1. Regulation Analysis using Restricted Boltzmann Machines
Network Modeling Seminar, 10/1/2013
Patrick Michl

2. Agenda
Biological Problem
Analysing the regulation of metabolism
Modeling
Implementation & Results

3. Biological Problem Analysing the regulation of metabolism
Signal
Regulation
Metabolism
A linear metabolic pathway of enzymes (E) …

4. Biological Problem Analysing the regulation of metabolism
Signal
Regulation
Metabolism
… is regulated by transcription factors (TF) …

5. Biological Problem Analysing the regulation of metabolism
Signal
Regulation
Metabolism
… which respond to signals (S)

6. Biological Problem Analysing the regulation of metabolism
Upregulated linear pathways …

7. Biological Problem Analysing the regulation of metabolism
… can appear in different patterns

8. Biological Problem Analysing the regulation of metabolism? ? E
Which transcription factors and signals cause this patterns …

9. Biological Problem Analysing the regulation of metabolism? ? ? ? E
… and how do they interact? (topological structure)

10. Agenda
Biological Problem
Analysing the regulation of metabolism
Network Modeling
Restricted Boltzmann Machines (RBM)
Validation & Implementation

11. Network Modeling Restricted Boltzmann Machines (RBM)TF E
Lets start with some pathway of our interest …

12. Network Modeling Restricted Boltzmann Machines (RBM)TF E
… and lists of interesting TFs and interesting SigMols

13. Network Modeling Restricted Boltzmann Machines (RBM)TF E
How to model the topological structure?

14. Network Modeling Restricted Boltzmann Machines (RBM)
Graphical Models
Graphical Models can preserve topological structures …

15. Network Modeling Restricted Boltzmann Machines (RBM)
Graphical Models
Directed Graph
Undirected Graph …
… but there are many types of graphical models

16. Network Modeling Restricted Boltzmann Machines (RBM)
Graphical Models
Directed Graph
Bayesian Networks
Undirected Graph …
The most common type is the Bayesian Network (BN) …

17. Network Modeling Restricted Boltzmann Machines (RBM)
Bayesian Networks a b c
P[a,b,c]
0.1 1
0.9
0.5 … a b ? c
Bayesian Networks use joint probabilities …

18. Network Modeling Restricted Boltzmann Machines (RBM)
Bayesian Networks a b c
P[a,b,c]
0.1 1
0.9
0.5 … a b c
… to represents conditional dependencies in an acyclic graph …

19. Network Modeling Restricted Boltzmann Machines (RBM)
Bayesian Networks b a d c
… but the regulation mechanism of a cell can be more complicated

20. Network Modeling Restricted Boltzmann Machines (RBM)
Graphical Models
Directed Graph
Bayesian Networks
Undirected Graph
Markov Random Fields …
Another type of graphical models are Markov Random Fields (MRF)…

21. Network Modeling Restricted Boltzmann Machines (RBM)
Markov Random Fields
Motivation (Ising Model)
A set of magnetic dipoles (spins)
is arranged in a graph (lattice)
where neighbors are
coupled with a given strengt
... which emerged with the Ising Model from statistical Physics …

22. Network Modeling Restricted Boltzmann Machines (RBM)
Markov Random Fields
Motivation (Ising Model)
A set of magnetic dipoles (spins)
is arranged in a graph (lattice)
where neighbors are
coupled with a given strengt
... which uses local energies to calculate new states …

23. Network Modeling Restricted Boltzmann Machines (RBM)
Markov Random Fields
Drawback
By allowing cyclic dependencies
the computational costs
explode b a d c
… the drawback are high computational costs …

24. Network Modeling Restricted Boltzmann Machines (RBM)
Graphical Models
Directed Graph
Bayesian Networks
Undirected Graph
Markov Random Fields
Restricted Boltzmann Machines (RBM) …
… which can be avoided by using Restricted Boltzmann Machines

25. Network Modeling Restricted Boltzmann Machines (RBM)
Neuron like units
RBMs are Artificial Neuronal Networks …

26. Network Modeling Restricted Boltzmann Machines (RBM)v1 v2 v3 v4 h1 h2 h3
… with two layers: visible units (v) and hidden units (h)

