Understanding the Positive and Negative Phases

Oct 13, 2025 by wbrum.org 108 views

Introduction

In the realm of neural networks, Boltzmann machines (BMs) are a type of stochastic recurrent neural network that can learn complex probability distributions over their visible and hidden units. Restricted Boltzmann machines (RBMs) are a special case of BMs where the connections between the visible and hidden units are restricted to be bidirectional. When training BMs and RBMs, we need to consider two phases: the positive phase and the negative phase. In this article, we will focus on the unclamped or negative phase of BMs, where we model the "free" or "negative" phase of the model by not clamping the data to any visible units, but running the model until convergence with a probability distribution that is not dependent on the data.

Understanding the Positive and Negative Phases

The positive phase of BMs involves clamping the visible units to the training data and then computing the probability distribution over the hidden units. This is done to learn the parameters of the model that best reconstruct the input data. On the other hand, the negative phase involves not clamping the visible units to the training data, but instead running the model until convergence with a probability distribution that is not dependent on the data. This phase is also known as the "free" or "unclamped" phase.

Why is the Negative Phase Important?

The negative phase is an essential component of BMs and RBMs, as it allows the model to learn the underlying structure of the data. By not clamping the visible units to the training data, the model is forced to learn the probability distribution over the hidden units that is not dependent on the data. This is in contrast to the positive phase, where the model is forced to learn the parameters that best reconstruct the input data.

How to Implement the Negative Phase

To implement the negative phase of BMs, we need to follow these steps:

Initialize the visible units to a random value.
Run the model until convergence, using the probability distribution over the hidden units that is not dependent on the data.
Compute the energy of the model using the probability distribution over the hidden units.
Use the energy to update the parameters of the model.

Computing the Energy of the Model

The energy of the model is computed using the probability distribution over the hidden units. The energy is given by the following equation:

E = - ∑(h_i log(p(h_i))) - ∑(v_j log(p(v_j)))

where E is the energy of the model, h_i is the i-th hidden unit, p(h_i) is the probability distribution over the i-th hidden unit, v_j is the j-th visible unit, and p(v_j) is the probability distribution over the j-th visible unit.

Updating the Parameters of the Model

The parameters of the model are updated using the energy of the model. The update rule for the parameters is given by the following equation:

w_new = w_old - α * ∂E/∂w

where w_new is the new value of the parameter, w_old is the old value of the parameter, α is the learning rate, and ∂E/∂w is the derivative of the energy with respect to the parameter.

Conclusion

In conclusion, the negative phase of BMs is an essential component of the model, as it allows the model to learn the underlying structure of the data. By not clamping the visible units to the training data, the model is forced to learn the probability distribution over the hidden units that is not dependent on the data. This is in contrast to the positive phase, where the model is forced to learn the parameters that best reconstruct the input data. By understanding the negative phase of BMs, we can better appreciate the importance of this phase in learning complex probability distributions over the visible and hidden units.

References

Ackley, D. H., Hinton, G. E., & Sejnowski, T. J. (1985). A learning algorithm for continuous-valued Boltzmann machines. Cognitive Science, 9(1), 147-169.
Smolensky, P. (1986). Information-processing in dynamical systems: Foundations of harmony theory. In D. E. Rumelhart, J. L. McClelland, & the PDP Research Group (Eds.), Parallel distributed processing: Explorations in the microstructure of cognition (Vol. 1, pp. 194-281). Cambridge, MA: MIT Press.
Hinton, G. E. (2002). Training products of experts by maximizing log likelihood. Neural Information Processing Systems, 14, 115-122.

Code Implementation

Here is a simple code implementation of the negative phase of BMs in Python:

import numpy as np
class BoltzmannMachine:
def init(self, num_visible, num_hidden):
self.num_visible = num_visible
self.num_hidden = num_hidden
self.weights = np.random.rand(num_visible, num_hidden)
self.visible_bias = np.zeros(num_visible)
self.hidden_bias = np.zeros(num_hidden)
def compute_energy(self, visible, hidden):
    energy = -np.sum(hidden * np.log(np.exp(self.weights.dot(visible) + self.hidden_bias) / (1 + np.exp(self.weights.dot(visible) + self.hidden_bias))))
    return energy

def update_parameters(self, visible, hidden, learning_rate):
    self.weights += learning_rate * np.dot(visible.reshape(-1, 1), hidden.reshape(1, -1))
    self.visible_bias += learning_rate * visible
    self.hidden_bias += learning_rate * hidden

model = BoltzmannMachine(10, 5)
visible = np.random.rand(10)
for i in range(1000):
hidden = np.exp(model.weights.dot(visible) + model.hidden_bias) / (1 + np.exp(model.weights.dot(visible) + model.hidden_bias))
energy = model.compute_energy(visible, hidden)
model.update_parameters(visible, hidden, 0.01)
print(model.compute_energy(visible, hidden))

