1. Sparse Modeling for Finding Representative Objects Ehsan Elhamifar Guillermo Sapiro Ren´e Vidal Johns Hopkins University University of 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 < Step1: k initial "means" are randomly selected from the data set < Step2: k clusters are created by every observation with the nearest mean < Step3: The centriod of each of the k clusters becomes the new mean < Step4: Steps 2 and 3 are repeated until convergence has been reached

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  Finding compact dictionaries to 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 Y: the matrix of data points C: the coefficient matrix enforce 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 k be 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 coefficient matrix corresponding to data from two subspaces >

16. Representatives of Subspaces

17. Practical Considerations and Extensions

18.  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 rsi value is close to 1  true representative the rsi is value close to 0

20. Practical Considerations and Extensions

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