Part 3: Image Classification using Sparse Coding: Advanced Topics

1. Part 3: Image Classification using Sparse Coding: Advanced Topics Kai Yu Dept. of Media Analytics NEC Laboratories America Andrew Ng Computer Science Dept. Stanford University

2. Outline of Part 3 • Why can sparse coding learn good features? • Intuition, topic model view, and geometric view • A theoretical framework: local coordinate coding • Two practical coding methods • Recent advances in sparse coding for image classification

3. Outline of Part 3 • Why can sparse coding learn good features? • Intuition, topic model view, and geometric view • A theoretical framework: local coordinate coding • Two practical coding methods • Recent advances in sparse coding for image classification

4. Intuition: why sparse coding helps classification? Figure from http://www.dtreg.com/svm.htm • The coding is a nonlinear feature mapping • Represent data in a higher dimensional space • Sparsity makes prominent patterns more distinctive

5. A “topic model” view to sparse coding Basis 1 Basis 2 Both figures adapted from CVPR10 tutorial by F. Bach, J. Mairal, J. Ponce and G. Sapiro • Each basis is a “direction” or a “topic”. • Sparsity: each datum is a linear combination of only a few bases. • Applicable to image denoising, inpainting, and super-resolution.

6. A geometric view to sparse coding Data manifold Data Basis • Each basis is somewhat like a pseudo data point – “anchor point” • Sparsity: each datum is a sparse combination of neighbor anchors. • The coding scheme explores the manifold structure of data.

7. MNIST Experiment: Classification using SC Try different values • 60K training, 10K for test • Let k=512 • Linear SVM on sparse codes

8. MNIST Experiment: Lambda = 0.0005 Each basis is like a part or direction.

9. MNIST Experiment: Lambda = 0.005 Again, each basis is like a part or direction.

10. MNIST Experiment: Lambda = 0.05 Now, each basis is more like a digit !

11. MNIST Experiment: Lambda = 0.5 Like clustering now!

12. Geometric view of sparse coding Error: 4.54% Error: 3.75% Error: 2.64% • When SC achieves the best classification accuracy, the learned bases are like digits – each basis has a clear local class association. • Implication: exploring data geometry may be useful for classification.

13. Distribution of coefficients (MNIST) Neighbor bases tend to get nonzero coefficients

14. Distribution of coefficient (SIFT, Caltech101) Similar observation here!

15. Recap: two different views to sparse coding • View 1 • Discover “topic” components • Each basis is a “direction” • Sparsity: each datum is a linear combination of several bases. • Related to topic model • View 2 • Geometric structure of data manifold • Each basis is an “anchor point” • Sparsity: each datum is a linear combination of neighbor anchors. • Somewhatlike a soft VQ (link to BoW) • Either can be valid for sparse coding under certain circumstances. • View 2 seems to be helpful to sensory data classification.

16. Outline of Part 3 • Why can sparse coding learn good features? • Intuition, topic model view, and geometric view • A theoretical framework: local coordinate coding • Two practical coding methods • Recent advances in sparse coding for image classification

17. Key theoretical question Classification Feature learning • Why unsupervised feature learning via sparse coding can help classification?

18. The image classification setting for analysis Implication: Learning an image classifier is a matter of learning nonlinear functions on patches. Dense local feature Sparse Coding Linear Pooling Linear SVM Function on images Function on patches

19. Illustration: nonlinear learning via local coding locally linear data points bases

20. How to learn a nonlinear function? Step 1: Learning the dictionary from unlabeled data

21. How to learn a nonlinear function? Step 2: Use the dictionary to encode data

22. How to learn a nonlinear function? Step 3: Estimate parameters Global linear weights to be learned Sparse codes of data • Nonlinear local learning via learning a global linear function.

23. Local Coordinate Coding (LCC): connect coding to nonlinear function learning Yu et al NIPS-09 If f(x) is (alpha, beta)-Lipschitz smooth The key message: A good coding scheme should 1. have a small coding error, 2. and also be sufficiently local Function approximation error Locality term Coding error

24. Outline of Part 3 • Why can sparse coding learn good features? • Intuition, topic model view, and geometric view • A theoretical framework: local coordinate coding • Two practical coding methods • Recent advances in sparse coding for image classification

25. Application of LCC theory • Fast Implementation with a large dictionary • A simple geometric way to improve BoW Wang et al, CVPR 10 Zhou et al, ECCV 10

26. Application of LCC theory • Fast Implementation with a large dictionary • A simple geometric way to improve BoW

27. The larger dictionary, the higher accuracy, but also the higher computation cost Yu et al NIPS-09 Yang et al CVPR 09 The same observation for Caltech-256, PASCAL, ImageNet, …

