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`pydbm` is Python library for building Restricted Boltzmann Machine(RBM), Deep Boltzmann Machine(DBM), Recurrent neural network Restricted Boltzmann Machine(RNN-RBM), LSTM Recurrent Temporal Restricted Boltzmann Machine(LSTM-RTRBM), and Shape Boltzmann Machine(Shape-BM).

Project description

Deep Learning Library: pydbm

pydbm is Python library for building Restricted Boltzmann Machine(RBM), Deep Boltzmann Machine(DBM), Long Short-Term Memory Recurrent Temporal Restricted Boltzmann Machine(LSTM-RTRBM), and Shape Boltzmann Machine(Shape-BM).

This is Cython version. pydbm_mxnet (MXNet version) is derived from this library.

Documentation

Full documentation is available on https://code.accel-brain.com/Deep-Learning-by-means-of-Design-Pattern/ . This document contains information on functionally reusability, functional scalability and functional extensibility.

Installation

Install using pip:

pip install pydbm

Or, you can install from wheel file.

pip install https://storage.googleapis.com/accel-brain-code/Deep-Learning-by-means-of-Design-Pattern/pydbm-{X.X.X}-cp36-cp36m-linux_x86_64.whl

Source code

The source code is currently hosted on GitHub.

Python package index(PyPI)

Installers for the latest released version are available at the Python package index.

Dependencies

  • numpy: v1.13.3 or higher.
  • cython: v0.27.1 or higher.

Description

The function of pydbm is building and modeling Restricted Boltzmann Machine(RBM) and Deep Boltzmann Machine(DBM). The models are functionally equivalent to stacked auto-encoder. The basic function is the same as dimensions reduction(or pre-training). And this library enables you to build many functional extensions from RBM and DBM such as Recurrent Temporal Restricted Boltzmann Machine(RTRBM), Recurrent Neural Network Restricted Boltzmann Machine(RNN-RBM), Long Short-Term Memory Recurrent Temporal Restricted Boltzmann Machine(LSTM-RTRBM), and Shape Boltzmann Machine(Shape-BM).

As more usecases, RTRBM, RNN-RBM, and LSTM-RTRBM can learn dependency structures in temporal patterns such as music, natural sentences, and n-gram. RTRBM is a probabilistic time-series model which can be viewed as a temporal stack of RBMs, where each RBM has a contextual hidden state that is received from the previous RBM and is used to modulate its hidden units bias. The RTRBM can be understood as a sequence of conditional RBMs whose parameters are the output of a deterministic RNN, with the constraint that the hidden units must describe the conditional distributions. This constraint can be lifted by combining a full RNN with distinct hidden units. In terms of this possibility, RNN-RBM and LSTM-RTRBM are structurally expanded model from RTRBM that allows more freedom to describe the temporal dependencies involved.

The usecases of Shape-BM are image segmentation, object detection, inpainting and graphics. Shape-BM is the model for the task of modeling binary shape images, in that samples from the model look realistic and it can generalize to generate samples that differ from training examples.

Image in the Weizmann horse dataset.

           

Binarized image.

       

Reconstructed image by Shape-BM.

The structure of RBM.

According to graph theory, the structure of RBM corresponds to a complete bipartite graph which is a special kind of bipartite graph where every node in the visible layer is connected to every node in the hidden layer. The state of this structure can be reflected by the energy function:

where is a bias in visible layer, is a bias in hidden layer, is an activity or a state in visible layer, is an activity or a state in hidden layer, and is a weight matrix in visible and hidden layer. The activities can be calculated as the below product, since the link of activations of visible layer and hidden layer are conditionally independent.

The learning equations of RBM.

Because of the rules of conditional independence, the learning equations of RBM can be introduced as simple form. The distribution of visible state which is marginalized over the hidden state is as following:

where is a partition function in statistical mechanics or thermodynamics. Let be set of observed data points, then . Therefore the gradients on the parameter of the log-likelihood function are

where is an expected value for . is a sigmoid function.

The learning equations of RBM are introduced by performing control so that those gradients can become zero.

Contrastive Divergence as an approximation method.

