Structure learning with Deep Neural Networks
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Talk about highly nonlinear association analysis in gene expression data by using deep neural networks. Given at 7th Network Modeling Workshop, Heidelberg, Germany, 2014
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Structure learning with Deep Neural Networks
- 1. Structure learning with deep neuronal networks 7th Network Modeling Workshop, 28/2/2014 Patrick Michl
- 2. Page 228.02.2014 Patrick Michl Network Modeling Agenda AutoencodersAutoencoders Biological ModelBiological Model Validation & ImplementationValidation & Implementation
- 3. Page 328.02.2014 Patrick Michl Network Modeling Real world data usually is high dimensional … x1 x2 Dataset Model Autoencoders
- 4. Page 428.02.2014 Patrick Michl Network Modeling … which makes structural analysis and modeling complicated! x1 x2 x1 x2 Dataset Model ? Autoencoders
- 5. Page 528.02.2014 Patrick Michl Network Modeling Dimensionality reduction techinques like PCA … x1 x2 PCA Dataset Model Autoencoders
- 6. Page 628.02.2014 Patrick Michl Network Modeling … can not preserve complex structures! x1 x2 PCA Dataset Model x1 x2 𝑥2=α 𝑥1+β Autoencoders
- 7. Page 728.02.2014 Patrick Michl Network Modeling Therefore the analysis of unknown structures … x1 x2 Dataset Model Autoencoders
- 8. Page 828.02.2014 Patrick Michl Network Modeling … needs more considerate nonlinear techniques! x1 x2 Dataset Model x1 x2 𝑥2=𝑓(𝑥1) Autoencoders
- 9. Page 928.02.2014 Patrick Michl Network Modeling Autoencoders are artificial neuronal networks … Autoencoder • Artificial Neuronal Network Autoencoders input data X output data X‘ Perceptrons Gaussian Units
- 10. Page 1028.02.2014 Patrick Michl Network Modeling Autoencoders are artificial neuronal networks … Autoencoder • Artificial Neuronal Network Autoencoders input data X output data X‘ Perceptrons Gaussian Units Perceptron 1 0 Gauss Units R
- 11. Page 1128.02.2014 Patrick Michl Network Modeling Autoencoders are artificial neuronal networks … Autoencoder • Artificial Neuronal Network Autoencoders input data X output data X‘ Perceptrons Gaussian Units
- 12. Page 1228.02.2014 Patrick Michl Network Modeling Autoencoder • Artificial Neuronal Network • Multiple hidden layers Autoencoders … with multiple hidden layers. Gaussian Units input data X output data X‘ Perceptrons (Visible layers) (Hidden layers)
- 13. Page 1328.02.2014 Patrick Michl Network Modeling Autoencoder • Artificial Neuronal Network • Multiple hidden layers Autoencoders Such networks are called deep networks. Gaussian Units input data X output data X‘ Perceptrons (Visible layers) (Hidden layers)
- 14. Page 1428.02.2014 Patrick Michl Network Modeling Autoencoder • Artificial Neuronal Network • Multiple hidden layers Autoencoders Such networks are called deep networks. Gaussian Units input data X output data X‘ Perceptrons (Visible layers) (Hidden layers)Definition (deep network) Deep networks are artificial neuronal networks with multiple hidden layers
- 15. Page 1528.02.2014 Patrick Michl Network Modeling Autoencoder Autoencoders Gaussian Units input data X output data X‘ Perceptrons (Visible layers) (Hidden layers) Such networks are called deep networks. • Deep network
- 16. Page 1628.02.2014 Patrick Michl Network Modeling Autoencoder Autoencoders Autoencoders have a symmetric topology … Gaussian Units input data X output data X‘ Perceptrons (Visible layers) (Hidden layers) • Deep network • Symmetric topology
- 17. Page 1728.02.2014 Patrick Michl Network Modeling Autoencoder Autoencoders … with an odd number of hidden layers. Gaussian Units input data X output data X‘ Perceptrons (Visible layers) (Hidden layers) • Deep network • Symmetric topology
- 18. Page 1828.02.2014 Patrick Michl Network Modeling Autoencoder Autoencoders The small layer in the center works lika an information bottleneck input data X output data X‘ • Deep network • Symmetric topology • Information bottleneck Bottleneck
- 19. Page 1928.02.2014 Patrick Michl Network Modeling Autoencoder Autoencoders ... that creates a low dimensional code for each sample in the input data. input data X output data X‘ • Deep network • Symmetric topology • Information bottleneck Bottleneck
- 20. Page 2028.02.2014 Patrick Michl Network Modeling Autoencoder Autoencoders The upper stack does the encoding … input data X output data X‘ • Deep network • Symmetric topology • Information bottleneck • Encoder Encoder
- 21. Page 2128.02.2014 Patrick Michl Network Modeling Autoencoder Autoencoders … and the lower stack does the decoding. input data X output data X‘ • Deep network • Symmetric topology • Information bottleneck • Encoder • Decoder Encoder Decoder
- 22. Page 2228.02.2014 Patrick Michl Network Modeling • Deep network • Symmetric topology • Information bottleneck • Encoder • Decoder Autoencoder Autoencoders … and the lower stack does the decoding. input data X output data X‘ Encoder Decoder Definition (deep network) Deep networks are artificial neuronal networks with multiple hidden layers Definition (autoencoder) Autoencoders are deep networks with a symmetric topology and an odd number of hiddern layers, containing a encoder, a low dimensional representation and a decoder.