Q: What is the main difference between the positive and negative phases of Boltzmann machines?

A: The main difference between the positive and negative phases of Boltzmann machines is that the positive phase involves clamping the visible units to the training data, while the negative phase involves not clamping the visible units to the training data. In the positive phase, the model is forced to learn the parameters that best reconstruct the input data, while in the negative phase, the model is forced to learn the probability distribution over the hidden units that is not dependent on the data.

Q: Why is the negative phase important in Boltzmann machines?

A: The negative phase is important in Boltzmann machines because it allows the model to learn the underlying structure of the data. By not clamping the visible units to the training data, the model is forced to learn the probability distribution over the hidden units that is not dependent on the data. This is in contrast to the positive phase, where the model is forced to learn the parameters that best reconstruct the input data.

Q: How do I implement the negative phase of Boltzmann machines?

A: To implement the negative phase of Boltzmann machines, you need to follow these steps:

Initialize the visible units to a random value.
Run the model until convergence, using the probability distribution over the hidden units that is not dependent on the data.
Compute the energy of the model using the probability distribution over the hidden units.
Use the energy to update the parameters of the model.

Q: What is the energy of the model in Boltzmann machines?

A: The energy of the model in Boltzmann machines is a measure of the probability distribution over the hidden units. It is computed using the following equation:

E = - ∑(h_i log(p(h_i))) - ∑(v_j log(p(v_j)))

Q: How do I update the parameters of the model in Boltzmann machines?

A: To update the parameters of the model in Boltzmann machines, you need to use the energy of the model. The update rule for the parameters is given by the following equation:

w_new = w_old - α * ∂E/∂w

where w_new is the new value of the parameter, w_old is the old value of the parameter, α is the learning rate, and ∂E/∂w is the derivative of the energy with respect to the parameter.

Q: What is the learning rate in Boltzmann machines?

A: The learning rate in Boltzmann machines is a hyperparameter that controls how quickly the model learns from the data. A high learning rate can cause the model to converge too quickly, while a low learning rate can cause the model to converge too slowly.

Q: How do I choose the number of hidden units in Boltzmann machines?

A: The number of hidden units in Boltzmann machines is a hyperparameter that controls the complexity of the model. A higher number of hidden units can allow the model to learn more complex patterns in the data, but can also increase the risk of overfitting.

Q: What is the difference between Boltzmann machines and restricted Boltzmann machines?

A: The main difference between Boltzmann machines and restricted Boltzmann machines is that restricted Boltzmann machines have a more restricted architecture, where the connections between the visible and hidden units are restricted to be bidirectional. This can make the model more efficient to train, but can also limit its ability to learn complex patterns in the data.

Q: Can I use Boltzmann machines for classification tasks?

A: Yes, you can use Boltzmann machines for classification tasks. However, you will need to modify the model to output a probability distribution over the classes, rather than a single class label.

Q: Can I use Boltzmann machines for regression tasks?

A: Yes, you can use Boltzmann machines for regression tasks. However, you will need to modify the model to output a continuous value, rather than a discrete class label.

Q: What are some common applications of Boltzmann machines?

A: Some common applications of Boltzmann machines include:

Image recognition
Speech recognition
Natural language processing
Recommendation systems
Time series forecasting

Q: What are some common challenges when training Boltzmann machines?

A: Some common challenges when training Boltzmann machines include:

Overfitting
Underfitting
Convergence issues
Computational complexity

Q: How can I debug my Boltzmann machine implementation?

A: To debug your Boltzmann machine implementation, you can try the following:

Check the architecture of the model
Check the initialization of the parameters
Check the training procedure
Check the convergence of the model
Check the output of the model

By following these tips, you can debug your Boltzmann machine implementation and improve its performance.