In relation to RBM, Contrastive Divergence(CD) is a method for approximation of the gradients of the log-likelihood. The procedure of this method is similar to Markov Chain Monte Carlo method(MCMC). However, unlike MCMC, the visbile variables to be set first in visible layer is not randomly initialized but the observed data points in training dataset are set to the first visbile variables. And, like Gibbs sampler, drawing samples from hidden variables and visible variables is repeated k times. Empirically (and surprisingly), k is considered to be 1.

The structure of DBM.

Salakhutdinov, R., Hinton, G. E. (2009). Deep boltzmann machines. In International conference on artificial intelligence and statistics (pp. 448-455). p451.

As is well known, DBM is composed of layers of RBMs stacked on top of each other. This model is a structural expansion of Deep Belief Networks(DBN), which is known as one of the earliest models of Deep Learning. Like RBM, DBN places nodes in layers. However, only the uppermost layer is composed of undirected edges, and the other consists of directed edges. DBN with R hidden layers is below probabilistic model:

where r = 0 points to visible layer. Considerling simultaneous distribution in top two layer,

and conditional distributions in other layers are as follows:

The pre-training of DBN engages in a procedure of recursive learning in layer-by-layer. However, as you can see from the difference of graph structure, DBM is slightly different from DBN in the form of pre-training. For instance, if r = 1, the conditional distribution of visible layer is

.

On the other hand, the conditional distribution in the intermediate layer is

where 2 has been introduced considering that the intermediate layer r receives input data from Shallower layer r-1 and deeper layer r+1. DBM sets these parameters as initial states.

DBM as a Stacked Auto-Encoder.

DBM is functionally equivalent to a Stacked Auto-Encoder, which is-a neural network that tries to reconstruct its input. To encode the observed data points, the function of DBM is as linear transformation of feature map below

.

On the other hand, to decode this feature points, the function of DBM is as linear transformation of feature map below

.

The reconstruction error should be calculated in relation to problem setting. This library provides a default method, which can be overridden, for error function that computes Mean Squared Error(MSE).

Structural expansion for RTRBM.

The RTRBM (Sutskever, I., et al. 2009) is a probabilistic time-series model which can be viewed as a temporal stack of RBMs, where each RBM has a contextual hidden state that is received from the previous RBM and is used to modulate its hidden units bias. Let be the hidden state in previous step t-1. The conditional distribution in hidden layer in time t is

where is weight matrix in each time steps. Then sampling of observed data points is is as following:

While the hidden units are binary during inference and sampling, it is the mean-field value that is transmitted to its successors.

Structural expansion for RNN-RBM.

The RTRBM can be understood as a sequence of conditional RBMs whose parameters are the output of a deterministic RNN, with the constraint that the hidden units must describe the conditional distributions and convey temporal information. This constraint can be lifted by combining a full RNN with distinct hidden units. RNN-RBM (Boulanger-Lewandowski, N., et al. 2012), which is the more structural expansion of RTRBM, has also hidden units .

Boulanger-Lewandowski, N., Bengio, Y., & Vincent, P. (2012). Modeling temporal dependencies in high-dimensional sequences: Application to polyphonic music generation and transcription. arXiv preprint arXiv:1206.6392., p4. Single arrows represent a deterministic function, double arrows represent the stochastic hidden-visible connections of an RBM.

The biases are linear function of . This hidden units are only connected to their direct predecessor and visible units in time t by the relation:

where and are weight matrixes.

Structural expansion for LSTM-RTRBM.

An example of the application to polyphonic music generation(Lyu, Q., et al. 2015) clued me in on how is it possible to connect RTRBM with LSTM. LSTM-RTRBM model integrates the ability of LSTM in memorizing and retrieving useful history information, together with the advantage of RBM in high dimensional data modelling. Like RTRBM, LSTM-RTRBM also has the recurrent hidden units. Let be previous hidden units. The conditional distribution of the current hidden layer is as following:

where is a weight matrix which indicates the connectivity between states at each time step in RBM. Now, sampling the observed data points in RTRBM is as follows.

"Adding LSTM units to RTRBM is not trivial, considering RTRBM’s hidden units and visible units are intertwined in inference and learning. The simplest way to circumvent this difficulty is to use bypass connections from LSTM units to the hidden units besides the existing recurrent connections of hidden units, as in LSTM-RTRBM."

