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Structure learning with Deep Neural Networks

Patrick Michl
Patrick Michl

The document discusses the structure and training of autoencoders, a type of deep neural network composed of multiple hidden layers and a symmetric topology. It explains the process of dimensionality reduction through training these networks to minimize the difference between input and output data, as well as the use of restricted Boltzmann machines for pretraining to enhance learning efficiency. Validation of results is emphasized, including the simulation of artificial datasets to assess the model's performance.

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Structure learning with deep neuronal networks 7th Network Modeling Workshop, 28/2/2014 Patrick Michl
Page 228.02.2014 Patrick Michl Network Modeling Agenda AutoencodersAutoencoders Biological ModelBiological Model Validation & ImplementationValidation & Implementation
Page 328.02.2014 Patrick Michl Network Modeling Real world data usually is high dimensional … x1 x2 Dataset Model Autoencoders
Page 428.02.2014 Patrick Michl Network Modeling … which makes structural analysis and modeling complicated! x1 x2 x1 x2 Dataset Model ?  Autoencoders
Page 528.02.2014 Patrick Michl Network Modeling Dimensionality reduction techinques like PCA … x1 x2 PCA Dataset Model Autoencoders
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
Page 728.02.2014 Patrick Michl Network Modeling Therefore the analysis of unknown structures … x1 x2 Dataset Model Autoencoders
Page 828.02.2014 Patrick Michl Network Modeling … needs more considerate nonlinear techniques! x1 x2 Dataset Model x1 x2 𝑥2=𝑓(𝑥1)  Autoencoders
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
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
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
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)
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)
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
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
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
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
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
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
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
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
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.
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
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
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
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.
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)
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.
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  
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
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
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
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!
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 …
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)
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
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
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
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
Page 4028.02.2014 Patrick Michl Network Modeling Autoencoders Autoencoder The top layer RBM transforms real value data into binary codes.   Top Training
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
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
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
Page 4428.02.2014 Patrick Michl Network Modeling Autoencoders Autoencoder Reduction   The next RBM layer maps the dependency encoding… Training
Page 4528.02.2014 Patrick Michl Network Modeling Autoencoders Autoencoder Reduction … from the upper layer … v   h1 v1 v2 v3 v4 h2 h3 Training
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
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
Page 4828.02.2014 Patrick Michl Network Modeling Autoencoders Autoencoder Unrolling The symmetric topology allows us to skip further training. Training
Page 4928.02.2014 Patrick Michl Network Modeling Autoencoders Autoencoder Unrolling The symmetric topology allows us to skip further training. Training
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
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)
Page 5228.02.2014 Patrick Michl Network Modeling Agenda Autoencoders Biological Model Validation & Implementation
Page 5328.02.2014 Patrick Michl Network Modeling Network Modeling Restricted Boltzmann Machines (RBM) How to model the topological structure? S E TF
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)
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 …
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)
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)
Page 5828.02.2014 Patrick Michl Network Modeling Agenda Autoencoder Biological Model Implementation & Results
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
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!
Page 6128.02.2014 Patrick Michl Network Modeling Results Artificial datasets We simulate data in three steps:
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
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
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
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,  
Page 6628.02.2014 Patrick Michl Network Modeling Simulation Results Noise   Step 2 Manipulate data
Page 6728.02.2014 Patrick Michl Network Modeling Simulation Results Step 3 Add noise: N(0,  
Page 6828.02.2014 Patrick Michl Network Modeling Results We analyse the data X with an RBM
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‘
Page 7028.02.2014 Patrick Michl Network Modeling Results We transform the data from X to X‘ And reduce the dimensionality
Page 7128.02.2014 Patrick Michl Network Modeling Results We analyse the transformed data X‘ with an RBM
Page 7228.02.2014 Patrick Michl Network Modeling Results Lets compare the models
Page 7328.02.2014 Patrick Michl Network Modeling Results Another Example with more nodes and larger autoencoder
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.
Page 7528.02.2014 Patrick Michl Network Modeling Acknowledgement eilsLABS Prof. Dr. Rainer König Prof. Dr. Roland Eils Network Modeling Group

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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