1. المشرف د.يــــاســـــــــر فـــــــؤاد By: ahmed badrealldeen
Sum-Product Networks
المشرف
د.يــــاســـــــــر فـــــــؤاد
By: ahmed badrealldeen 1

2. introduction
The key limiting factor in graphical model inference and learning is the complexity of the partition function. We thus ask the question: what are general conditions under which the partition function is tractable? The answer leads to a new kind of deep architecture, which we call sum product networks (SPNs). SPNs are directed acyclic graphs with variables as leaves, sums and products as internal nodes. We show that if an SPN is complete and consistent it represents the partition function and all marginal of some graphical model,and give semantics to its nodes .

3. What is a Sum-Product Network
What is a Sum-Product Network? * Poon and Domingos, _ Acyclic directed graph of sums and products * Leaves can be indicator variables or univariate distributions

4. SPN Architecture Specific type of deep neural network
– Activation function: product
Advantage:
– Clear semantics and well understood theory

5. This Talk: Sum-Product Networks
Compactly represent partition function using a deep network
Sum-Product Networks
Graphical Models
Existing Tractable Models

6. Learn optimal way to reuse computation, etc.
Sum-Product Networks
Graphical Models
Existing Tractable Models

7. Outline Sum-product networks (SPNs) Learning SPNs Experimental results
Conclusion 7

8. Why Is Inference Hard? Bottleneck: Summing out variables
E.g.: Partition function
Sum of exponentially many products

9. Alternative RepresentationX1 X2
P(X) 1
0.4
0.2
0.1
0.3
P(X) = 0.4  I[X1=1]  I[X2=1]
+ 0.2  I[X1=1]  I[X2=0]
+ 0.1  I[X1=0]  I[X2=1]
+ 0.3  I[X1=0]  I[X2=0]

10. Shorthand for IndicatorsX1 X2
P(X) 1
0.4
0.2
0.1
0.3
P(X) = 0.4  X1  X2
+ 0.2  X1  X2
+ 0.1  X1  X2
+ 0.3  X1  X2    

11. Easy: Set both indicators to 1
Sum Out Variables
e: X1 = 1 X1 X2
P(X) 1
0.4
0.2
0.1
0.3
P(e) = 0.4  X1  X2
+ 0.2  X1  X2
+ 0.1  X1  X2
+ 0.3  X1  X2      
Set X1 = 1, X1 = 0, X2 = 1, X2 = 1
Easy: Set both indicators to 1

12. SUM PRODUCT NETWORK Graphical Representation X1 X2
P(X) 1
0.4
0.2
0.1
0.3
0.4
0.3
0.2
0.1       X1 X1 X2 X2

13. But … Exponentially Large
Example: Parity
Uniform distribution over states with even number of 1’s
2N-1                 
N2N-1      X1 X1 X2 X2 X3 X3 X4 X4 X5 X5
Can we make this more compact 13 13

14. But … Exponentially Large
Can we make this more compact?
Example: Parity
Uniform distribution over states of even number of 1’s                       X1 X1 X2 X2 X3 X3 X4 X4 X5 X5 14 14

15. Use a Deep Network Example: Parity
Uniform distribution over states of even number of 1’s
Induce many hidden layers
Reuse partial computation 15

16. Sum-Product Networks (SPNs)
Rooted DAG
Nodes: Sum, product, input indicator
Weights on edges from sum to children 
0.7
0.3      
0.4
0.9
0.7
0.2
0.6
0.1
0.3
0.8   X1 X1 X2 X2 16

17. Valid SPN SPN is valid if S(e) = Xe S(X) for all e
Valid  Can compute marginals efficiently
Partition function Z can be computed by setting all indicators to 1 17

18. Valid SPN: General Conditions
Theorem: SPN is valid if it is complete & consistent
Consistent: Under product, no variable in one child and negation in another
Complete: Under sum, children cover the same set of variables
Incomplete
Inconsistent
S(e)  Xe S(X)
S(e)  Xe S(X) 18

19. Semantics of Sums and Products
Product  Feature  Form feature hierarchy
Sum  Mixture (with hidden var. summed out) i i  
wij
wij
Sum out Yi j ……  …… ……  …… j 
I[Yi = j]