27. Network Modeling Restricted Boltzmann Machines (RBM)v1 v2 v3 v4 h1 h2 h3
Visible units are strictly connected with hidden units

28. Network Modeling Restricted Boltzmann Machines (RBM)
𝑉 ≔set of visible units
𝑥 𝑣 ≔value of unit 𝑣,∀𝑣∈𝑉
𝑥 𝑣 ∈𝑅, ∀𝑣∈𝑉
In our model the visible units have continuous values …

29. Network Modeling Restricted Boltzmann Machines (RBM)
𝑉 ≔set of visible units
𝑥 𝑣 ≔value of unit 𝑣,∀𝑣∈𝑉
𝑥 𝑣 ∈𝑅, ∀𝑣∈𝑉
𝐻 ≔set of hidden units
𝑥 ℎ ≔value of unit ℎ, ∀ℎ∈𝐻
𝑥 ℎ ∈{0, 1}, ∀ℎ∈𝐻
… and the hidden units binary values

30. Network Modeling Restricted Boltzmann Machines (RBM)
𝑥 𝑣 ~𝑁 𝑏 𝑣 + ℎ 𝑤 𝑣ℎ 𝑥 ℎ , 𝜎 𝑣 ,∀𝑣∈𝑉
𝜎 𝑣 ≔ std. dev. of unit 𝑣
𝑏 𝑣 ≔bias of unit 𝑣
𝑤 𝑣ℎ ≔weight of edge (𝑣,ℎ)
Visible units are modeled with gaussians to encode data …

31. Network Modeling Restricted Boltzmann Machines (RBM)
𝑥 𝑣 ~𝑁 𝑏 𝑣 + ℎ 𝑤 𝑣ℎ 𝑥 ℎ , 𝜎 𝑣 ,∀𝑣∈𝑉
𝜎 𝑣 ≔ std. dev. of unit 𝑣
𝑏 𝑣 ≔bias of unit 𝑣
𝑤 𝑣ℎ ≔weight of edge (𝑣,ℎ)
𝑥 ℎ ~sigmoid 𝑏 ℎ + 𝑣 𝑤 𝑣ℎ 𝑥 𝑣 𝜎 𝑣 ,∀ℎ∈𝐻
𝑏 ℎ ≔bias of unit ℎ
… and hidden units with simoids to encode dependencies

32. Network Modeling Restricted Boltzmann Machines (RBM)
Learning in Restricted Boltzmann Machines
Task: Find dependencies in data
↔ Find configuration of parameters with maximum likelihood (to data)
The challenge is to find the configuration of the parameters …

33. Network Modeling Restricted Boltzmann Machines (RBM)
Learning in Restricted Boltzmann Machines
Task: Find dependencies in data
↔ Find configuration of parameters with maximum likelihood (to data)
In RBMs configurations of parameters have probabilities,
that can be defined by local energies
Local Energy 1 2
𝐸 𝑣 ≔− ℎ 𝑤 𝑣ℎ 𝑥 𝑣 𝜎 𝑣 𝑥 ℎ + ( 𝑥 𝑣 − 𝑏 𝑣 ) 𝜎 𝑣 2
𝐸 ℎ ≔− 𝑣 𝑤 𝑣ℎ 𝑥 𝑣 𝜎 𝑣 𝑥 ℎ + 𝑥 ℎ 𝑏 ℎ
Like in the Ising model the units states correspond to local energies …

34. Network Modeling Restricted Boltzmann Machines (RBM)
Learning in Restricted Boltzmann Machines
Task: Find dependencies in data
↔ Find configuration of parameters with maximum likelihood (to data)
↔ Minimize global energy (to data)
Global Energy
𝐸≔ 𝑣 𝐸 𝑣 + ℎ 𝐸 ℎ =− 𝑣 ℎ 𝑤 𝑣ℎ 𝑥 𝑣 𝜎 𝑣 𝑥 ℎ + 𝑣 ( 𝑥 𝑣 − 𝑏 𝑣 ) 𝜎 𝑣 ℎ 𝑤 𝑣ℎ 𝑥 𝑣 𝜎 𝑣 𝑥 ℎ
… which sum to a global energy, which is our objective function