- 23. Page 2328.02.2014 Patrick Michl Network Modeling Autoencoder Autoencoders Autoencoders can be used to reduce the dimension of data … input data X output data X‘ Problem: dimensionality of data Idea: 1. Train autoencoder to minimize the distance between input X and output X‘ 2. Encode X to low dimensional code Y 3. Decode low dimensional code Y to output X‘ 4. Output X‘ is low dimensional
- 24. Page 2428.02.2014 Patrick Michl Network Modeling Autoencoder Autoencoders … if we can train them! input data X output data X‘ Problem: dimensionality of data Idea: 1. Train autoencoder to minimize the distance between input X and output X‘ 2. Encode X to low dimensional code Y 3. Decode low dimensional code Y to output X‘ 4. Output X‘ is low dimensional
- 25. Page 2528.02.2014 Patrick Michl Network Modeling Autoencoder Autoencoders In feedforward ANNs backpropagation is a good approach. input data X output data X‘ Training Backpropagation
- 26. Page 2628.02.2014 Patrick Michl Network Modeling Backpropagation Autoencoder Autoencoders input data X output data X‘ Training Definition (autoencoder) Backpropagation (1) The distance (error) between current output X‘ and wanted output Y is computed. This gives a error function error In feedforward ANNs backpropagation is a good approach.
- 27. Page 2728.02.2014 Patrick Michl Network Modeling Backpropagation Autoencoder Autoencoders In feedforward ANNs backpropagation is the choice input data X output data X‘ Training Definition (autoencoder) Backpropagation (1) The distance (error) between current output X‘ and wanted output Y is computed. This gives a error function Example (linear neuronal unit with two inputs)
- 28. Page 2828.02.2014 Patrick Michl Network Modeling Backpropagation Autoencoder Autoencoders input data X output data X‘ Training Definition (autoencoder) Backpropagation (1) The distance (error) between current output X‘ and wanted output Y is computed. This gives a error function (2) By calculating we get a vector that shows in a direction which decreases the error (3) We update the parameters to decrease the error In feedforward ANNs backpropagation is a good approach.
- 29. Page 2928.02.2014 Patrick Michl Network Modeling Backpropagation Autoencoder Autoencoders In feedforward ANNs backpropagation is the choice input data X output data X‘ Training Definition (autoencoder) Backpropagation (1) The distance (error) between current output X‘ and wanted output Y is computed. This gives a error function (2) By calculating we get a vector that shows in a direction which decreases the error (3) We update the parameters to decrease the error (4) We repeat that
- 30. Page 3028.02.2014 Patrick Michl Network Modeling Autoencoder Autoencoders … the problem are the multiple hidden layers! input data X output data X‘ Training Backpropagation Problem: Deep Network
- 31. Page 3128.02.2014 Patrick Michl Network Modeling Autoencoder Autoencoders input data X output data X‘ Training Backpropagation is known to be slow far away from the output layer … Backpropagation Problem: Deep Network • Very slow training
- 32. Page 3228.02.2014 Patrick Michl Network Modeling Autoencoder Autoencoders input data X output data X‘ Training … and can converge to poor local minima. Backpropagation Problem: Deep Network • Very slow training • Maybe bad solution
- 33. Page 3328.02.2014 Patrick Michl Network Modeling Autoencoder Autoencoders input data X output data X‘ Training Backpropagation Problem: Deep Network • Very slow training • Maybe bad solution Idea: Initialize close to a good solution The task is to initialize the parameters close to a good solution!
