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Optimization Problem Based on L 2,1 -norms

Optimization Problem Based on L 2,1 -norms. Xiaohong Chen 19-10-2012. Outline. Efficient and robust feature selection via joint l 2,1 -norm minimzation Robust and discriminative distance for multi-instance learning Its application…. Outline.

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Optimization Problem Based on L 2,1 -norms

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  1. Optimization Problem Based on L2,1-norms Xiaohong Chen 19-10-2012

  2. Outline • Efficient and robust feature selection via joint l2,1-norm minimzation • Robust and discriminative distance for multi-instance learning • Its application…

  3. Outline • Efficient and robust feature selection via joint l2,1-norm minimization • Robust and discriminative distance for multi-instance learning • Its application…

  4. Efficient and robust feature selection via joint l2,1-norm minimzation

  5. Robust Feature Selection Based on l21-norm Given training data {x1, x2,…, xn} and the associated class labels {y1,y2,…, yn} Least square regression solves the following optimizaiton problem to obtain the projection matrix W Add a regularization R(W) to the robust version of LS,

  6. Robust Feature Selection Based on l21-norm Possible regularizations

  7. Robust Feature Selection Based on l21-norm

  8. Robust Feature Selection Based on l21-norm Denote (14)

  9. Robust Feature Selection Based on l21-norm Then we have (19)

  10. The iterative algorithm to solve problem (14) Theorem1:The algorithm will monotonically decrease the objective of the problem in Eq.(14) in each iteration, and converge to the global optimum of the problem.

  11. u u Proof of theorem1

  12. Proof of theorem1

  13. (1)+(2) (1) (2)

  14. Outline • Efficient and robust feature selection via joint l2,1-norm minimization • Robust and discriminative distance for multi-instance learning • Its application…

  15. Robust and discriminative distance for multi-instance learning

  16. Multi-instance learning 多示例学习中，训练集由若干个具有概念标记的包(bag)组成， 每个包包含若干个没有概念标记的示例。若一个包中至少有 一个正例，则该包被标记为正(positive)，若一个包中所以示 例都是反例，则该包被标记为反(negative),通过对训练包的学 习，希望学习系统尽可能正确地对训练集之外的包的概念标 记进行预测。

  17. The illustration of MIL

  18. Notations Given N training bags and K conceptual classes. Each bag contains a number of instances Given the class memberships of the input data, denoted as

  19. Notations First, we represent every class as a super-bag that comprises the instances of all its training , where

  20. Objective to learn class specific distance metrics For a given class, Ck,, we solve the following optimization problem:

  21. Algorithm and its analysis

  22. Algorithm and its analysis

  23. Algorithm and its analysis

  24. Algorithm and its analysis On the other hand,

  25. Algorithm and its analysis

  26. Algorithm and its analysis

  27. Algorithm and its analysis Therefore, the objective value of the problem of (6) is decreased in each iteration till convergences.

  28. Outline • Efficient and robust feature selection via joint l2,1-norm minimzation • Robust and discriminative distance for multi-instance learning • Its application…

  29. Its application For example:

  30. Reference • [1]F.Nie, D.Xu, X.Cai, and C.Ding. Efficient and robust feature selection via • joint l2,1-norm minimzation. NIPS 2010. • [2] H.Wang, F.Nie and H.Huang. Robust and discriminative distance for multi- • instance learning, CVPR 2012: 2919-2924

  31. Thanks! Q&A

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