35. Network Modeling Restricted Boltzmann Machines (RBM)
Learning in Restricted Boltzmann Machines
Task: Find dependencies in data
↔ Find configuration of parameters with maximum likelihood (to data)
↔ Minimize global energy (to data)
↔ Perform stochastic gradient descent on 𝜎 𝑣 , 𝑏 𝑣 , 𝑏 ℎ , 𝑤 𝑣ℎ (to data)
The optimization can be done using stochastic gradient descent …

36. Network Modeling Restricted Boltzmann Machines (RBM)
Learning in Restricted Boltzmann Machines
Task: Find dependencies in data
↔ Find configuration of parameters with maximum likelihood (to data)
↔ Minimize global energy (to data)
↔ Perform stochastic gradient descent on 𝜎 𝑣 , 𝑏 𝑣 , 𝑏 ℎ , 𝑤 𝑣ℎ (to data)
Gradient Descent on RBMs
The bipartite graph structure allows
constrastive divergency learning,
using Gibbs-sampling
… which has an efficient learning algorithmus

37. Network Modeling Restricted Boltzmann Machines (RBM)TF E
How to model our initial structure as an RBM?

38. Network Modeling Restricted Boltzmann Machines (RBM)TF E
We define S and E as visible Layer …

39. Network Modeling Restricted Boltzmann Machines (RBM)TF
We define S and E as visible Layer …

40. Network Modeling Restricted Boltzmann Machines (RBM)TF
… and TF as hidden Layer

41. Agenda
Biological Problem
Analysing the regulation of metabolism
Network Modeling
Restricted Boltzmann Machines (RBM)
Implementation & Results
python::metapath

42. Results
Validation of the results
Information about the true regulation
Information about the descriptive power of the data

43. Results
Validation of the results
Information about the true regulation
Information about the descriptive power of the data
Without this infomation validation can only be done, using simulated data!

44. Results
Simulation 1
First of all we need to understand how the modell handles
dependencies and noise
To demonstrate this we create very simple data with a simple structure

45. What can we expect from this model?
Simulation 1 S TF E
What can we expect from this model?

46. … as RBM we get 8 visible and 2 hidden units, fully connected
Simulation 1 S E TF
… as RBM we get 8 visible and 2 hidden units, fully connected

47. Let‘s feed the machine with samples …
Simulation 1
Data S E
1,0,0,1
1,0,0,0
1,1,0,0
1,0,1,0
1,0,1,1
0,0,0,0
0,1,0,0
0,0,1,0
0,0,0,1
Let‘s feed the machine with samples …

48. .. to get the calculated parameters (especially the weight matrix)
Simulation 1
Weight matrix
TF1
TF2 S1
0,3
0,8 S2
0,5
0,6 S3
1,0
0,1 S4 E1
0,0 E2 E3 E4
0,2
.. to get the calculated parameters (especially the weight matrix)

49. The weights are visualized by the intensity of the edges
Simulation 1
Weight matrix S
TF1
TF2 S1
0,3
0,8 S2
0,5
0,6 S3
1,0
0,1 S4 E1
0,0 E2 E3 E4
0,2 TF E
The weights are visualized by the intensity of the edges

50. Now we can compare the results with the samples
Simulation 1
Learning samples S S E
1,0,0,1
1,0,0,0
1,1,0,0
1,0,1,0
1,0,1,1
0,0,0,0
0,1,0,0
0,0,1,0
0,0,0,1 TF E
Now we can compare the results with the samples

51. There‘s a strong dependency between S3 an E1
Simulation 1
Learning samples S S E
1,0,0,1
1,0,0,0
1,1,0,0
1,0,1,0
1,0,1,1
0,0,0,0
0,1,0,0
0,0,1,0
0,0,0,1 TF E
There‘s a strong dependency between S3 an E1