Lyu, Q., Wu, Z., & Zhu, J. (2015, October). Polyphonic music modelling with LSTM-RTRBM. In Proceedings of the 23rd ACM international conference on Multimedia (pp. 991-994). ACM., p.993.

Therefore it is useful to introduce a distinction of channel which means the sequential information. indicates the direct connectivity in RBM, while can be defined as a concept representing the previous time step combination in the LSTM units. Let and be the hidden units indicating short-term memory and long-term memory, respectively. Then sampling the observed data points in LSTM-RTRBM can be re-described as follows.

Structural expansion for Shape-BM.

The concept of Shape Boltzmann Machine (Eslami, S. A., et al. 2014) provided inspiration to this library. This model uses below has two layers of hidden variables: and . The visible units v arethe pixels of a binary image of size . In the visible layer we enforce local receptive fields by connecting each hidden unit in only to a subset of the visible units, corresponding to one of four rectangular patches. In order to encourage boundary consistency each patch overlaps its neighbor by pixels and so has side lengths of and . In this model, the weight matrix in visible and hidden layer correspond to conectivity between the four sets of hidden units and patches, however the visible biases are not shared.

Eslami, S. A., Heess, N., Williams, C. K., & Winn, J. (2014). The shape boltzmann machine: a strong model of object shape. International Journal of Computer Vision, 107(2), 155-176., p156.

Eslami, S. A., Heess, N., Williams, C. K., & Winn, J. (2014). The shape boltzmann machine: a strong model of object shape. International Journal of Computer Vision, 107(2), 155-176., p156.

The Shape-BM is a DBM in three layer. The learning algorithm can be completed by optimization of

where .

The commonality/variability analysis in order to practice object-oriented design.

From perspective of commonality/variability analysis in order to practice object-oriented design, the concepts of RBM and DBM paradigms can be organized as follows.

While each model is common in that it is constituted by stacked RBM, its approximation methods and activation functions are variable depending on the problem settings. Considering the commonality, it is useful to design based on Builder Pattern, which separates the construction of RBM object from its representation so that the same construction process can create different representations such as DBM, RTRBM, RNN-RBM, and Shape-BM. On the other hand, to deal with the variability, Strategy Pattern, which provides a way to define a family of algorithms such as approximation methods and activation functions, is useful design method, which is encapsulate each one as an object, and make them interchangeable from the point of view of functionally equivalent.

Usecase: Building the deep boltzmann machine for feature extracting.

Import Python and Cython modules.

# The `Client` in Builder Pattern
from pydbm.dbm.deep_boltzmann_machine import DeepBoltzmannMachine
# The `Concrete Builder` in Builder Pattern.
from pydbm.dbm.builders.dbm_multi_layer_builder import DBMMultiLayerBuilder
# Contrastive Divergence for function approximation.
from pydbm.approximation.contrastive_divergence import ContrastiveDivergence
# Logistic Function as activation function.
from pydbm.activation.logistic_function import LogisticFunction
# Tanh Function as activation function.
from pydbm.activation.tanh_function import TanhFunction
# ReLu Function as activation function.
from pydbm.activation.relu_function import ReLuFunction

Instantiate objects and call the method.

dbm = DeepBoltzmannMachine(
    DBMMultiLayerBuilder(),
    # Dimention in visible layer, hidden layer, and second hidden layer.
    [train_arr.shape[1], 10, train_arr.shape[1]],
    # Setting objects for activation function.
    [ReLuFunction(), LogisticFunction(), TanhFunction()],
    # Setting the object for function approximation.
    [ContrastiveDivergence(), ContrastiveDivergence()], 
    # Setting learning rate.
    0.05,
    # Setting dropout rate.
    0.5
)

# Execute learning.
dbm.learn(
    train_arr,
     # If approximation is the Contrastive Divergence, this parameter is `k` in CD method.
    training_count=1,
    # Batch size in mini-batch training.
    batch_size=200,
    # if `r_batch_size` > 0, the function of `dbm.learn` is a kind of reccursive learning.
    r_batch_size=-1 
)

If you do not want to execute the mini-batch training, the value of batch_size must be -1. And r_batch_size is also parameter to control the mini-batch training but is refered only in inference and reconstruction. If this value is more than 0, the inferencing is a kind of reccursive learning with the mini-batch training.

