Research on Weight Initialization Method for Restricted Boltzmann Machines Based on Statistical Mechanical Analysis

Academic Background

In deep learning, the initialization of neural network weights significantly impacts the training effectiveness of models. Particularly in feed-forward neural networks, several dataset-free weight initialization methods have been proposed, such as LeCun, Xavier (or Glorot), and He initializations. These methods randomly determine the initial values of weight parameters based on specific distributions (e.g., Gaussian or uniform distributions) without using training datasets. However, similar weight initialization methods have not yet been developed for Restricted Boltzmann Machines (RBMs). RBMs are probabilistic neural networks consisting of two layers and are widely used in collaborative filtering, dimensionality reduction, classification, anomaly detection, and deep learning. Since weight initialization in RBMs significantly affects learning efficiency, developing a dataset-free weight initialization method for RBMs is of great importance.

In this paper, the authors propose a weight initialization method for Bernoulli-Bernoulli RBMs based on statistical mechanical analysis. This method determines the standard deviation of the Gaussian distribution for weight initialization by maximizing the layer correlation (LC) between the two layers, thereby improving learning efficiency.

Source of the Paper

This paper is co-authored by Muneki Yasuda, Ryosuke Maeno, and Chako Takahashi. Muneki Yasuda is from the Graduate School of Science and Engineering at Yamagata University, Japan, Ryosuke Maeno is from Techno Provide Inc., and Chako Takahashi is also from Yamagata University. The paper was published in 2025 in the journal Neural Networks, Volume 187, Article Number 107297.

Research Process

1. Research Objectives and Hypotheses

The goal of this study is to propose a dataset-free weight initialization method for Bernoulli-Bernoulli RBMs. The authors hypothesize that maximizing the layer correlation (LC) between the visible and hidden layers in RBMs can improve the learning efficiency of the model. Specifically, the weight parameters are randomly initialized from a Gaussian distribution with zero mean, and the standard deviation σ is determined by maximizing the LC.

2. Statistical Mechanical Analysis

Based on mean-field analysis and the replica method in statistical mechanics, the authors derive the expression for layer correlation. Through analysis, they find that the standard deviation σ corresponding to the maximum LC is related to the network structure (e.g., the size ratio α of the layers) and the type of hidden layer ({0,1} or {-1,1}). Specifically, when the sizes of the visible and hidden layers are the same, the hidden layer consists of {-1,1} binary variables, and all bias parameters are zero, the proposed weight initialization method is identical to the Xavier initialization method.

3. Numerical Experiments

To validate the effectiveness of the proposed weight initialization method, the authors conducted numerical experiments using a toy dataset and real-world datasets (including the Dry Bean dataset, Urban Land Cover dataset, and MNIST dataset). The main objective of the experiments is to evaluate the impact of different initialization methods on the learning efficiency of RBMs, i.e., the growth rate of the training log-likelihood.

3.1 Toy Dataset Experiment

The authors first conducted experiments on an artificially generated toy dataset. The dataset was generated from four base patterns, with 100 data points generated from each pattern, resulting in a total of 400 data points. In the experiments, the size of the visible layer in the RBM was 20, and the sizes of the hidden layers were 10, 20, and 30 (i.e., α=0.5, 1, 1.5). The authors compared the learning effects under different standard deviations σ (including σ=β_max/4, β_max/2, β_max, 2β_max, and 4β_max). The experimental results show that the initialization method using σ=β_max achieves the best learning performance after 200 training epochs.

3.2 Real-World Dataset Experiments

The authors further conducted experiments on three real-world datasets: the Dry Bean dataset, Urban Land Cover dataset, and MNIST dataset. In the Dry Bean dataset experiment, the authors used 10,000 data points, each containing 16 features. The size of the visible layer in the RBM was 16, and the sizes of the hidden layers were 16 and 32 (i.e., α=1, 2). The experimental results show that the initialization method using σ=β_max achieves the best or second-best learning performance after 200 training epochs.

In the Urban Land Cover dataset experiment, the authors used 500 data points, each containing 147 features. The size of the visible layer in the RBM was 147, and the size of the hidden layer was 200 (i.e., α≈1.36). The experimental results show that the initialization method using σ=β_max achieves the best or second-best learning performance after 100 training epochs.

In the MNIST dataset experiment, the authors used 3,000 data points, each containing 784 features. The size of the visible layer in the RBM was 784, and the size of the hidden layer was 500 (i.e., α≈0.64). The experimental results show that the initialization method using σ=β_max achieves the best or second-best learning performance after 100 training epochs.

Research Results and Conclusions

1. Main Results

Through statistical mechanical analysis and numerical experiments, the authors obtained the following main results: - The proposed weight initialization method determines the standard deviation σ of the Gaussian distribution by maximizing the layer correlation (LC), thereby improving the learning efficiency of RBMs. - Under specific conditions (i.e., when the sizes of the visible and hidden layers are the same, the hidden layer consists of {-1,1} binary variables, and all bias parameters are zero), the proposed initialization method is identical to the Xavier initialization method. - Numerical experiments show that the initialization method using σ=β_max achieves the best learning performance on both the toy dataset and real-world datasets.

2. Research Significance

The significance of this study lies in proposing a dataset-free weight initialization method for Bernoulli-Bernoulli RBMs. This method, based on statistical mechanical analysis, determines the standard deviation for weight initialization by maximizing layer correlation, thereby improving the learning efficiency of the model. This method not only has theoretical value but also has broad application prospects, particularly in deep learning, dimensionality reduction, and anomaly detection.

Research Highlights

• Innovation: This study is the first to propose a dataset-free weight initialization method for RBMs, filling a research gap in this field.
• Theoretical Support: Through mean-field analysis and the replica method in statistical mechanics, the authors derived the expression for layer correlation, providing a theoretical basis for weight initialization.
• Experimental Validation: Numerical experiments on the toy dataset and multiple real-world datasets validate the effectiveness of the proposed method, demonstrating its advantages in improving the learning efficiency of RBMs.

Future Research Directions

The authors propose four future research directions: 1. Extending the weight initialization method to Gaussian-Bernoulli RBMs. 2. Developing an initialization method that utilizes information from a given dataset. 3. Deriving an explicit expression for β_max as a function of α, c, and the type of hidden layer. 4. Further exploring the relationship between the proposed method and the Xavier initialization method to validate the hypothesis’s reasonability.