- 34. Page 3428.02.2014 Patrick Michl Network Modeling Autoencoder Autoencoders input data X output data X‘ Training Backpropagation Problem: Deep Network • Very slow training • Maybe bad solution Idea: Initialize close to a good solution • Pretraining Therefore the training of autoencoders has a pretraining phase …
- 35. Page 3528.02.2014 Patrick Michl Network Modeling Autoencoder Autoencoders input data X output data X‘ Training Backpropagation Problem: Deep Network • Very slow training • Maybe bad solution Idea: Initialize close to a good solution • Pretraining • Restricted Boltzmann Machines … which uses Restricted Boltzmann Machines (RBMs)
- 36. Page 3628.02.2014 Patrick Michl Network Modeling Autoencoder Autoencoders input data X output data X‘ Training Backpropagation Problem: Deep Network • Very slow training • Maybe bad solution Idea: Initialize close to a good solution • Pretraining • Restricted Boltzmann Machines … which uses Restricted Boltzmann Machines (RBMs) Restricted Boltzmann Machine • RBMs are Markov Random Fields
- 37. Page 3728.02.2014 Patrick Michl Network Modeling Autoencoder Autoencoders input data X output data X‘ Training Backpropagation Problem: Deep Network • Very slow training • Maybe bad solution Idea: Initialize close to a good solution • Pretraining • Restricted Boltzmann Machines … which uses Restricted Boltzmann Machines (RBMs) Restricted Boltzmann Machine • RBMs are Markov Random Fields Markov Random Field Every unit influences every neighbor The coupling is undirected Motivation (Ising Model) A set of magnetic dipoles (spins) is arranged in a graph (lattice) where neighbors are coupled with a given strengt
- 38. Page 3828.02.2014 Patrick Michl Network Modeling Autoencoder Autoencoders input data X output data X‘ Training Backpropagation Problem: Deep Network • Very slow training • Maybe bad solution Idea: Initialize close to a good solution • Pretraining • Restricted Boltzmann Machines … which uses Restricted Boltzmann Machines (RBMs) Restricted Boltzmann Machine • RBMs are Markov Random Fields • Bipartite topology: visible (v), hidden (h) • Use local energy to calculate the probabilities of values Training: contrastive divergency (Gibbs Sampling) h1 v1 v2 v3 v4 h2 h3
- 39. Page 3928.02.2014 Patrick Michl Network Modeling Autoencoder Autoencoders input data X output data X‘ Training Backpropagation Problem: Deep Network • Very slow training • Maybe bad solution Idea: Initialize close to a good solution • Pretraining • Restricted Boltzmann Machines … which uses Restricted Boltzmann Machines (RBMs) Restricted Boltzmann Machine Gibbs Sampling
- 40. Page 4028.02.2014 Patrick Michl Network Modeling Autoencoders Autoencoder The top layer RBM transforms real value data into binary codes. Top Training
- 41. Page 4128.02.2014 Patrick Michl Network Modeling Autoencoders Autoencoder Top Therefore visible units are modeled with gaussians to encode data … h2 v1 v2 v3 v4 h3 h4 h5h1 Training
- 42. Page 4228.02.2014 Patrick Michl Network Modeling Autoencoders Autoencoder Top … and many hidden units with simoids to encode dependencies h2 v1 v2 v3 v4 h3 h4 h5h1 Training
- 43. Page 4328.02.2014 Patrick Michl Network Modeling Autoencoders Autoencoder Top The objective function is the sum of the local energies. Local Energy 𝐸 𝑣 ≔ −∑ h 𝑤 𝑣h 𝑥 𝑣 𝜎 𝑣 𝑥 h + ( 𝑥 𝑣 − 𝑏 𝑣 )2 2 𝜎 𝑣 2 h2 v1 v2 v3 v4 h3 h4 h5h1 Training
- 44. Page 4428.02.2014 Patrick Michl Network Modeling Autoencoders Autoencoder Reduction The next RBM layer maps the dependency encoding… Training
- 45. Page 4528.02.2014 Patrick Michl Network Modeling Autoencoders Autoencoder Reduction … from the upper layer … v h1 v1 v2 v3 v4 h2 h3 Training
- 46. Page 4628.02.2014 Patrick Michl Network Modeling Autoencoders Autoencoder Reduction … to a smaller number of simoids … h h1 v1 v2 v3 v4 h2 h3 Training
- 47. Page 4728.02.2014 Patrick Michl Network Modeling Autoencoders Autoencoder Reduction … which can be trained faster than the top layer Local Energy 𝐸𝑣 ≔−∑ h 𝑤 𝑣h 𝑥 𝑣 𝑥h+𝑥h 𝑏h 𝐸h ≔−∑ 𝑣 𝑤 𝑣h 𝑥 𝑣 𝑥h+𝑥 𝑣 𝑏 𝑣 h1 v1 v2 v3 v4 h2 h3 Training
- 48. Page 4828.02.2014 Patrick Michl Network Modeling Autoencoders Autoencoder Unrolling The symmetric topology allows us to skip further training. Training
- 49. Page 4928.02.2014 Patrick Michl Network Modeling Autoencoders Autoencoder Unrolling The symmetric topology allows us to skip further training. Training
- 50. Page 5028.02.2014 Patrick Michl Network Modeling After pretraining backpropagation usually finds good solutions Autoencoders Autoencoder Training • Pretraining Top RBM (GRBM) Reduction RBMs Unrolling • Finetuning Backpropagation