20. Handling Continuous Variables
Sum  Integral over input
Simplest case: Indicator  Gaussian
SPN compactly defines a very large mixture of Gaussians

21. SPNs can compactly represent many more distributions
SPNs Everywhere
Graphical models
Existing tractable models, e.g.:
hierarchical mixture model, thin junction tree, etc.
SPNs can compactly represent
many more distributions 21

22. SPNs can represent, combine, and learn the optimal way
SPNs Everywhere
Graphical models
Inference methods, e.g.:
Junction-tree algorithm, message passing, …
SPNs can represent, combine, and learn the optimal way 22

23. SPNs Everywhere Graphical models
SPNs can be more compact by leveraging determinism,
context-specific independence,
etc. 23

24. SPN: First approach for learning directly from data
SPNs Everywhere
Graphical models
Models for efficient inference
E.g., arithmetic circuits,
AND/OR graphs, case-factor diagrams
SPN: First approach for
learning directly from data 24

25. SPNs Everywhere Graphical models Models for efficient inference
General, probabilistic convolutional network
Sum: Average-pooling
Max: Max-pooling 25

26. Product: Production rule
SPNs Everywhere
Graphical models
Models for efficient inference
General, probabilistic convolutional network
Grammars in vision and language
E.g., object detection grammar,
probabilistic context-free grammar
Sum: Non-terminal
Product: Production rule 26

27. Outline Sum-product networks (SPNs) Learning SPNs Experimental results
Conclusion 27

28. The Challenge In principle, can use gradient descent
But … gradient quickly dilutes
Similar problem with EM
Hard EM overcomes this problem 28

29. Outline Sum-product networks (SPNs) Learning SPNs Experimental results
Conclusion 29

30. Task: Image Completion
Very challenging
Good for evaluating deep models
Methodology:
Learn a model from training images
Complete unseen test images
Measure mean square errors

31. SPN Architecture          ...... ...... ...... ...... x
Region    
...... 
......  
...... 
...... 
Pixel x

32. Systems SPN DBM [Salakhutdinov & Hinton, 2010]
DBN [Hinton & Salakhutdinov, 2006]
PCA [Turk & Pentland, 1991]
Nearest neighbor [Hays & Efros, 2007] 32

33. Preprocessing for DBN Did not work well despite best effort
Followed Hinton & Salakhutdinov [2006]
Reduced scale: 64  64  25  25
Used much larger dataset:  120,000 images
Reduced-scale  Artificially lower errors

34. Caltech: Mean-Square Errors
LEFT
BOTTOM
SPN
3551
3270
DBM
9043
9792
DBN
4778
4492
PCA
4234
4465
Nearest Neighbor
4887
5505

35. SPN vs. DBM / DBN SPN is order of magnitude faster
No elaborate preprocessing, tuning
Reduced errors by 30-60%
Learned up to 46 layers
SPN
DBM / DBN
Learning
2-3 hours
Days
Inference
< 1 second
Minutes or hours

36. Scatter Plot: Complete Left

37. Scatter Plot: Complete Bottom

38. Olivetti: Mean-Square Errors
LEFT
BOTTOM
SPN
942
918
DBM
1866
2401
DBN
2386
1931
PCA
1076
1265
Nearest Neighbor
1527
1793

39. End-to-End Comparison
Approximate
General
Graphical Models
Approximate
Data
Performance
Given same computation budget,
which approach has better performance?
Approximate
Sum-Product Networks
Exact
Data
Performance

40. Deep Learning Stack many layers
E.g.: DBN [Hinton & Salakhutdinov,2006]
CDBN [Lee et al.,2009]
DBM [Salakhutdinov & Hinton,2010]
Potentially much more powerful than shallow architectures [Bengio, 2009]
But …
Inference is even harder
Learning requires extensive effort 40

41. Conclusion Sum-product networks (SPNs)
DAG of sums and products
Compactly represent partition function
Learn many layers of hidden variables
Exact inference: Linear time in network size
Deep learning: Online hard EM
Substantially outperform state of the art on image completion 41