52. S1, S2 and S4 do almost not affect the metabolism …
Simulation 1
Learning samples S S E
1,0,0,1
1,0,0,0
1,1,0,0
1,0,1,0
1,0,1,1
0,0,0,0
0,1,0,0
0,0,1,0
0,0,0,1 TF E
S1, S2 and S4 do almost not affect the metabolism …

53. … so we can forget them and get S1,TF1 for our regulation model
Simulation 1 S TF E
… so we can forget them and get S1,TF1 for our regulation model

54. We can also take a look at the causal mechanism …
Simulation 1
Weight matrix
TF1
TF2 S1
0,3
0,8 S2
0,5
0,6 S3
1,0
0,1 S4 E1
0,0 E2 E3 E4
0,2
We can also take a look at the causal mechanism …

55. The edge (S3, TF1) dominates TF1 …
Simulation 1
Weight matrix
TF1
TF2 S1
0,3
0,8 S2
0,5
0,6 S3
1,0
0,1 S4 E1
0,0 E2 E3 E4
0,2
The edge (S3, TF1) dominates TF1 …

56. Also E1 seems to have an effect on S3 (fewer than S3 on E1)
Simulation 1
TF1 e1
TF1 e4 s3
TF1
TF1
TF1 e3 e2
Also E1 seems to have an effect on S3 (fewer than S3 on E1)

57. Of course we want to compare the method with Bayesian Networks
Results
Comparing to Bayesian Networks
For this purpose we simulate data in three steps
Of course we want to compare the method with Bayesian Networks

58. Of course we want to compare the method with Bayesian Networks
Results
Comparing to Bayesian Networks
Step 1
Choose number of Genes (E+S) and
create random bimodal distributed data
Of course we want to compare the method with Bayesian Networks

59. Of course we want to compare the method with Bayesian Networks
Results
Comparing to Bayesian Networks
Step 1
Choose number of Genes (E+S) and
create random bimodal distributed data Step 2
Manipulate data in a fixed order
Of course we want to compare the method with Bayesian Networks

60. Of course we want to compare the method with Bayesian Networks
Results
Comparing to Bayesian Networks
Step 1
Choose number of Genes (E+S) and
create random bimodal distributed data Step 2
Manipulate data in a fixed order
Step 3
Add noise to manipulated data and normalize data
Of course we want to compare the method with Bayesian Networks

61. Of course we want to compare the method with Bayesian Networks
Results
Comparing to Bayesian Networks
Idea
‚melt down‘ the bimodal distribution from very sharp to very noisy
Try to find the original causal structure with BN and RBM
Measure Accuracy by counting the right and wrong dependencies
Of course we want to compare the method with Bayesian Networks

62. Of course we want to compare the method with Bayesian Networks
Results
Simulation 2
Step 1: Number of visible nodes 8 (4E, 4S)
Create intergradient datasets from sharp to noisy bimodal distribution
𝜎 1 =0.0, 𝜎 1 =0.3, 𝜎 3 =0.9, 𝜎 4 =1.2, 𝜎 4 =1.5
Step 2 + 3: Data Manipulation + add noise
𝑒 1 =0.5 𝑠 𝑠 2 +𝑁(𝜇=0, 𝜎)
𝑒 2 =0.5 𝑠 𝑠 3 +𝑁(𝜇=0, 𝜎)
𝑒 3 =0.5 𝑠 𝑠 4 +𝑁(𝜇=0, 𝜎)
𝑒 4 =0.5 𝑠 𝑠 1 +𝑁(𝜇=0, 𝜎)
Of course we want to compare the method with Bayesian Networks

63. Results
Simulation 2
RBM
Model
(𝜎=0.0)

64. Results
Simulation 2
Causal
Mechanism
(𝜎=0.0)

65. Results
Simulation 2
Comparison
BN / RBM

66. Conclusion
Conclusion
RBMs are more stable against noise compared to BNs. It has to be assumed that RBMs have high predictive power regarding the regulation mechanisms of cells
The drawback are high computational costs Since RBMs are getting more popular (Face recognition / Voice recognition, Image transformation). Many new improvements in facing the computational costs have been made.

67. Acknowledgement
eilsLABS PD Dr. Rainer König Prof. Dr Roland Eils Network Modeling Group