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A New Approach To The Multiclass Classification Problem Category Vector Space
[object Object],[object Object],[object Object],[object Object],Agenda
[object Object],[object Object],[object Object],[object Object],[object Object],Classification Problem Problem
Object recognition Automated protein classification 300-600  Digit recognition Phoneme recognition [ Waibel, Hanzawa, Hinton,Shikano, Lang 1989 ] http://www.glue.umd.edu/~zhelin/recog.html ,[object Object],Multiclass Applications Large Category Space Problem 100 50 10
[object Object],[object Object],[object Object],[object Object],Other Multiclass Applications Problem
[object Object],Classification Setup Question :  design a classification rule  y = f(x) such that, given a new x, this predicts y with minimal probability of error Training and test data drawn i.i.d. from fixed but unknown probability distribution  D Labeled training set Problem
+ + + + _ + _ _ _ _ + + _ ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],Support Vector Machines (SVMs) Problem
[object Object],[object Object],[object Object],[object Object],+ + + + _ + _ _ _ _ + + _ w b Definition Of SVM Classifiers Problem
Definition Of A Margin ,[object Object],[object Object],Problem BAD
[object Object],[object Object],Maximize The Margin ,[object Object],[object Object],Problem GOOD
[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],Combining Binary Classifiers Problem
[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],Example 1 Motivation
[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],Example 2 Motivation
[object Object],[object Object],[object Object],[object Object],[object Object],Problems With Combining Binary Classifiers Motivation
[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],Category Vector Spaces Solution Note: Extensions to other multi-category learning applications are straightforward assuming we can map category labels to coordinate. Motivation
Solution: The columns of  W  are the top  D  eigenvectors (corresponding to the largest eigenvalues) of Multiclass Fisher Related Idea D  categories and a projected set of features defined by the MC-FLD maximizes where Given the feature vectors ,[object Object],[object Object],[object Object],[object Object],Discussion
[object Object],[object Object],[object Object],[object Object],[object Object],Discussion Disadvantage Of Multiclass Fisher
[object Object],[object Object],[object Object],[object Object],[object Object],Depicts basic categories Description of Category Vector Spaces Discussion
[object Object],[object Object],[object Object],[object Object],Advantages Of Category Vector Space  Discussion
[object Object],[object Object],[object Object],[object Object],Technical Challenges Discussion
[object Object],[object Object],[object Object],[object Object],[object Object],Regression In Category Space subject to the constraints Discussion
[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],Classification In Category Space subject to the constraints Discussion
[object Object],Integrated Classification and Regression Objective Function Discussion
[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],Multi-Category GRBF Preliminary Results Solution
[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],Experimental Setup White Basis =  y 1 = [1,0,0] T  Black Basis =  y 2 = [0,1,0] T  Asian Basis =  y 3 = [0,0,1] T  Preliminary Results
Training set images: Top row: Asian, Middle row: Black, Bottom row: White Race Classification Training Images Preliminary Results
Training set images mapped into the category vector space Category Space For Training Images Preliminary Results
[object Object],[object Object],[object Object],[object Object],Race Classification Testing Images Preliminary Results
[object Object],Category Space Testing Images Preliminary Results
[object Object],[object Object],[object Object],Pairwise Projection Of Category Space Testing Images ,[object Object],[object Object],Preliminary Results
[object Object],[object Object],[object Object],Ambiguous faces mapped into the category space. Note how they cluster together. Ambiguity Testing Preliminary Results
Experiment With MPEG-7 Database Butterfly Bat Bird Preliminary Results
Experiment With MPEG-7 Database Fly Chicken Batbird Preliminary Results
3 Class Training Preliminary Results
3 Class Testing Preliminary Results
4 Class Training Preliminary Results
4 Class Testing Preliminary Results
[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],Summary
Questions & Discussion Thank You
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Category vectorspaceessex

  • 1. A New Approach To The Multiclass Classification Problem Category Vector Space
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  • 26. Training set images: Top row: Asian, Middle row: Black, Bottom row: White Race Classification Training Images Preliminary Results
  • 27. Training set images mapped into the category vector space Category Space For Training Images Preliminary Results
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  • 32. Experiment With MPEG-7 Database Butterfly Bat Bird Preliminary Results
  • 33. Experiment With MPEG-7 Database Fly Chicken Batbird Preliminary Results
  • 34. 3 Class Training Preliminary Results
  • 35. 3 Class Testing Preliminary Results
  • 36. 4 Class Training Preliminary Results
  • 37. 4 Class Testing Preliminary Results
  • 38.
