This document describes a method for rank aggregation using nuclear norm minimization. It proposes modeling the ranking problem as a skew-symmetric matrix completion problem and solving it using nuclear norm regularization. The nuclear norm heuristic provides a convex relaxation to the hard rank minimization problem. Experiments on synthetic data show the nuclear norm approach outperforms simple averaging baselines. Future work includes comparing to other rank aggregation methods, analyzing recovery under noisy conditions, and developing faster SVD solvers.

1. Rank aggregation via nuclear norm minimization
David F. Gleich
Purdue University
@dgleich
Lek-Heng Lim
University of Chicago
KDD2011
San Diego, CA
Lek funded by NSF CAREER award (DMS-1057064); David funded by DOE John von Neumann and NSERC
David F. Gleich (Purdue) KDD 2011 1/20

2. Which is a better list of good DVDs?
Lord of the Rings 3: The Return of … Lord of the Rings 3: The Return of …
Lord of the Rings 1: The Fellowship Lord of the Rings 1: The Fellowship
Lord of the Rings 2: The Two Towers Lord of the Rings 2: The Two Towers
Lost: Season 1 Star Wars V: Empire Strikes Back
Battlestar Galactica: Season 1 Raiders of the Lost Ark
Fullmetal Alchemist Star Wars IV: A New Hope
Trailer Park Boys: Season 4 Shawshank Redemption
Trailer Park Boys: Season 3 Star Wars VI: Return of the Jedi
Tenchi Muyo! Lord of the Rings 3: Bonus DVD
Shawshank Redemption The Godfather
Standard Nuclear Norm
rank aggregation based rank aggregation
(the mean rating) (not matrix completion on the
netflix rating matrix)
David F. Gleich (Purdue) KDD 2011 2/20

3. Rank Aggregation
Given partial orders on subsets of items, rank aggregation is
the problem of finding an overall ordering.
Voting Find the winning candidate
Program committees Find the best papers given reviews
Dining Find the best restaurant in San Diego (subject to a budget?)
David F. Gleich (Purdue) KDD 2011 3/20

4. Ranking is really hard
John Kemeny Dwork, Kumar, Naor,
Ken Arrow Sivikumar
All rank aggregations
involve some measure A good ranking is the
of compromise “average” ranking under NP hard to compute
a permutation distance Kemeny’s ranking
David F. Gleich (Purdue) KDD 2011 4/20

5. Embody chair
John Cantrell (flickr)
Given a hard problem,
what do you do?
Numerically relax!
It’ll probably be easier.
David F. Gleich (Purdue) KDD 2011 5/20

6. Suppose we had scores
Let be the score of the ith movie/song/paper/team to rank
Suppose we can compare the ith to jth:
Then is skew-symmetric, rank 2.
Also works for with an extra log.
Numerical ranking is intimately intertwined
with skew-symmetric matrices
Kemeny and Snell, Mathematical Models in Social Sciences (1978)
David F. Gleich (Purdue) KDD 2011 6/20

7. Using ratings as comparisons
Ratings induce various skew-symmetric matrices.
Arithmetic Mean
Log-odds
David 1988 – The Method of Paired Comparisons
David F. Gleich (Purdue) KDD 2011 7/20

8. Extracting the scores
Given with all entries, then 107
is the Borda
Movie Pairs
105
count, the least-squares
solution to
How many do we have? 101
Most.
101 105
Number of Comparisons
Do we trust all ?
Not really. Netflix data 17k movies,
500k users, 100M ratings–
99.17% filled
David F. Gleich (Purdue) KDD 2011 8/20

9. Only partial info? Complete it!
Let be known for We trust these scores.
Goal Find the simplest skew-symmetric matrix that matches
the data
noiseless
noisy
Both of these are NP-hard too.
David F. Gleich (Purdue) KDD 2011 9/20

10. Solution Go Nuclear
From a French nuclear test in 1970, image from http://picdit.wordpress.com/2008/07/21/8-insane-nuclear-explosions/
David F. Gleich (Purdue) KDD 2011 10

11. The nuclear norm
The analog of the 1-norm or for matrices.
For vectors For matrices
Let be the SVD.
is NP-hard while
is convex and gives the same best convex under-
answer “under appropriate estimator of rank on unit ball.
circumstances”
David F. Gleich (Purdue) KDD 2011 11/20

12. Only partial info? Complete it!
Let be known for We trust these scores.
Goal Find the simplest skew-symmetric matrix that matches
the data
NP hard
Heuristic
Convex
David F. Gleich (Purdue) KDD 2011 12/20

13. Solving the nuclear norm problem
Use a LASSO formulation 1.
2. REPEAT
3. = rank-k SVD of
4.
5.
6. UNTIL
Jain et al. propose SVP for
this problem without
Jain et al. NIPS 2010
David F. Gleich (Purdue) KDD 2011 13/20

14. Skew-symmetric SVDs
Let be an skew-symmetric matrix with
eigenvalues ,
where and . Then the SVD of is
given by
for and given in the proof.
Proof Use the Murnaghan-Wintner form and the SVD of a
2x2 skew-symmetric block
This means that SVP will give us the skew-
symmetric constraint “for free”
David F. Gleich (Purdue) KDD 2011 14/20

15. Exact recovery results
David Gross showed how to recover Hermitian matrices.
i.e. the conditions under which we get the exact
Note that is Hermitian. Thus our new result!
Gross arXiv 2010.
David F. Gleich (Purdue) KDD 2011 15/20

16. Recovery Discussion and Experiments
Confession If , then just look at differences from
a connected set. Constants? Not very good.
Intuition for the truth.
David F. Gleich (Purdue) KDD 2011 16/20

17. The Ranking Algorithm
0. INPUT (ratings data) and c
(for trust on comparisons)
1. Compute from
2. Discard entries with fewer than
c comparisons
3. Set to be indices and
values of what’s left
4. = SVP( )
5. OUTPUT
David F. Gleich (Purdue) KDD 2011 17/20

18. Synthetic Results
Construct an Item Response Theory model. Vary number of
ratings per user and a noise/error level
Our Algorithm! The Average Rating
David F. Gleich (Purdue) KDD 2011 18/20

19. Conclusions and Future Work
“aggregate, then complete” 1. Compare against others
Rank aggregation with 2. Noisy recovery! More
the nuclear norm is realistic sampling.
principled
easy to compute 3. Skew-symmetric Lanczos
The results are much better based SVD?
than simple approaches.
David F. Gleich (Purdue) KDD 2011 19/20

20. Google nuclear ranking gleich