And the feature points can be extracted by this method.

print(dbm.get_feature_point(layer_number=1))

Usecase: Extracting all feature points for dimensions reduction(or pre-training)

Import Python and Cython modules.

# `StackedAutoEncoder` is-a `DeepBoltzmannMachine`.
from pydbm.dbm.deepboltzmannmachine.stacked_auto_encoder import StackedAutoEncoder
# The `Concrete Builder` in Builder Pattern.
from pydbm.dbm.builders.dbm_multi_layer_builder import DBMMultiLayerBuilder
# Contrastive Divergence for function approximation.
from pydbm.approximation.contrastive_divergence import ContrastiveDivergence
# Logistic Function as activation function.
from pydbm.activation.logistic_function import LogisticFunction

Instantiate objects and call the method.

# Setting objects for activation function.
activation_list = [LogisticFunction(), LogisticFunction(), LogisticFunction()]
# Setting the object for function approximation.
approximaion_list = [ContrastiveDivergence(), ContrastiveDivergence()]

dbm = StackedAutoEncoder(
    DBMMultiLayerBuilder(),
    [target_arr.shape[1], 10, target_arr.shape[1]],
    activation_list,
    approximaion_list,
    0.05, # Setting learning rate.
    0.5   # Setting dropout rate.
)

# Execute learning.
dbm.learn(
    target_arr,
    1, # If approximation is the Contrastive Divergence, this parameter is `k` in CD method.
    batch_size=200,  # Batch size in mini-batch training.
    r_batch_size=-1  # if `r_batch_size` > 0, the function of `dbm.learn` is a kind of reccursive learning.
)

Extract reconstruction error rate.

You can check the reconstruction error rate. During the approximation of the Contrastive Divergence, the mean squared error(MSE) between the observed data points and the activities in visible layer is computed as the reconstruction error rate.

Call get_reconstruct_error_arr method as follow.

reconstruct_error_arr = dbm.get_reconstruct_error_arr(layer_number=0)

layer_number corresponds to the index of approximaion_list. And reconstruct_error_arr is the np.ndarray of reconstruction error rates.

Extract the result of dimention reduction

And the result of dimention reduction can be extracted by this property.

pre_trained_arr = dbm.feature_points_arr

Extract weights obtained by pre-training.

If you want to get the pre-training weights, call get_weight_arr_list method.

weight_arr_list = dbm.get_weight_arr_list()

weight_arr_list is the list of weights of each links in DBM. weight_arr_list[0] is 2-d np.ndarray of weights between visible layer and first hidden layer.

Extract biases obtained by pre-training.

Call get_visible_bias_arr_list method and get_hidden_bias_arr_list method in the same way.

visible_bias_arr_list = dbm.get_visible_bias_arr_list()
hidden_bias_arr_list = dbm.get_hidden_bias_arr_list()

visible_bias_arr_list and hidden_bias_arr_list are the list of biases of each links in DBM.

Transfer learning in DBM.

DBMMultiLayerBuilder can be given weight_arr_list, visible_bias_arr_list, and hidden_bias_arr_list obtained by pre-training.

dbm = StackedAutoEncoder(
    DBMMultiLayerBuilder(
        # Setting pre-learned weights matrix.
        weight_arr_list,
        # Setting pre-learned bias in visible layer.
        visible_bias_arr_list,
        # Setting pre-learned bias in hidden layer.
        hidden_bias_arr_list
    ),
    [next_target_arr.shape[1], 10, next_target_arr.shape[1]],
    activation_list,
    approximaion_list,
    # Setting learning rate.
    0.05,
    # Setting dropout rate.
    0.0
)

# Execute learning.
dbm.learn(
    next_target_arr,
    1, # If approximation is the Contrastive Divergence, this parameter is `k` in CD method.
    batch_size=200,  # Batch size in mini-batch training.
    r_batch_size=-1  # if `r_batch_size` > 0, the function of `dbm.learn` is a kind of reccursive learning.
)

Performance

Run a program: demo_stacked_auto_encoder.py

time python demo_stacked_auto_encoder.py

The result is follow.

real    1m35.472s
user    1m32.300s
sys     0m3.136s

Detail

This experiment was performed under the following conditions.

