in a column have Binomial distribution)
These overrepresented rows and columns alter the spectrum of
: average number of entries per row =
: average number of entries per column =

10. Trimming

12. Main results

13. Rank-1 case

14. Relation to other work
Candes and Recht [9] introduced the incoherent model for M
Within this model, it was proved that if E is random, convex relaxation correctly reconstructs M as long as
From information theoretic point of view, it is clear that
should reconstruct M with arbitrary precision. This point was raised in [9], and proved in [20]. Theorem 1.1 confirms this and shows that simple trimming plus SVD can achieve this objective
The bound on for exact recovery provided by Theorem 1.2 improves upon the corresponding bound in [9] for
[9] E. J. Candès and B. Recht, Exact Matrix Completion via Convex Optimization
[20] R. H. Keshavan, A. Montanari, and S. Oh, “Learning low rank matrices from entries,” presented at the Allerton Conf. on Commun., Control and Computing, Sep

15. Simulations: Reconstruction Rate
The matrix is considered reconstructed if the relative error
The reconstruction rate is the fraction of instances for which the matrix was reconstructed
11/5/10

16. [31] A. Singer and M. Cucuringu, Uniqueness of Low-Rank Matrix Completion by Rigidity Theory, 2009 [Online]. Available: arXiv:

17. Conclusions