1. Zhu Han University of Houston Thanks for Dr. Hung Nguyen’s Slides
Signal processing and Networking for Big Data Applications Lecture 19: Tensor Basics
Zhu Han
University of Houston
Thanks for Dr. Hung Nguyen’s Slides

2. Outline 1. Basic concepts 2. Tensor operations 3. Tensor analysis
4. Applications
5. Tensor Voting
6. Conclusions

3. 1. What is a tensor?
Tensor is a generalization of an n-dimensional array
Vector as a special case of Tensor
Vector

4. Matrix as a special case of Tensor

5. Multiple dimensional array in most of programming languages
3rd order tensor

7. Tensors of arbitrary order

8. Dynamic Data Model (time, source, destination, port)
keyword
Author
time
(time, source, destination, port)
(time, author, keyword)

9. Two important points
Traditional matrix-based data analysis in inherently two-dimensional -> limit to apply to multi-dimensional data

10. 2. Tensor operations
Basis calculus

11. Vectorization

12. Matricize X(d)
Unfold a tensor into a matrix
5 7
6 8

14. source Multiply a tensor with a matrix port port source group source
destination
destination
group
group
source

15. 3. Tensor analysis
Tensor decomposition: generalize concept of low rank from matrices to tensors
Result: Resulting tensor has just a few nonzero columns in each lateral slice.
term
doc
author
term
author
doc

16. Reminder: SVD  m n U VT  A m n

17. Reminder: SVD n
1u1v1
2u2v2 A  + m

18. Rank of a tensor

19. Goal: extension to >=3 modes
I x R
K x R A B
J x R C
R x R x R
I x J x K
+…+ = =

20. Tucker Decomposition - intuition
I x J x K = A
I x R B
J x S C
K x T
R x S x T
author x keyword x conference
A: author x author-group
B: keyword x keyword-group
C: conf. x conf-group
G: how groups relate to each other

21. Tucker 3 Decomposition

22. PARAFAC Decomposition

23. In the presence of missing data
Tensor completion

24. 4. Applications
GOAL: to differentiate between left and right hand stimulation

25. In the presence of missing data
Tensor completion

26. Surveillance
original
Low rank
sparse

28. Analyzing Publication Data: Doc x Doc x Similarity Representation

29. Traffic engineering
Dest. port
125
... 80
IP source
IP destination

31. Tensor Voting Tensor Voting Framework Normal space Tensor inference
Token refinement
Token decomposition
Results and conclusion

32. Tensor Voting Framework
Objective: infer “hidden” objects—gaps and broken parts.
Gestalt principles: the presence of each input token implies a hypothesis that structure passes through it.
Proximity
Closure
Continuity
Inclined to infer those structures by red lines:
Proximity
Closure
Continuity

33. Tensor Voting Framework-Normal Space
Normal Space: encodes structure information; spanned by normal vectors.
Tensor: describes the linear relation of vectors; outer product.
stick tensor – surface element
plate tensor – curve element
ball tensor – point element
Structure types in 3D: surfaces; curves; volumes.
White arrows: normal vectors
Blue regions: normal space
Red shape: structure element

34. Tensor Voting Framework-Tensor Inference
Consider a voter point p on a surface
Normal is a single vector with saliency (magnitude)
Information propagates to its neighboring votee point x
The most likely smooth path: arc of the osculating circle
Tensor(structure information) received at x:

35. Tensor Voting Framework-Tensor Inference
voter projects structure information via its tensor weighted by decay function (DF) to neighboring votees.
sum up all the tensors received by each votee. ϴ P1
NP1 P2 s l Pk
NPk_P1
NPk_P2
NP2
Voting procedure in 2D with stick vote
Decompose voter’s tensor as stick tensors
Vote as the fundamental stick vote.
Decompose voted tensors at the votee to reconstruct the same normal space as voter’s

36. Tensor Voting Framework-Token Refinement
Problem: prior knowledge of structure type(normal space) & its saliencies unknown.
Token refinement procedure is thus needed:
Initialize each token with unit ball tenor(no direction preference)
Tensor voting between tokens
Decompose resulting tensor for each token indicating direction preference
Initial Ball Tensors in 2D
After Token Refinement in 2D

37. Inference Algorithm 1> Input initial trace with missing parts
2> Perform token refinement
3> Mark sparse voting region to define potential votees
4> Perform tensor voting to infer structures
5> Decompose results & add determined votees to the previous trace
6> Re-input the thinned trace and iterate the whole procedure starting with 2> No
Iterate enough times?
Yes
7> Output final trace result

38. Results and Conclusions
Inference results:
Number of pixels
σ values
Polluted trace as algorithm input
Tensor voting with σ=1
Tensor voting with σ=2
Compared with victim method:

39. Results and Conclusions
Provide an efficient approach to infer the human mobility trace, given that the observed location data exist missing parts.
No user instructions are required to identify which part to be inferred.
Discovering missing positions and accomplishing inference are done in automatic fashion.
Sparse per-votee implementation scheme reduces computation load.
Achieve relatively high accuracy.

40. 5. Conclusions Tensor is a multidimensional array
Tensor decomposition (factorization) can be considered higher- order generalization of matrix SVD or PCA
Wide applications in data reconstruction, cluster analysis, compression.