  • 40. References [1] H. Guo. Diffeomorphic point matching with applications in medical image analysis . PhD thesis, University of Florida, Gainesville, FL, 2005. Ph.D. Thesis. [2] J. Zhang. New information theoretic distance measures and algorithms for multimodality image registration . PhD thesis, University of Florida, Gainesville, FL, 2005. Ph.D. Thesis. [3] A. A. Kumthekar. Affine image registration using minimum spanning tree entropies. Master’s thesis, University of Florida, Gainesville, FL, 2004. M. S. Thesis. [4] A. Rajwade, A. Banerjee, and A. Rangarajan. A new method of probability density estimation with application to mutual information-based image registration. In IEEE Computer Vision and Pattern Recognition (CVPR) , volume 2, pages 1769–1776, 2006. [5] A. Peter and A. Rangarajan. A new closed form information metric for shape analysis. In Medical Image Computing and Computer Assisted Intervention (MICCAI part 1) , Springer LNCS 4190, pages 249–256. 2006. [6] A. S. Roy, A. Gopinath, and A. Rangarajan. Deformable density matching for 3D non-rigid registration of shapes. In Medical Image Computing and Computer Assisted Intervention (MICCAI part 1) , Springer LNCS 4791, pages 942–949. 2007. [7] F.Wang, B. Vemuri, and A. Rangarajan. Groupwise point pattern registration using a novel CDF-based Jensen Shannon divergence. In IEEE Computer Vision and Pattern Recognition (CVPR) , volume 1, pages 1283–1288, 2006. [8] L. Garcin, A. Rangarajan, and L. Younes. Non-rigid registration of shapes via diffeomorphic point matching and clustering. In IEEE Conf. on Image Processing , volume 5, pages 3299–3302, 2004. [9] F. Wang, B.C. Vemuri, A. Rangarajan, I.M. Schmalfuss, and S.J. Eisenschenk. Simultaneous nonrigid registration of multiple point sets and atlas construction. In European Conference on Computer Vision (ECCV) , pages 551–563, 2006. [10] H. Guo, A. Rangarajan, and S. Joshi. 3D diffeomorphic shape registration on hippocampal datasets. In James S. Duncan and Guido Gerig, editors, Medical Image Computing and Computer Assisted Intervention (MICCAI) , pages 984–991. 2005.
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Hinweis der Redaktion

  1. Mapping in such a way that the system has reasonable learning and generalization properties; Training and test set errors are low. Regardless of what features are extracted.
  2. Instead of getting musical time samples containing just a single note. For example, to represent the fact that we have patterns belonging to basic categories {C,D, etc.} and patterns corresponding to compound categories such as certain tritones (highly dissonant intervals in a scale), chords etc. Tritone is not represented in the set of labels. Unstable since it could potentially lead to an exponentially large label space. If the set of labels is truncated because of exponential size considerations, then the intolerance to ambiguity returns in the form of un-represented intervals and chords.
  3. Rather than conceive of categories as nominal labels. Returning to our music sequence classification example. To achieve this, we would have to first map the raw musical sequences to the 12 dimensional vector space and then (if a support vector machine approach is used), maximize the margin of the mapped vectors in the category space.
  4. Attempt at taking categories or labels and building a vector space out of them with one intriguing exception: the multiple class Fisher linear discriminant (MCFLD)
  5. Actually, these are merely hypercone-like since cutting them with a hyperplane need not necessarily be hyperspheres. However, we will continue to call the decision boundaries hypercones for the sake of simplicity. The extension to the case of compound categories is straightforward with the hypercone surrounding the axis of the compound category vector.
  6. After we have learned a mapping from the original feature space to the category vector space If there are compound categories, such as intervals and chords in the music sequence example and mixed races in the race classification example, and we have expert information regarding the compound categories, then these patterns can be directly represented in the category space and if necessary, we can maximize the compound category margin as well as the margins for the basic categories.
  7. Zi is unknown. For a compound category—say a chord, it does not have to be a category basis vector. Since the mapped patterns zi are unknown, this immediately suggests that the classification and regression problems are linked Given a set of patterns xi and their counterpart mapped patterns zi we can fit a regression function. Given a set of mapped patterns zi and their associated label vectors y (which will not all be unique), we can maximize their margins and learn a classification. These problems are coupled Consequently, we combine problems one and two above into an integrated classification and regression objective function.
  8. Illustration of SVM regression, showing the regression curve. Also shown are examples of the slack variables. Illustration of the slack variables >=0. Data points with circles around them are support vectors Standard L1 norm-based SVR objective function
  9. The principal advantage in adopting the SVR formulation is that it can be well integrated with the SVM multi-category objective function