Rank aggregation via nuclear norm minimization
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1 Rank aggregation via nuclear norm minimization David F. Gleich Purdue Lek-Heng Lim University of Chicago KDD2011 San Diego, CA Lek funded by NSF CAREER award (DMS ); David funded by DOE John von Neumann and NSERC David F. Gleich (Purdue) KDD /20
2 Which is a better list of good DVDs? Lord of the Rings 3: The Return of Lord of the Rings 1: The Fellowship Lord of the Rings 2: The Two Towers Lost: Season 1 Battlestar Galactica: Season 1 Fullmetal Alchemist Trailer Park Boys: Season 4 Trailer Park Boys: Season 3 Tenchi Muyo! Shawshank Redemption Lord of the Rings 3: The Return of Lord of the Rings 1: The Fellowship Lord of the Rings 2: The Two Towers Star Wars V: Empire Strikes Back Raiders of the Lost Ark Star Wars IV: A New Hope Shawshank Redemption Star Wars VI: Return of the Jedi Lord of the Rings 3: Bonus DVD The Godfather Standard rank aggregation (the mean rating) Nuclear Norm based rank aggregation (not matrix completion on the netflix rating matrix) David F. Gleich (Purdue) KDD /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 /20
4 Ranking is really hard Ken Arrow John Kemeny Dwork, Kumar, Naor, Sivikumar All rank aggregations involve some measure of compromise A good ranking is the average ranking under a permutation distance NP hard to compute Kemeny s ranking David F. Gleich (Purdue) KDD /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 /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 /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 /20
8 Movie Pairs Extracting the scores Given with all entries, then is the Borda count, the least-squares solution to How many Most. Do we trust all? Not really. do we have? Number of Comparisons Netflix data 17k movies, 500k users, 100M ratings 99.17% filled David F. Gleich (Purdue) KDD /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 /20
10 Solution Go Nuclear From a French nuclear test in 1970, image from David F. Gleich (Purdue) KDD
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 answer under appropriate circumstances best convex underestimator of rank on unit ball. David F. Gleich (Purdue) KDD /20
12 Heuristic 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 Convex David F. Gleich (Purdue) KDD /20
13 Solving the nuclear norm problem Use a LASSO formulation REPEAT 3. = rank-k SVD of 4. Jain et al. propose SVP for this problem without UNTIL Jain et al. NIPS 2010 David F. Gleich (Purdue) KDD /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 skewsymmetric constraint for free David F. Gleich (Purdue) KDD /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 David F. Gleich (Purdue) KDD /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 /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 /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 /20
19 Conclusions and Future Work aggregate, then complete 1. Compare against others Rank aggregation with the nuclear norm is principled easy to compute The results are much better than simple approaches. 2. Noisy recovery! More realistic sampling. 3. Skew-symmetric Lanczos based SVD? David F. Gleich (Purdue) KDD /20
20 Google nuclear ranking gleich
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