Tensor Methods: A New Paradigm for Training Probabilistic Models and Feature Learning: Tensors are rich structures for modeling complex higher order relationships in data rich domains such as social networks, computer vision, internet of things, and so on. Tensor decomposition methods are embarrassingly parallel and scalable to enormous datasets. They are guaranteed to converge to the global optimum and yield consistent estimates of parameters for many probabilistic models such as topic models, community models, hidden Markov models, and so on. I will show the results of these methods for learning topics from text data, communities in social networks, disease hierarchies from healthcare records, cell types from mouse brain data, etc. I will also demonstrate how tensor methods can yield rich discriminative features for classification tasks and can serve as an alternative method for training neural networks.

1. Tensor Methods: A New Paradigm for
Probabilistic Models and Feature Learning
Anima Anandkumar
U.C. Irvine

2. Learning with Big Data

3. Data vs. Information

4. Data vs. Information

5. Data vs. Information
Missing observations, gross corruptions, outliers.

6. Data vs. Information
Missing observations, gross corruptions, outliers.
Learning useful information is finding needle in a haystack!

7. Matrices and Tensors as Data Structures
Multi-modal and multi-relational data.
Matrices: pairwise relations. Tensors: higher order relations.
Multi-modal data figure from Lise Getoor slides.

8. Spectral Decomposition of Tensors
M2 =
i
λiui ⊗ vi
= + ....
Matrix M2 λ1u1 ⊗ v1 λ2u2 ⊗ v2

9. Spectral Decomposition of Tensors
M2 =
i
λiui ⊗ vi
= + ....
Matrix M2 λ1u1 ⊗ v1 λ2u2 ⊗ v2
M3 =
i
λiui ⊗ vi ⊗ wi
= + ....
Tensor M3 λ1u1 ⊗ v1 ⊗ w1 λ2u2 ⊗ v2 ⊗ w2
We have developed efficient methods to solve tensor decomposition.

10. Strengths of Tensor Methods
Fast and accurate, orders of magnitude faster than previous methods.
Embarrassingly parallel and suited for cloud systems, e.g.Spark.
Exploit optimized linear algebra libraries.
Exploit parallelism of GPU systems.

11. Strengths of Tensor Methods
Fast and accurate, orders of magnitude faster than previous methods.
Embarrassingly parallel and suited for cloud systems, e.g.Spark.
Exploit optimized linear algebra libraries.
Exploit parallelism of GPU systems.
10
2
10
3
10
−1
10
0
10
1
10
2
10
3
10
4
Number of communities k
Runningtime(secs)
MATLAB Tensor Toolbox(CPU)
CULA Standard Interface(GPU)
CULA Device Interface(GPU)
Eigen Sparse(CPU)

12. Outline
1 Introduction
2 Learning Probabilistic Models
3 Experiments
4 Feature Learning with Tensor Methods
5 Conclusion

13. Latent variable models
Incorporate hidden or latent variables.
Information structures: Relationships between latent variables and
observed data.

14. Latent variable models
Incorporate hidden or latent variables.
Information structures: Relationships between latent variables and
observed data.
Basic Approach: mixtures/clusters
Hidden variable is categorical.

15. Latent variable models
Incorporate hidden or latent variables.
Information structures: Relationships between latent variables and
observed data.
Basic Approach: mixtures/clusters
Hidden variable is categorical.
Advanced: Probabilistic models
Hidden variables have more general distributions.
Can model mixed membership/hierarchical
groups.
x1 x2 x3 x4 x5
h1
h2 h3

16. Challenges in Learning LVMs
Computational Challenges
Maximum likelihood: non-convex optimization. NP-hard.
Practice: Local search approaches such as gradient descent, EM,
Variational Bayes have no consistency guarantees.
Can get stuck in bad local optima. Poor convergence rates and hard
to parallelize.
Tensor methods yield guaranteed learning for LVMs

17. Unsupervised Learning of LVMs
GMM HMM
h1 h2 h3
x1 x2 x3
ICA
h1 h2 hk
x1 x2 xd
Multiview and Topic Models

18. Overall Framework for Unsupervised Learning
= +
....
Unlabeled
Data
Probabilistic
admixture
models
Tensor
Method
Inference

19. Outline
1 Introduction
2 Learning Probabilistic Models
3 Experiments
4 Feature Learning with Tensor Methods
5 Conclusion

20. Demo for Learning Gaussian Mixtures

21. NYTimes Demo

22. Experimental Results on Yelp and DBLP
Lowest error business categories & largest weight businesses
Rank Category Business Stars Review Counts
1 Latin American Salvadoreno Restaurant 4.0 36
2 Gluten Free P.F. Chang’s China Bistro 3.5 55
3 Hobby Shops Make Meaning 4.5 14
4 Mass Media KJZZ 91.5FM 4.0 13
5 Yoga Sutra Midtown 4.5 31