28. Locality-constrained linear coding a fast implementation of LCC Wang et al, CVPR 10 • Dictionary Learning: k-means (or hierarchical k-means) • Coding for X, • Step 1 – ensure locality: find the K nearest bases • Step 2 – ensure low coding error:

29. Competitive in accuracy, cheap in computation Comparable with sparse coding This is one of the two major algorithms applied by NEC-UIUC team to achieve the No.1 position in ImageNet challenge 2010! Sparse coding Significantly better than sparse coding Wang et al CVPR 10

30. Application of the LCC theory • Fast Implementation with a large dictionary • A simple geometric way to improve BoW

31. Interpret “BoW + linear classifier” Piece-wise local constant (zero-order) data points cluster centers

32. Super-vector coding: a simple geometric way to improve BoW (VQ) Zhou et al, ECCV 10 Piecewise local linear (first-order) Local tangent data points cluster centers

33. Super-vector coding: a simple geometric way to improve BoW (VQ) If f(x) is beta-Lipschitz smooth, and Local tangent Quantization error Function approximation error

34. Super-vector coding: learning nonlinear function via a global linear model Let be the VQ coding of This is one of the two major algorithms applied by NEC-UIUC team to achieve the No.1 position in PASCAL VOC 2009! Global linear weights to be learned Super-vector codes of data

35. Summary of Geometric Coding Methods (Fast) Local Coordinate Coding Vector Quantization (BoW) Super-vector Coding • All lead to higher-dimensional,sparse, and localized coding • All explore geometric structure of data • New coding methods are suitable for linear classifiers. • Their implementations are quite straightforward.

36. Things not covered here • Improved LCC using Local Tangent, Yu & Zhang, ICML10 • Mixture of Sparse Coding, Yang et al ECCV 10 • Deep Coding Network, Lin et al NIPS 10 • Pooling methods • Max-pooling works well in practice, but appears to be ad-hoc. • An interesting analysis on max-pooling, Boureau et al. ICML 2010 • We are working on a linear pooling method, which has a similar effect as max-pooling. Some preliminary results already in the super-vector coding paper, Zhou et al, ECCV2010.

37. Outline of Part 3 • Why can sparse coding learn good features? • Intuition, topic model view, and geometric view • A theoretical framework: local coordinate coding • Two practical coding methods • Recent advances in sparse coding for image classification

38. Fast approximation of sparse coding via neural networks Gregor & LeCun, ICML-10 • The method aims at improving sparse coding speed in coding time, not training speed, potentially make sparse coding practical for video. • Idea: Given a trained sparse coding model, use its input outputs as training data to train a feed-forward model • They showed a speedup of X20 faster. But not evaluated on real video data.

39. Group sparse coding Bengio et al, NIPS 09 • Sparse coding is on patches, the image representation is unlikely sparse. • Idea: enforce joint sparsity via L1/L2 norm on sparse codes of a group of patches. • The resultant image representation becomes sparse, which can save the memory cost, but the classification accuracy decreases.

40. Learning hierarchical dictionary Jenatton, Mairal, Obozinski, and Bach, 2010 A node can be active only if its ancestors are active.

41. Reference Image Classification using Super-Vector Coding of Local Image Descriptors, Xi Zhou, Kai Yu, Tong Zhang, and Thomas Huang. In ECCV 2010. Efficient Highly Over-Complete Sparse Coding using a Mixture Model, Jianchao Yang, Kai Yu, and Thomas Huang. In ECCV 2010. Learning Fast Approximations of Sparse Coding, Karol Gregor and Yann LeCun. In ICML 2010. Improved Local Coordinate Coding using Local Tangents, Kai Yu and Tong Zhang. In ICML 2010. Sparse Coding and Dictionary Learning for Image Analysis, Francis Bach,  Julien Mairal, Jean Ponce, and Guillermo Sapiro. CVPR 2010 Tutorial Supervised translation-invariant sparse coding, Jianchao Yang, Kai Yu, and Thomas Huang, In CVPR 2010. Learning locality-constrained linear coding for image classification, Jingjun Wang, Jianchao Yang, Kai Yu, Fengjun Lv, Thomas Huang, and Yihong Gong. In CVPR 2010. Group Sparse Coding, Samy Bengio, Fernando Pereira, Yoram Singer, and Dennis  Strelow, In NIPS*2009. Nonlinear learning using local coordinate coding, Kai Yu, Tong Zhang, and Yihong Gong. In NIPS*2009. Linear spatial pyramid matching using sparse coding for image classification, Jianchao Yang, Kai Yu, Yihong Gong, and Thomas Huang. In CVPR 2009. Efficient sparse coding algorithms. Honglak Lee, Alexis Battle, Raina Rajat and Andrew Y.Ng. In NIPS*2007.