1. Robust Optimization and Applications in Machine Learning

2. Part 4: Sparsity in Unsupervised Learning

3. Unsupervised learning

4. Sparse PCA: outline

5. Principal Component Analysis

6. PCA for visualization

7. First principal component

8. Sparse PCA: outline

9. Why sparse factors?

10. PCA: rank-one case

11. sparse PCA: rank-one case

12. Sparse PCA: outline

13. SDP relaxation

15. Dual problem

16. Sparsity and robustness

17. Sparse PCA decomposition?

18. Sparse PCA: outline

19. First-order algorithm

20. Sparse PCA: outline

21. PITPROPS data

22. PITPROPS data: numerical results

23. Financial example

24. Covariance matrix

25. Second factor

26. Gene expression data

27. Clustering of gene expression data

28. Conclusions on sparse PCA

29. Part 4: Sparsity in Unsupervised Learning

30. Sparse Gaussian networks: outline

31. Gaussian network problem

32. Correlation-based approach

33. Approach based on the precision matrix

34. Example

35. Relevance network vs. graphical model

36. Can we check this?

37. Sparse inverse covariance and conditional independence

38. Related work

39. Maximum-likelihood estimation

40. Problems with ordinary MLE

41. MLE with cardinality penalty

42. Convex relaxation

43. Link with robustness

44. Properties of estimate

45. Algorithms: challenges

46. First- vs. second-order algorithms

47. Black- vs. grey-box first-order algorithms

48. Algorithms: problem structure

49. Nesterov’s smooth minimization algorithm

50. Nesterov’s method

51. Putting the problem in Nesterov’s format

52. Making the problem smooth

53. Optimal scheme for smooth minimization

54. Application to our problem

55. Dual block-coordinate descent

56. Properties of dual block-coordinate descent

57. Link with LASSO

58. Example

59. Inverse covariance estimates

60. Average error on zeros

61. Computing time

62. Classification error

63. Recovering structure

64. Part 4: summary

65. References