Machine type
  • vCPU: 2
  • memory: 8GB
  • CPU Platform: Intel Ivy Bridge
Observation Data Points

The observated data is the result of np.random.uniform(size=(10000, 10000)).

Number of units
  • Visible layer: 10000
  • hidden layer(feature point): 10
  • hidden layer: 10000
Activation functions
  • visible: Logistic Function
  • hidden(feature point): Logistic Function
  • hidden: Logistic Function
Approximation
  • Contrastive Divergence
Hyper parameters
  • Learning rate: 0.05
  • Dropout rate: 0.5
Feature points
0.190599  0.183594  0.482996  0.911710  0.939766  0.202852  0.042163
0.470003  0.104970  0.602966  0.927917  0.134440  0.600353  0.264248
0.419805  0.158642  0.328253  0.163071  0.017190  0.982587  0.779166
0.656428  0.947666  0.409032  0.959559  0.397501  0.353150  0.614216
0.167008  0.424654  0.204616  0.573720  0.147871  0.722278  0.068951
.....
Reconstruct error
 [ 0.08297197  0.07091231  0.0823424  ...,  0.0721624   0.08404181  0.06981017]

Usecase: Building the RTRBM for recursive learning.

Import Python and Cython modules.

# `Builder` in `Builder Patter`.
from pydbm.dbm.builders.rt_rbm_simple_builder import RTRBMSimpleBuilder
# RNN and Contrastive Divergence for function approximation.
from pydbm.approximation.rt_rbm_cd import RTRBMCD
# Logistic Function as activation function.
from pydbm.activation.logistic_function import LogisticFunction
# Softmax Function as activation function.
from pydbm.activation.softmax_function import SoftmaxFunction

Instantiate objects and execute learning.

# `Builder` in `Builder Pattern` for RTRBM.
rtrbm_builder = RTRBMSimpleBuilder()
# Learning rate.
rtrbm_builder.learning_rate = 0.00001
# Set units in visible layer.
rtrbm_builder.visible_neuron_part(LogisticFunction(), arr.shape[1])
# Set units in hidden layer.
rtrbm_builder.hidden_neuron_part(LogisticFunction(), 3)
# Set units in RNN layer.
rtrbm_builder.rnn_neuron_part(LogisticFunction())
# Set graph and approximation function.
rtrbm_builder.graph_part(RTRBMCD())
# Building.
rbm = rtrbm_builder.get_result()

# Learning.
rbm.learn(
    arr,
    training_count=1, 
    batch_size=200
)

The rbm has a np.ndarray of graph.visible_activity_arr. The graph.visible_activity_arr is the inferenced feature points. This value can be observed as data point.

# Execute recursive learning.
inferenced_arr = rbm.inference(
    test_arr,
    training_count=1, 
    r_batch_size=-1
)

Returned value inferenced_arr is generated by input parameter test_arr.

Usecase: Building the RNN-RBM for recursive learning.

Import Python and Cython modules.

# `Builder` in `Builder Patter`.
from pydbm.dbm.builders.rnn_rbm_simple_builder import RNNRBMSimpleBuilder
# RNN and Contrastive Divergence for function approximation.
from pydbm.approximation.rtrbmcd.rnn_rbm_cd import RNNRBMCD
# Logistic Function as activation function.
from pydbm.activation.logistic_function import LogisticFunction
# Softmax Function as activation function.
from pydbm.activation.softmax_function import SoftmaxFunction

Instantiate objects.

