See All by Looking at A Few

1. Sparse Modeling for Finding Representative Objects See All by Looking at A Few EhsanElhamifar Guillermo SapiroRen´eVidal Johns Hopkins University Universityof Minnesota Johns Hopkins University

2. Outline • Introduction • Problem Formulation • Geometry of Representatives • Representatives of Subspaces • Practical Considerations and Extensions • Experimental Results

3. Introduction • Two important problem related to large database • Reducing the Feature-space dimension • Reducing the Object-space dimension

4. Introduction • Kmeans < Step3: The centriod of each of the k clusters becomes the new mean < Step1: k initial "means" are randomly selected from the data set < Step4: Steps 2 and 3 are repeated until convergence has been reached < Step2: k clusters are created by every observation with the nearest mean

5. Introduction • Kmedoids • A variant of Kmeans < Step3

6. Introduction • Rank Revealing QR(RRQR) • a matrix decomposition algorithm based on the QR factorization • T. Chan. Rank revealing qr factorizations. Lin. Alg. and its Appl.,1987. 1

7. Introduction • Consider the optimization problem • Wish to find at most k << N representatives that best reconstruct the data collection is the coefficient matrix counts the number of nonzero rows of C

8. Introduction • Applications to video summarization

9. Problem Formulation • Findingcompactdictionariesto represent data • minimizing the objective function D: the dictionary X: the coefficient matrix

10. Problem Formulation • Finding Representative Data • Consider a modification to the dictionary learning framework enforce Y: the matrix of data points C: the coefficient matrix : the i-th row of C I( · ) : the indicator function Counts the number of nonzero rows of C

11. Problem Formulation • This is an NP-hard problem • A standard relaxation of this optimization is obtained as An appropriately chosen parameter

12. Problem Formulation • Indicates the representatives as the nonzero rows of C

13. Geometry of Representatives • minimizes the number of representatives that can reconstruct the collection of data points up to an ε error • Set ε = 0

14. Geometry of Representatives • Theorem • H be the convex hull of the columns of Y • kbe the number of vertices of H the optimal solution : a permutation matrix : the k-dimensional identity matrix : the elements of Δ lie in [0, 1)

15. Representatives of Subspaces • Assume that the data lie in a union of affine subspaces of • the number of representatives from each subspace is greater than or equal to its dimension coefficient matrix corresponding to data from two subspaces >

16. Representatives of Subspaces • Theorem • If the data points are drawn from a union of independent subspaces • then the solution of this finds at least dim( ) representatives from each subspace

17. Practical Considerations and Extensions • Dealing with Outliers • we denote the inliers by Y the outliers by • The solution has the structure

18. Practical Considerations and Extensions • Dealing with Outliers • Among the rows of the coefficient matrix • the true data • many nonzero elements • Outlier • just one nonzero element

19. Practical Considerations and Extensions • Define the row-sparsity index of each candidate representative • outliers • the rsivalue is close to 1 • true representative • the rsiis value close to 0

20. Practical Considerations and Extensions • Dealing with New Observations • Let Y be the collection of points that has already been in the dataset • be the new points that are added to the dataset

21. Practical Considerations and Extensions • Dealing with New Observations • : the representatives of Y

22. Experimental Results • Video Summarization • Using Lagrange multipliers

23. Experimental Results • Investigate the effect of changing the parameter λ

24. Experimental Results • Classification Using Representatives • Evaluate the performance • Sparse Modeling Representative Selection (SMRS) - proposed algorithm • Kmedoids • Rank Revealing QR (RRQR) • simple random selection of training data (Rand)

25. Experimental Results • Several standard classification algorithms • Nearest Neighbor (NN) • Nearest Subspace (NS) • Sparse Representation-based Classification (SRC) • Linear Support Vector Machine (SVM)

26. Experimental Results • run on the USPS digits database and the Extended YaleB face database • USPS digit database • YaleB face database SRC and NS work well when the data in each class lie in a union of low-dimensional subspaces NN often needs to have enough samples from its nearest neighbor

27. Experimental Results • Outlier Rejection • a dataset of N = 1, 024 images • (1 − ρ) fraction of the images are randomly selected from the Extended YaleB face database • ρ fraction of random images downloaded from the internet

28. Experimental Results • Outlier Rejection