Modelling and Visualization of High Dimensional Data. Sample Examination Paper

Transcription

1 Duration not specified UNIVERSITY OF MANCHESTER SCHOOL OF COMPUTER SCIENCE Modelling and Visualization of High Dimensional Data Sample Examination Paper Examination date not specified Time: Examination time not specified The use of electronic calculators is permitted provided they are not programmable and do not store text. Page 1 of 5 [Next page ]

2 Section A COMP61021 Answer all questions from this section. 1. In machine learning, curse of dimensionality always exists whenever high-dimensional data need to be dealt with regardless of learning paradigms. True or False? 2. In general, feature selection always outperforms feature extraction for dimension reduction. True or False? 3. For any two matrices A and B, A B B A in general where is the matrix multiplication operator. True or False? 4. Self-organising maps (SOM) is a biologically inspired algorithm for density estimation and visualisation. However, SOM is subject to a limitation that its result can be visualised only if the data are in a K-dimensional space (K = 1,2,3). True or False? Page 2 of 5

3 5. Isometric feature mapping (ISOMAP) and locally linear embedding (LLE) are two typical manifold learning algorithms. A major difference between them is that ISOMAP can be used in nonlinear manifold learning but LLE is only applicable to linear manifold learning. True or False? 6. What is the inner product of two vectors u = ( ) T and v = ( ) T? A. 2 B. 10 C. 0 D If the size of three matrices A, B and C are k n, m n and m k respectively, what is the size of the product AB T C? A. k k. B. k m. C. k n. D. m n. Page 3 of 5 [Next page ]

4 8. If A is a n n real and symmetric matrix, state which of the following statements is incorrect. A. All eigenvalues of A are real numbers. B. Eigenvectors associated with different eigenvalues of A are orthogonal. C. A has n non-zero eigenvalues. D. A 2 is a n n real and symmetric matrix. 9. Principal component analysis (PCA) is an effective dimension reduction method. State which of the following statements on the first component achieved by the PCA is correct. A. The first component specifies an orientation in the original space where a given data set has the maximum variance. B. The first component specifies an orientation in a low-dimensional space where a given data set has the least variance. C. The first component is the eigenvector corresponding to the maximum eigenvalue of the data matrix representing a given data set. D. None of the above is correct. 10. Multi-dimensional scaling (MDS) is a dimension reduction method used to preserve the distance information underlying a high-dimensional source space in a specified low-dimensional target space. For MDS learning, a cost function is required. State which of the following statements is correct. A. Stress is commonly used to describe the cost function used in an MDS B. Disparity is commonly used to describe the cost function used in an MDS C. Utility is commonly used to describe the cost function used in an MDS D. None of the above terms have been used to describe the cost function used in an MDS Page 4 of 5

5 Section B COMP61021 Answer all questions from this section. 1. Linear Discriminative Analysis (LDA) is a popular supervised dimension reduction method. For a labelled data set of C classes (C > 2), describe within-class, between-class and total scatter matrices, S W, S B and S T, in LDA with the same notation used in the lecture notes. (6 marks) 2. Isometric feature mapping (ISOMAP) is a nonlinear manifold learning Describe the main steps of ISOMAP algorithm in detail. (6 marks) 3. For a given data set X = {x 1,,x N }, we construct two matrices A = N 1 1 X T X and B = N 1 1 XXT. Show that two matrices A and B share the same eigenvalues. Note that singular value decomposition (SVD) is NOT allowed to be used in your proof. (8 marks) Page 5 of 5 END OF EXAMINATION