# `Builder` in `Builder Pattern` for RNN-RBM.
rnnrbm_builder = RNNRBMSimpleBuilder()
# Learning rate.
rnnrbm_builder.learning_rate = 0.00001
# Set units in visible layer.
rnnrbm_builder.visible_neuron_part(LogisticFunction(), arr.shape[1])
# Set units in hidden layer.
rnnrbm_builder.hidden_neuron_part(LogisticFunction(), 3)
# Set units in RNN layer.
rnnrbm_builder.rnn_neuron_part(LogisticFunction())
# Set graph and approximation function.
rnnrbm_builder.graph_part(RNNRBMCD())
# Building.
rbm = rnnrbm_builder.get_result()

The function of learning and inferencing is equivalent to rbm of RTRBM.

Usecase: Building the LSTM-RTRBM for pre-learning, and Transfer learning in LSTM.

Import Python and Cython modules.

# `Builder` in `Builder Patter`.
from pydbm.dbm.builders.lstm_rt_rbm_simple_builder import LSTMRTRBMSimpleBuilder
# LSTM and Contrastive Divergence for function approximation.
from pydbm.approximation.rtrbmcd.lstm_rt_rbm_cd import LSTMRTRBMCD
# Logistic Function as activation function.
from pydbm.activation.logistic_function import LogisticFunction
# Tanh Function as activation function.
from pydbm.activation.tanh_function import TanhFunction

Instantiate objects.

# `Builder` in `Builder Pattern` for LSTM-RTRBM.
rnnrbm_builder = LSTMRTRBMSimpleBuilder()
# Learning rate.
rnnrbm_builder.learning_rate = 1e-05
# Set units in visible layer.
rnnrbm_builder.visible_neuron_part(LogisticFunction(), 50)
# Set units in hidden layer.
rnnrbm_builder.hidden_neuron_part(LogisticFunction(), 100)
# Set units in RNN layer.
rnnrbm_builder.rnn_neuron_part(TanhFunction())
# Set graph and approximation function.
rnnrbm_builder.graph_part(LSTMRTRBMCD())
# Building.
rbm = rnnrbm_builder.get_result()

Pre-learning by LSTM-RTRBM.

The function of learning and inferencing is equivalent to rbm of RTRBM. Like RTRBM and RNN-RBM, call the method learning to learn the observed data points.

# Learning.
rbm.learn(
    observed_arr,
    training_count=100, 
    batch_size=20
)

Extract pre-Learned parameters.

rbm, generated by LSTMRTRBMSimpleBuilder, has a method extract_transfered_params to extract pre-learned parameters.

graph = rbm.extract_transfered_params(
    input_neuron_count=50,
    hidden_neuron_count=100,
    output_neuron_count=25
)

Building LSTM units.

The object graph has the pre-learned parameters. To do tranfer learning in LSTM model, import Python and Cython modules for building LSTM units.

# Cost function for logging and verification.
from pydbm.rnn.loss.mean_squared_error import MeanSquaredError
# Verification.
from pydbm.rnn.verification.verificate_function_approximation import VerificateFunctionApproximation
# LSTM model.
from pydbm.rnn.lstm_model import LSTMModel

Transfer learning based on pre-learned parameters in LSTM.

Under the same conditions as LSTM-RTRBM's pre-learning, instantiate objects and call the methods for transfer learning.

lstm_model = LSTMModel(
    # By delegating `graph` to this method, it is possible to do transfer learning with pre-learned parameters.
    graph=graph,
    # The number of epochs in mini-batch training.
    epochs=1000,
    # The batch size.
    batch_size=20,
    # Learning rate.
    learning_rate=1e-05,
    # Attenuate the `learning_rate` by a factor of this value every `attenuate_epoch`.
    learning_attenuate_rate=0.1,
    # Attenuate the `learning_rate` by a factor of `learning_attenuate_rate` every `attenuate_epoch`.
    attenuate_epoch=50,
    # Regularization for weights matrix in hidden layer(RNN layers)
    # to repeat multiplying the weights matrix and `0.9` until
    # $\sum_{j=0}^{n}w_{ji}^2 < weight\_limit$.
    weight_limit=0.05,
    # The probability of dropout.
    dropout_rate=0.5,
    # Refereed maxinum step `t` in Backpropagation Through Time(BPTT).
    # If `0`, this class referes all past data in BPTT.
    bptt_tau=16,
    # Size of Test data set. If this value is `0`, the validation will not be executed.
    test_size_rate=0.3,
    # Verification function.
    verificatable_result=VerificateFunctionApproximation(MeanSquaredError())
)
# Learning.
lstm_model.learn(observed_arr, target_arr)
# Logging the loss.
lstm_model.verificatable_result.logs_df.to_csv("pre_learn.csv", index=False)

Standard learning in LSTM.

