IMP: A Message-Passing Algorithm for Matrix Completion
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1 IMP: A Message-Passing Algorithm for Matrix Completion Henry D. Pfister (joint with Byung-Hak Kim and Arvind Yedla) Electrical and Computer Engineering Texas A&M University ISTC 2010 Brest, France September 9, 2010 Kim, Yedla and Pfister (Texas A&M) IMP for Matrix Completion ISTC / 17
2 Outline 1 Matrix Completion via Iterative Information Processing 2 Factor Graph Model for Matrix Completion 3 IMP Algorithm and Performance Results 4 Conclusions and Open Problems Kim, Yedla and Pfister (Texas A&M) IMP for Matrix Completion ISTC / 17
3 Matrix Completion via Iterative Information Processing Matrix Completion Problem In its simplest form, the problem is to recover a matrix from a small sample of its entries,... Imagine now that we only observe a few entries of a data matrix. Then is it possible to accurately or even exactly guess the entries that we have not seen?... - Candes and Plan 09 Kim, Yedla and Pfister (Texas A&M) IMP for Matrix Completion ISTC / 17
4 Matrix Completion via Iterative Information Processing Motivation: The Netflix Challenge ( 06-09) An overwhelming portion of the user-movie matrix (e.g., 99%) is unknown and the few observed movie ratings are noisy! Kim, Yedla and Pfister (Texas A&M) IMP for Matrix Completion ISTC / 17
5 Matrix Completion via Iterative Information Processing Recent Work on Matrix Completion Efficient Models and Practical Algorithms Low rank matrix models and clustering models Convex relaxation (SDP) and Bayesian approaches Exploration of the Fundamental Limits Main Differences Relationship between sparse observation and the recovery of missing entries Cold-start problem in recommender systems 1 Focus on the matrix where the entries (drawn from a finite alphabet) are modeled by a factor graph 2 MP based algorithm to learn the missing information and eventually estimate missing entries Kim, Yedla and Pfister (Texas A&M) IMP for Matrix Completion ISTC / 17
6 Matrix Completion via Iterative Information Processing Our Goal: Matrix Completion via Iterative Processing Basic Approach Establish a factor-graph model Estimate parameters of model Use for inference via message-passing (IMP) Benefits of our factor-graph model Establishes a generative model for data matrices Sparse observations reduce complexity Drawbacks of this approach Mixes clustering and message-passing Difficult to analyze behavior Kim, Yedla and Pfister (Texas A&M) IMP for Matrix Completion ISTC / 17
7 Factor Graph Model for Matrix Completion Factor Graph Model (1): Conditional Independence Movies V Ratings R U Users V 0 V 1 V 2 V 3 V 4 V 5 V 6 permutation permutation U 0 U 1 U 2 U 3 U 4 U 5 U 6 V M U N Assumption The movie ratings R nm (user n, movie m) are conditionally independent given the user group U n and the movie group V m Pr (R O U, V) w (R nm U n, V m ) (n,m) O where w(r u, v) Pr(R nm = r U n = u, V m = v) Kim, Yedla and Pfister (Texas A&M) IMP for Matrix Completion ISTC / 17
8 Factor Graph Model for Matrix Completion Factor Graph Model (2): Computational Tree Unknown Rating Known Rating Movie Group User Group Kim, Yedla and Pfister (Texas A&M) IMP for Matrix Completion ISTC / 17
9 Factor Graph Model for Matrix Completion Factor Graph Model (3): Matrix Decomposition MMSE estimates can be written as a matrix factorization In contrast to the standard low-rank matrix model, this adds non-negativity and normalization constraints Kim, Yedla and Pfister (Texas A&M) IMP for Matrix Completion ISTC / 17
10 IMP Algorithm and Performance Results Inference by Message-Passing (IMP) Algorithm (1) Step I. Initialize Group Rating Probabilities w(r u, v) Cluster users (and movies) using variable-dimension vector quantization (VDVQ), which is a variant of VQ where the codebook vectors are full, but training vectors are sparse. Training is based on based on the generalized Lloyd algorithm (GLA) and the distance computed only on overlapping entries. For user clustering, the K codebook vectors can be thought of as K critics which have rated every movie. After clustering users/movies each into user/movie groups, estimate w(r u, v) from the frequencies of each user-group / movie-group / rating triple. Kim, Yedla and Pfister (Texas A&M) IMP for Matrix Completion ISTC / 17
11 IMP Algorithm (2) IMP Algorithm and Performance Results Movies V V 0 V 1 V 2 V 3 V 4 V 5 V 6 permutation V M x (i) Ratings R O permutation Notation U Users U 0 U 1 U 2 U 3 U 4 U 5 U 6 Let T (i) m n be m n computation tree after i iterations, then message from movie m to user n is x (i) m n Pr(V m = v T (i) message from user n to movie m is y (i) n m Pr(U n = u T (i) n m) U N y (i) m n) and Kim, Yedla and Pfister (Texas A&M) IMP for Matrix Completion ISTC / 17
12 IMP Algorithm (3) IMP Algorithm and Performance Results Step II. Message-Passing Update of Group Vectors 1 Initialization of user/movie to prior group probabilities m (v)=p V (v), y (0) n (u)=p U (u) x (0) 2 Recursive update for user/movie group probabilities x (i+1) m n(v) x m (0) (v) w (r u, v) y (i) y (i+1) k U m\n u n m(u) y n (0) (u) w (r u, v) x (i) k V n\m v k m (u) k n (v) U m : all users who rated movie m V n : all movies whose rating by user n was observed 3 Update w(r u, v) for group ratings Kim, Yedla and Pfister (Texas A&M) IMP for Matrix Completion ISTC / 17
13 IMP Algorithm and Performance Results IMP Algorithm (4) Step III. Approximate Matrix Completion After i iterations of message-passing update, Output probability of rating R nm given observed ratings ˆp (i+1) R nm R (r) x (i+1) O m n(v)w (r u, v) y n m(u) (i+1) u,v Minimize the mean-squared prediction error with ˆr (i+1) n,m = r R r ˆp (i+1) R nm R O (r) Kim, Yedla and Pfister (Texas A&M) IMP for Matrix Completion ISTC / 17
14 IMP Algorithm and Performance Results IMP Improves the Cold-Start Problem RMSE Netflix Data Matrix 2 IMP OptSpace SET SVT Movie Average Average Number of Observed Ratings per User Dataset Description Netflix submatrix of 5,035 movies and 5,017 users by avoiding movies and users with less than 3 ratings 16% of the users and 41% of the movies have less than 10 ratings Kim, Yedla and Pfister (Texas A&M) IMP for Matrix Completion ISTC / 17
15 IMP Algorithm and Performance Results Generation of Synthetic Datasets RMSE Synthetic Dataset 2 IMP Movie Average Lower Bound Average Number of Observed Ratings per User Dataset Description Synthetic Dataset generated after learning Netflix Data Matrix with 16 movie/user groups and randomly subsampled Kim, Yedla and Pfister (Texas A&M) IMP for Matrix Completion ISTC / 17
16 Conclusion Conclusion Main Contributions 1 Factor Graph Model: Probabilistic low-rank matrix model 2 IMP Algorithm: Combines clustering with message-passing Avenues for Future Research 1 Combine with clustering via message-passing to get a fully iterative approach 2 Obtain performance analysis of IMP algorithm via density evolution Kim, Yedla and Pfister (Texas A&M) IMP for Matrix Completion ISTC / 17
17 Thank you Kim, Yedla and Pfister (Texas A&M) IMP for Matrix Completion ISTC / 17
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