23. Experimental Results on Yelp and DBLP
Lowest error business categories & largest weight businesses
Rank Category Business Stars Review Counts
1 Latin American Salvadoreno Restaurant 4.0 36
2 Gluten Free P.F. Chang’s China Bistro 3.5 55
3 Hobby Shops Make Meaning 4.5 14
4 Mass Media KJZZ 91.5FM 4.0 13
5 Yoga Sutra Midtown 4.5 31
Top-5 bridging nodes (businesses)
Business Categories
Four Peaks Brewing Restaurants, Bars, American, Nightlife, Food, Pubs, Tempe
Pizzeria Bianco Restaurants, Pizza, Phoenix
FEZ Restaurants, Bars, American, Nightlife, Mediterranean, Lounges, Phoenix
Matt’s Big Breakfast Restaurants, Phoenix, Breakfast& Brunch
Cornish Pasty Co Restaurants, Bars, Nightlife, Pubs, Tempe

24. Experimental Results on Yelp and DBLP
Lowest error business categories & largest weight businesses
Rank Category Business Stars Review Counts
1 Latin American Salvadoreno Restaurant 4.0 36
2 Gluten Free P.F. Chang’s China Bistro 3.5 55
3 Hobby Shops Make Meaning 4.5 14
4 Mass Media KJZZ 91.5FM 4.0 13
5 Yoga Sutra Midtown 4.5 31
Top-5 bridging nodes (businesses)
Business Categories
Four Peaks Brewing Restaurants, Bars, American, Nightlife, Food, Pubs, Tempe
Pizzeria Bianco Restaurants, Pizza, Phoenix
FEZ Restaurants, Bars, American, Nightlife, Mediterranean, Lounges, Phoenix
Matt’s Big Breakfast Restaurants, Phoenix, Breakfast& Brunch
Cornish Pasty Co Restaurants, Bars, Nightlife, Pubs, Tempe
Error (E) and Recovery ratio (R)
Dataset ˆk Method Running Time E R
DBLP sub(n=1e5) 500 ours 10,157 0.139 89%
DBLP sub(n=1e5) 500 variational 558,723 16.38 99%
DBLP(n=1e6) 100 ours 5407 0.105 95%

25. Discovering Gene Profiles of Neuronal Cell Types
Learning mixture of point processes of cells through tensor methods.
Components of mixture are candidates for neuronal cell types.

26. Discovering Gene Profiles of Neuronal Cell Types
Learning mixture of point processes of cells through tensor methods.
Components of mixture are candidates for neuronal cell types.

27. Hierarchical Tensors for Healthcare Analytics
= + .... = + .... = + ....
= + ....

28. Hierarchical Tensors for Healthcare Analytics
= + .... = + .... = + ....
= + ....
CMS dataset: 1.6 million patients, 15.8 million events.
Mining disease inferences from patient records.

29. Outline
1 Introduction
2 Learning Probabilistic Models
3 Experiments
4 Feature Learning with Tensor Methods
5 Conclusion

30. Feature Learning For Efficient Classification
Find good transformations of input for improved classification
Figures used attributed to Fei-Fei Li, Rob Fergus, Antonio Torralba, et al.

31. Tensor Methods for Training Neural Networks
First order score function m-th order score function
Input pdf is p(x)
Sm(x) :=
(−1)m∇(m)p(x)
p(x)
Capture local variations in
data.
Algorithm for Training Neural Networks
Estimate score functions using autoencoder.
Decompose tensor E[y ⊗ Sm(x)] to obtain weights, for m ≥ 3.
Recursively estimate score function of autoencoder and repeat.

32. Tensor Methods for Training Neural Networks
Second order score function m-th order score function
Input pdf is p(x)
Sm(x) :=
(−1)m∇(m)p(x)
p(x)
Capture local variations in
data.
Algorithm for Training Neural Networks
Estimate score functions using autoencoder.
Decompose tensor E[y ⊗ Sm(x)] to obtain weights, for m ≥ 3.
Recursively estimate score function of autoencoder and repeat.

33. Tensor Methods for Training Neural Networks
Third order score function m-th order score function
Input pdf is p(x)
Sm(x) :=
(−1)m∇(m)p(x)
p(x)
Capture local variations in
data.
Algorithm for Training Neural Networks
Estimate score functions using autoencoder.
Decompose tensor E[y ⊗ Sm(x)] to obtain weights, for m ≥ 3.
Recursively estimate score function of autoencoder and repeat.

34. Demo: Training Neural Networks

35. Combining Probabilistic Models with Deep Learning
Multi-object Detection in Computer vision
Deep learning is able to extract good features, but not context.
Probabilistic models capture contextual information.
Hierarchical models + pre-trained deep learning features.
State-of-art results on Microsoft COCO.

36. Outline
1 Introduction
2 Learning Probabilistic Models
3 Experiments
4 Feature Learning with Tensor Methods
5 Conclusion

37. Conclusion: Tensor Methods for Learning
Tensor Decomposition
Efficient sample and computational complexities
Better performance compared to EM, Variational Bayes etc.
In practice
Scalable and embarrassingly parallel: handle large datasets.
Efficient performance: perplexity or ground truth validation.

38. My Research Group and Resources
Furong H. Majid J. Hanie S. Niranjan U.N.
Forough A. Tejaswi N. Hao L. Yang S.
ML summer school lectures available at
http://newport.eecs.uci.edu/anandkumar/MLSS.html