If you want to execute learning not based on pre-learned parameters in order to compare initialization strategies, initialize graph and delegate it to LSTMModel as follows.

# LSTM Graph which is-a `Synapse`.
from pydbm.synapse.recurrenttemporalgraph.lstm_graph import LSTMGraph

# Init.
graph = LSTMGraph()

# Activation function in LSTM.
graph.observed_activating_function = TanhFunction()
graph.input_gate_activating_function = LogisticFunction()
graph.forget_gate_activating_function = LogisticFunction()
graph.output_gate_activating_function = LogisticFunction()
graph.hidden_activating_function = TanhFunction()
graph.output_activating_function = LogisticFunction()

# Initialization strategy.
# This method initialize each weight matrices and biases in Gaussian distribution: `np.random.normal(size=hoge) * 0.01`.
graph.create_rnn_cells(
    input_neuron_count=50,
    hidden_neuron_count=100,
    output_neuron_count=25
)

# Delegate `graph` to `LSTMModel`.
lstm_model = LSTMModel(
    graph=graph,
    hoge=fuga
)

Toy problem: Sine Wave Prediction.

It can be easy to implement a comparison of initialization strategies by introducing simple toy problem, namely Sine Wave Prediction. In this demonstration, the training dataset is generated as follows.

# The number of sampled data.
sample_n = 1000
# The length of one cycle.
cycle_len = 100
# Amplitude of Sine Wave.
amp = 0.05
# The rate of the length of predicted data.
target_size = 0.5

# The list of observed data points.
observed_arr_list = []
# The list of objective data.
target_arr_list = []

for _ in range(sample_n):
    sin_arr = np.sin(2.0 * np.pi * np.arange(0, cycle_len))
    noise_arr = amp * np.random.normal(loc=0.0, scale=0.5, size=sin_arr.shape[0])
    sin_arr += noise_arr

    observed_arr_list.append(sin_arr[:int(sin_arr.shape[0] * (1 - target_size))])
    target_arr_list.append(sin_arr[int(sin_arr.shape[0] * (1 - target_size)):])

# Reshape for `LSTMModel`.
observed_arr = np.array(observed_arr_list)
observed_arr = observed_arr.reshape(observed_arr.shape[0], observed_arr.shape[1], 1)
target_arr = np.array(target_arr_list)

Then execute pre-learning the dataset by LSTM-RTRBM and learning same dataset by LSTM under the same conditions, where hidden_neuron_count is 100, LSTM-RTRBM's training_count is 100, LSTM's epochs is 1000, batch_size is 20, learning_rate is 1e-05, learning_attenuate_rate is 0.1, attenuate_epoch is 50, and test_size_rate is 0.3, and logging the loss function: Mean Square Error(MSE). The result of with-and-without comparison were as follows.

Without pre-learning (Standard learning in LSTM).

With pre-learning (LSTM-RTRBM's pre-learning and LSTM's learning).

Difference demonstrates the function.

The difference of MSE observed from the simple toy problem demonstrates that the function of LSTM-RTRBM's pre-learning is to improve performance of LSTM's learning. On the context of paradigm of Deep learning and Transfer learning, the initialization strategies, based on not only a mere Gaussian noise but also unsupervised pre-learning of each layer, have been shown to be important both for supervised and unsupervised learning of Machine learning. In particular, LSTM-RTRBM makes it possible to do pre-learning and Transfer learning for probabilistic time series patterns.

Usecase: Image segmentation by Shape-BM.

First, acquire image data and binarize it.

from PIL import Image
img = Image.open("horse099.jpg")
img

If you think the size of your image datasets may be large, resize it to an arbitrary size.

img = img.resize((90, 90))

Convert RGB images to binary images.

img_bin = img.convert("1")
img_bin

Set up hyperparameters.

filter_size = 5
overlap_n = 4

learning_rate = 0.01

filter_size is the 'filter' size. This value must be more than 4. And overlap_n is hyperparameter specific to Shape-BM. In the visible layer, this model has so-called local receptive fields by connecting each first hidden unit only to a subset of the visible units, corresponding to one of four square patches. Each patch overlaps its neighbor by overlap_n pixels (Eslami, S. A., et al, 2014).

