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Coupled Matrix Factorizations using OptimizationPowerPoint Presentation

Coupled Matrix Factorizations using Optimization

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### Coupled Matrix Factorizations using Optimization

Daniel M. Dunlavy, Tamara G. Kolda, Evrim Acar

Sandia National Laboratories

SIAM Conference on Computational Science and Engineering

March 4, 2009

Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company,for the United States Department of Energy’s National Nuclear Security Administration under contract DE-AC04-94AL85000.

SAND2009-2389C

1/17

Motivating Problems

- Data with multiple types of two-way relationships
- Bibliometric analysis
- author-document, term-document, author-venue, etc.
- Can we predict potential co-authors?

- Movie ratings
- movie-actor, user-movie, actor-award
- Can we predict useful movie ratings for other users?

- Bibliometric analysis
- Consistent dimensionality reduction
- Improved interpretation through non-negativity constraints

2/17

Some Related Work

matrices of same size

- Simultaneous factor analysis
- Gramian matrices [Levin, 1966]
- Test score covariance matrices over time [Millsap, et al., 1988]

- Simultaneous diagonalization
- Population differentiation in biology [Thorpe, 1988]
- Blind source separation [Ziehe et al., 2004]

- Generalized SVD
- Damped or constrained least squares [Van Loan, 1976]
- Microarray data analysis [Alter, et al., 2003]
- Multimicrophone speech filtering [Doclo and Moonen, 2002]

- Simultaneous Non-negative Matrix Factorization
- Gene clustering in microarray data [Badea, 2007; 2008]

- Tensor decompositions
- Data mining, chemometrics, neuroscience[Kolda, Acar, Bro, Park, Zhang, Berry, Chen, Martin, CSE09]

matrices of same size

only 2 matrices

slow

at least one common dimension

3/17

Method: CNMF-ALS

- CNMF-ALS: Alternating Least Squares [Extends Berry, et al., 2006]

linear least squares

+ simple projection to constraint boundary

5/17

Method: CNMF-MULT

- CNMF-MULT: Multiplicative Updates [Badea, 2007; Badea, 2008; extends Lee and Seung, 2001]

6/17

Method: CNMF-OPT

- CNMF-OPT: Projective Nonlinear CG, More-Thuente LS[Extends Acar, Kolda, and Dunlavy, 2009 and Lin, 2007]

7/17

Future Work

- Extending other promising methods to CNMF
- Block principal pivoting based NMF [Park, et al. 2008]
- Projected gradient NMF [Lin, 2007]
- Projected Newton NMF [Kim, et al., 2008]

- CNMF-OPT extensions
- Sparse data, regularization [Acar, Kolda, and Dunlavy, 2009]
- Sparsity constraints [Park, et al. 2008]

- Numerical experiments
- Scale to larger data sets
- Comparisons on real data sets [Park, et al. 2008]

- Alternate models / problem formulations
- Coupling matrix and tensor decompositions (CNMF/CNTF)

15/17

Conclusions

- Coupled matrix factorizations
- Method for computing factorizations consistent along common dimensions in data

- Results
- CNMF-OPT
- Fast and accurate
- Overfactors well and handles noise well

- Fast and accurate
- CNMF-ALS
- Fast, but not accurate
- Overfactoring is a big challenge

- Fast, but not accurate
- CNMF-MULT
- Accurate, but may be too slow (similar to NMF results)

- CNMF-OPT
- Future Work
- Identified several promising paths forward

16/17

Thank You

Coupled Matrix Factorizations using Optimization

Danny Dunlavy

http://www.cs.sandia.gov/~dmdunla

17/17

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