Multimedia Databases

1. Multimedia Databases LSI and SVD

2. Text - Detailed outline • text • problem • full text scanning • inversion • signature files • clustering • information filtering and LSI

3. Information Filtering + LSI • [Foltz+,’92] Goal: • users specify interests (= keywords) • system alerts them, on suitable news-documents • Major contribution: LSI = Latent Semantic Indexing • latent (‘hidden’) concepts

4. Information Filtering + LSI Main idea • map each document into some ‘concepts’ • map each term into some ‘concepts’ ‘Concept’:~ a set of terms, with weights, e.g. • “data” (0.8), “system” (0.5), “retrieval” (0.6) -> DBMS_concept

5. Information Filtering + LSI Pictorially: term-document matrix (BEFORE)

6. Information Filtering + LSI Pictorially: concept-document matrix and...

7. Information Filtering + LSI ... and concept-term matrix

8. Information Filtering + LSI Q: How to search, eg., for ‘system’?

9. Information Filtering + LSI A: find the corresponding concept(s); and the corresponding documents

10. Information Filtering + LSI A: find the corresponding concept(s); and the corresponding documents

11. Information Filtering + LSI Thus it works like an (automatically constructed) thesaurus: we may retrieve documents that DON’T have the term ‘system’, but they contain almost everything else (‘data’, ‘retrieval’)

12. SVD - Detailed outline • Motivation • Definition - properties • Interpretation • Complexity • Case studies • Additional properties

13. SVD - Motivation • problem #1: text - LSI: find ‘concepts’ • problem #2: compression / dim. reduction

14. SVD - Motivation • problem #1: text - LSI: find ‘concepts’

15. SVD - Motivation • problem #2: compress / reduce dimensionality

16. Problem - specs • ~10**6 rows; ~10**3 columns; no updates; • random access to any cell(s) ; small error: OK

17. SVD - Motivation

18. SVD - Motivation

19. SVD - Detailed outline • Motivation • Definition - properties • Interpretation • Complexity • Case studies • Additional properties

20. SVD - Definition A[n x m] = U[n x r]L [ r x r] (V[m x r])T • A: n x m matrix (eg., n documents, m terms) • U: n x r matrix (n documents, r concepts) • L: r x r diagonal matrix (strength of each ‘concept’) (r : rank of the matrix) • V: m x r matrix (m terms, r concepts)

21. SVD - Definition • A = ULVT - example:

22. SVD - Properties THEOREM [Press+92]:always possible to decomposematrix A into A = ULVT , where • U,L,V: unique (*) • U, V: column orthonormal (ie., columns are unit vectors, orthogonal to each other) • UTU = I; VTV = I (I: identity matrix) • L: eigenvalues are positive, and sorted in decreasing order

23. SVD - Example • A = ULVT - example: retrieval inf. lung brain data CS x x = MD

24. SVD - Example • A = ULVT - example: retrieval CS-concept inf. lung MD-concept brain data CS x x = MD

25. SVD - Example doc-to-concept similarity matrix • A = ULVT - example: retrieval CS-concept inf. lung MD-concept brain data CS x x = MD

26. SVD - Example • A = ULVT - example: retrieval ‘strength’ of CS-concept inf. lung brain data CS x x = MD

27. SVD - Example • A = ULVT - example: term-to-concept similarity matrix retrieval inf. lung brain data CS-concept CS x x = MD

28. SVD - Example • A = ULVT - example: term-to-concept similarity matrix retrieval inf. lung brain data CS-concept CS x x = MD

29. SVD - Detailed outline • Motivation • Definition - properties • Interpretation • Complexity • Case studies • Additional properties

30. SVD - Interpretation #1 ‘documents’, ‘terms’ and ‘concepts’: • U: document-to-concept similarity matrix • V: term-to-concept sim. matrix • L: its diagonal elements: ‘strength’ of each concept

31. SVD - Interpretation #2 • best axis to project on: (‘best’ = min sum of squares of projection errors)

32. SVD - Motivation

33. minimum RMS error SVD - interpretation #2 SVD: gives best axis to project v1

34. SVD - Interpretation #2

35. x x = v1 SVD - Interpretation #2 • A = ULVT - example:

36. SVD - Interpretation #2 • A = ULVT - example: variance (‘spread’) on the v1 axis x x =

37. SVD - Interpretation #2 • A = ULVT - example: • UL gives the coordinates of the points in the projection axis x x =

38. x x = SVD - Interpretation #2 • More details • Q: how exactly is dim. reduction done?

39. SVD - Interpretation #2 • More details • Q: how exactly is dim. reduction done? • A: set the smallest eigenvalues to zero: x x =

40. SVD - Interpretation #2 x x ~

41. SVD - Interpretation #2 x x ~

42. SVD - Interpretation #2 x x ~

43. SVD - Interpretation #2 ~

44. SVD - Interpretation #2 Equivalent: ‘spectral decomposition’ of the matrix: x x =

45. SVD - Interpretation #2 Equivalent: ‘spectral decomposition’ of the matrix: l1 x x = u1 u2 l2 v1 v2

46. l1 l2 u1 u2 vT1 vT2 SVD - Interpretation #2 Equivalent: ‘spectral decomposition’ of the matrix: m = + +... n

47. l1 l2 u1 u2 vT1 vT2 SVD - Interpretation #2 ‘spectral decomposition’ of the matrix: m r terms = + +... n n x 1 1 x m

48. l1 l2 u1 u2 vT1 vT2 SVD - Interpretation #2 approximation / dim. reduction: by keeping the first few terms (Q: how many?) m = + +... n assume: l1 >= l2 >= ...

49. l1 l2 u1 u2 vT1 vT2 SVD - Interpretation #2 A (heuristic - [Fukunaga]): keep 80-90% of ‘energy’ (= sum of squares of li ’s) m = + +... n assume: l1 >= l2 >= ...

50. SVD - Interpretation #3 • finds non-zero ‘blobs’ in a data matrix x x =