Please note that the recommended ratio of filter_size and overlap_n is 5:4. It is not a constraint demanded by pure theory of Shape Boltzmann Machine itself but is a kind of limitation to simplify design and implementation in this library.

And import Python and Cython modules.

# The `Client` in Builder Pattern
from pydbm.dbm.deepboltzmannmachine.shape_boltzmann_machine import ShapeBoltzmannMachine
# The `Concrete Builder` in Builder Pattern.
from pydbm.dbm.builders.dbm_multi_layer_builder import DBMMultiLayerBuilder

Instantiate objects and call the method.

dbm = ShapeBoltzmannMachine(
    DBMMultiLayerBuilder(),
    learning_rate=learning_rate,
    overlap_n=overlap_n,
    filter_size=filter_size
)

img_arr = np.asarray(img_bin)
img_arr = img_arr.astype(np.float64)

# Execute learning.
dbm.learn(
    # `np.ndarray` of image data.
    img_arr,
    # If approximation is the Contrastive Divergence, this parameter is `k` in CD method.
    training_count=1,
    # Batch size in mini-batch training.
    batch_size=300,
    # if `r_batch_size` > 0, the function of `dbm.learn` is a kind of reccursive learning.
    r_batch_size=-1,
    # Learning with the stochastic gradient descent(SGD) or not.
    sgd_flag=True
)

Extract dbm.visible_points_arr as the observed data points in visible layer. This np.ndarray is segmented image data.

inferenced_data_arr = dbm.visible_points_arr.copy()
inferenced_data_arr = 255 - inferenced_data_arr
Image.fromarray(np.uint8(inferenced_data_arr))

References

  • Ackley, D. H., Hinton, G. E., & Sejnowski, T. J. (1985). A learning algorithm for Boltzmann machines. Cognitive science, 9(1), 147-169.
  • Boulanger-Lewandowski, N., Bengio, Y., & Vincent, P. (2012). Modeling temporal dependencies in high-dimensional sequences: Application to polyphonic music generation and transcription. arXiv preprint arXiv:1206.6392.
  • Eslami, S. A., Heess, N., Williams, C. K., & Winn, J. (2014). The shape boltzmann machine: a strong model of object shape. International Journal of Computer Vision, 107(2), 155-176.
  • Hinton, G. E. (2002). Training products of experts by minimizing contrastive divergence. Neural computation, 14(8), 1771-1800.
  • Le Roux, N., & Bengio, Y. (2008). Representational power of restricted Boltzmann machines and deep belief networks. Neural computation, 20(6), 1631-1649.
  • Lyu, Q., Wu, Z., Zhu, J., & Meng, H. (2015, June). Modelling High-Dimensional Sequences with LSTM-RTRBM: Application to Polyphonic Music Generation. In IJCAI (pp. 4138-4139).
  • Lyu, Q., Wu, Z., & Zhu, J. (2015, October). Polyphonic music modelling with LSTM-RTRBM. In Proceedings of the 23rd ACM international conference on Multimedia (pp. 991-994). ACM.
  • Salakhutdinov, R., & Hinton, G. E. (2009). Deep boltzmann machines. InInternational conference on artificial intelligence and statistics (pp. 448-455).
  • Srivastava, N., Hinton, G., Krizhevsky, A., Sutskever, I., & Salakhutdinov, R. (2014). Dropout: a simple way to prevent neural networks from overfitting. The Journal of Machine Learning Research, 15(1), 1929-1958.
  • Sutskever, I., Hinton, G. E., & Taylor, G. W. (2009). The recurrent temporal restricted boltzmann machine. In Advances in Neural Information Processing Systems (pp. 1601-1608).
  • Zaremba, W., Sutskever, I., & Vinyals, O. (2014). Recurrent neural network regularization. arXiv preprint arXiv:1409.2329.

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