- 51. Page 5128.02.2014 Patrick Michl Network Modeling The algorithmic complexity of RBM training depends on the network size Autoencoders Autoencoder Training • Complexity: O(inw) i: number of iterations n: number of nodes w: number of weights • Memory Complexity: O(w)
- 52. Page 5228.02.2014 Patrick Michl Network Modeling Agenda Autoencoders Biological Model Validation & Implementation
- 53. Page 5328.02.2014 Patrick Michl Network Modeling Network Modeling Restricted Boltzmann Machines (RBM) How to model the topological structure? S E TF
- 54. Page 5428.02.2014 Patrick Michl Network Modeling We define S and E as visible data Layer … S E TF Network Modeling Restricted Boltzmann Machines (RBM)
- 55. Page 5528.02.2014 Patrick Michl Network Modeling S E TF Network Modeling Restricted Boltzmann Machines (RBM) We identify S and E with the visible layer …
- 56. Page 5628.02.2014 Patrick Michl Network Modeling S E … and the TFs with the hidden layer in a RBM TF Network Modeling Restricted Boltzmann Machines (RBM)
- 57. Page 5728.02.2014 Patrick Michl Network Modeling S E The training of the RBM gives us a model TF Network Modeling Restricted Boltzmann Machines (RBM)
- 58. Page 5828.02.2014 Patrick Michl Network Modeling Agenda Autoencoder Biological Model Implementation & Results
- 59. Page 5928.02.2014 Patrick Michl Network Modeling Results Validation of the results • Needs information about the true regulation • Needs information about the descriptive power of the data
- 60. Page 6028.02.2014 Patrick Michl Network Modeling Results Validation of the results • Needs information about the true regulation • Needs information about the descriptive power of the data Without this infomation validation can only be done, using artificial datasets!
- 61. Page 6128.02.2014 Patrick Michl Network Modeling Results Artificial datasets We simulate data in three steps:
- 62. Page 6228.02.2014 Patrick Michl Network Modeling Results Artificial datasets We simulate data in three steps Step 1 Choose number of Genes (E+S) and create random bimodal distributed data
- 63. Page 6328.02.2014 Patrick Michl Network Modeling Results Artificial datasets We simulate data in three steps Step 1 Choose number of Genes (E+S) and create random bimodal distributed data Step 2 Manipulate data in a fixed order
- 64. Page 6428.02.2014 Patrick Michl Network Modeling Results Artificial datasets We simulate data in three steps 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
- 65. Page 6528.02.2014 Patrick Michl Network Modeling Simulation Results Step 1 Number of visible nodes 8 (4E, 4S) Create random data: Random {-1, +1} + N(0,
- 66. Page 6628.02.2014 Patrick Michl Network Modeling Simulation Results Noise Step 2 Manipulate data
- 67. Page 6728.02.2014 Patrick Michl Network Modeling Simulation Results Step 3 Add noise: N(0,
- 68. Page 6828.02.2014 Patrick Michl Network Modeling Results We analyse the data X with an RBM
- 69. Page 6928.02.2014 Patrick Michl Network Modeling Results We train an autoencoder with 9 hidden layers and 165 nodes: Layer 1 & 9: 32 hidden units Layer 2 & 8: 24 hidden units Layer 3 & 7: 16 hidden units Layer 4 & 6: 8 hidden units Layer 5: 5 hidden units input data X output data X‘
- 70. Page 7028.02.2014 Patrick Michl Network Modeling Results We transform the data from X to X‘ And reduce the dimensionality
- 71. Page 7128.02.2014 Patrick Michl Network Modeling Results We analyse the transformed data X‘ with an RBM
- 72. Page 7228.02.2014 Patrick Michl Network Modeling Results Lets compare the models
- 73. Page 7328.02.2014 Patrick Michl Network Modeling Results Another Example with more nodes and larger autoencoder
- 74. Page 7428.02.2014 Patrick Michl Network Modeling Conclusion Conclusion • Autoencoders can improve modeling significantly by reducing the dimensionality of data • Autoencoders preserve complex structures in their multilayer perceptron network. Analysing those networks (for example with knockout tests) could give more structural information • The drawback are high computational costs Since the field of deep learning is getting more popular (Face recognition / Voice recognition, Image transformation). Many new improvements in facing the computational costs have been made.
- 75. Page 7528.02.2014 Patrick Michl Network Modeling Acknowledgement eilsLABS Prof. Dr. Rainer König Prof. Dr. Roland Eils Network Modeling Group