1 Introduction
Deep learning is a subset of machine learning algorithms that construct the Deep Neural Networks (DNNs) to solve complex problems
[1]. Although it has achieved great success in various fields, such as speech recognition [2] and image classification [3], the internal logic of deep learning is still not convincingly explained and DNNs have been regarded as "black boxes" [4].Based on an underlying premise that DNNs establish a complex probabilistic model [5, 6, 7, 8], numerous theories, such as the representation learning [9, 10, 11], the Information Bottleneck (IB) theory [12, 13, 14, 15], have been proposed to explore the working mechanism of deep learning. Though the proposed theories reveal some important properties of deep learning, such as hierarchy [9, 10] and sufficiency [12, 15], a fundamental problem is that the proposed theories cannot be directly validated by empirical experiments due to the fact that the distributions of the benchmark datasets, e.g., MNIST, are unknown. For example, hierarchy is an important property of DNNs, but we still cannot explicitly formulate the hierarchy property and directly validate it by empirical experiments.
To solve this problem, we propose a novel algorithm for generating a synthetic dataset obeying a Gaussian distribution based on the NIST ^{1}^{1}1https://www.nist.gov/srd/nistspecialdatabase19 dataset of handwritten digits by class. In particular, the synthetic dataset has the same characteristics as the benchmark dataset MNIST [16]. Specifically, the synthetic dataset consists of 70,000 grayscale images in 10 classes (digits from 0 to 9). Each class has 6,000 training images and 1,000 testing images. Fig. 1 shows three synthetic images. Therefore, we can easily apply various DNNs on the synthetic dataset like MNIST. Since all the grayscale images are sampled from a known distribution, the synthetic dataset obeys the Gaussian distribution.
This paper is organized as follows. Section 2 describes the specific method for generating the synthetic dataset obeying a known Gaussian distribution and Section 3 shows that the synthetic dataset can be easily applied to most commonly used DNNs. Section 4 demonstrates that given the synthetic dataset, we can verify some important properties of deep learning, e.g., hierarchy, based on the recent proposed probabilistic explanation of hidden layers of DNNs [17, 18].
2 The method for generating the synthetic dataset
An underlying assumption of deep learning is that the given training dataset
is composed of i.i.d. samples from a joint distribution
, where describes the prior knowledge of , describes the connection between and , and indicate the parameters of . Since we can easily formulate given , is the key of explicitly formulating .Unlike previous works using a complex probabilistic model to formulate [19, 20] for a given dataset, we first generate a random dataset obeying a Gaussian distribution and then use the generated random dataset to construct a synthetic image based on the mask derived from a benchmark dataset. Since each data in the random dataset obeys , we can conclude that the synthetic image also obeys based on the spatial stationary property, i.e., .
More specifically, the method includes seven steps: (i) generating a random vector
by sampling the Gaussian distribution for constructing a synthetic image with dimension ; (ii) converting an image of the NIST dataset into a binary image; (iii) extracting the central part of the binary image and the dimension of the derived image is ; (iv) downsampling the derived image in the previous step to obtain a binary image with dimension ; (v) generating the mask of the binary digits image based on the Canny edge detection algorithm [21], and the mask indicates four parts of the binary image: outside, outside boundary, inside boundary and inside; (vi) deriving an ordered vector by sorting in the descending order and decomposing into four parts, i.e., , where corresponds to the outside, the inside boundary, the outside boundary, and the inside. (vii) generating a synthetic image by randomly placing each pixel in the four subvectors into a random position within the corresponding masks.The method for generating synthetic image is summarized in Algorithm 1, and Fig. 3 visualizes the relationship between and their corresponding masks.
3 Experiments
In this section, we demonstrate that the synthetic dataset can be easily used to DNNs. First, we design a simple but comprehensive Convolutional Neural Network (abbr. CNN1) for classifying the synthetic dataset. CNN1 has five hidden layers: two convolutional layers, two ReLU operator, and two max pooling layers. Table
1 summarizes the architecture of CNN1.We take 30 training epochs to train CNN1 for classifying the synthetic dataset, and the learning rate is 0.008. Fig.
2 shows the performance of CNN1 on the synthetic dataset. We can see that CNN1 achieves zero training error after 20 training epochs, the testing error is also very small. Overall, we can conclude that the synthetic dataset can be applied to DNNsR.V.  Layer  Description  CNN1 
Input  
Conv ()  
Maxpool + ReLU  
Conv ()  
Maxpool + ReLU  
Fully connected  
Output(softmax) 

R.V. is the random variable of the hidden layer(s).
4 Conclusion
In this work, we propose a novel method for generating a synthetic dataset. In contrast to the commonly used benchmark datasets with unknown distribution, the synthetic dataset has a explicit distribution, i.e., Gaussian distribution. In particular, it has the same characteristics of the benchmark dataset MNIST. As a result, we can easily apply Deep Neural Networks (DNNs) on the synthetic dataset.
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