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High-Performance Implementation of Positive Matrix Completion for SDPs

High-Performance Implementation of Positive Matrix Completion for SDPs. Makoto Yamashita (Tokyo Institute of Technology) Kazuhide Nakata (Tokyo Institute of Technology). The research was financially supported by the Sasakawa Sientific Research Grant from The Japan Science Society.

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High-Performance Implementation of Positive Matrix Completion for SDPs

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  1. High-Performance Implementation of Positive Matrix Completion for SDPs Makoto Yamashita (Tokyo Institute of Technology) Kazuhide Nakata (Tokyo Institute of Technology) The research was financially supported by the SasakawaSientific Research Grant from The Japan Science Society. INFORMS Annual Meeting 2013 2013/10/6-9 Minneapolis Convention Center, Minneapolis, USA

  2. Sparsity in SDPs Notation Only blue elements are involved in inner-product. However, we also have SDP condition. INFORMS Annual Meeting 2013

  3. Structural Sparsity in Spin-Glass SDPs • Each node interacts with only its 6 neighbors • Only the blue elements are involved in inner-product ⇒ Exploit this structural sparsity • SDP condition ⇒ Positive Matrix Completion INFORMS Annual Meeting 2013

  4. Idea of Matrix-Completion type Interior-Point Method Complement Blacks Without Blacks INFORMS Annual Meeting 2013

  5. Outline of this talk • Introduction of Matrix-Completion IPM • Speed-up by new factorization formula • Multiple threads computation • Numerical results • This talk corresponds to the new version of SDPA-C (SDPA with the Completion) • Available at http://sdpa.sf.net/ INFORMS Annual Meeting 2013

  6. Standard form of SDP • Primal-Dual form • . INFORMS Annual Meeting 2013

  7. Framework of IPM INFORMS Annual Meeting 2013

  8. Keywords in Matrix-Completion • How many elements are necessary? ⇒ Aggregate Sparsity Pattern • How to convert into smaller matrices? ⇒ Chordal Graph & Maximal Cliques • How to complete ?⇒ The form of INFORMS Annual Meeting 2013

  9. Aggregate Sparsity Pattern • Non-zero pattern in the dual side 2 1 graph Example 4 3 5 7 6 INFORMS Annual Meeting 2013

  10. Chordal Graph • Chodal, if every cycle longer than 3 has a chord • The variable matrix is decomposed by the maximal cliques. • Maximal Cliques (Clique, if there is an edge between any pair of the verticies.) length4 length5 Chordal 2 2 1 1 4 4 3 3 5 5 7 7 6 6 INFORMS Annual Meeting 2013

  11. Decomopostion of Blue: aggregate, Red:Chordal 2 1 4 3 The entire matrix can be positive definite. Grone et al. 1984 5 7 6 INFORMS Annual Meeting 2013

  12. An example of Matrix Completion 1 2 3 Non-singular& Transpose Non-singular Positive Definite INFORMS Annual Meeting 2013

  13. A remarkable property of the matrix completion • The matrix is fully-dense, but its inverse is sparse. • . INFORMS Annual Meeting 2013

  14. The factorization of the variable matrix Point:: INFORMS Annual Meeting 2013

  15. Schurcomplement matrixwith the sparse matrices INFORMS Annual Meeting 2013

  16. Outline of this talk • Introduction of Matrix-Completion IPM • Speed-up by new factorization formula • Mutliple threads computation • Numerical results INFORMS Annual Meeting 2013

  17. Review of Matrix Factorization • . • . • Point INFORMS Annual Meeting 2013

  18. INFORMS Annual Meeting 2013

  19. INFORMS Annual Meeting 2013

  20. Speed-up by the new factorization Max-clique SDP The computation time is shrunk, but there is still room to improve. Parallel computation by multiple threads INFORMS Annual Meeting 2013

  21. Multiple threadedSchur complement matrix Each column is independent from others. The thread that becomes idle computes the next column. 4 3 2 1 2 1 4 3 INFORMS Annual Meeting 2013

  22. The effect of multiple threads The number in （ ）is threads 5.31 times speed-up Max-clique INFORMS Annual Meeting 2013

  23. New version of SDPA-C INFORMS Annual Meeting 2013

  24. Test Environments and Test Problems • CPU Xeon X5365(3.0GHz), Memory 48GB, Red Hat Linux • Test Problem 1SDP relaxation of Max-Clique Problem on lattice (p,q) • Test Problem 2Spin-glasscomputation in quantum chemistry INFORMS Annual Meeting 2013

  25. Max clique SDPs (p=400,q=10) #Clique438, Average Size 29.89, Max Size 59 New SDPA-C is the fastest. INFORMS Annual Meeting 2013

  26. Spin-glass SDPs The unit of computation time is second. • The computation cost grows up mildly in SDPA-C. • The clique size is almost constant. • For larger SDPs, SDPA-C is more efficient. INFORMS Annual Meeting 2013

  27. Conclusion and future works • Speed-up of Matrix-Completion IPMby the new factorization formula • More effective for larger problems • Speed-up by multiple threads. • We should automatically select the standard IPM or Matrix-Completion IPM INFORMS Annual Meeting 2013

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