# Empirical Remarks on the Translational Equivariance of Convolutional Layers
###### Abstract  
In general, convolutional neural networks (CNNs) maintain some level of translational invariance. However, the convolutional layer itself is translational-equivariant. The pooling layers provide some level of invariance. In object recognition, invariance is more important than equivariance. In this paper, we investigate how vulnerable CNNs without pooling or augmentation are to translation in object recognition. For CNNs that are specialized in learning local textures but vulnerable to learning global geometric information, we propose a method to explicitly transform an image into a global feature image and then provide it as an input to neural networks. In our experiments on a modified MNIST dataset, we demonstrate that the recognition accuracy of a conventional baseline network significantly decreases from 98% to less than 60% even in the case of 2-pixel translation. We also demonstrate that the proposed method is far superior to the baseline network in terms of performance improvement.  
Keywords:translational equivariant; translational invariant; complementary information; object recognition; convolutional neural networks +
Footnote †: journal:  
proposed the use of the style transfer network [16] for texture augmentation while preserving the main geometric shape. The success of this method indicates that CNNs are biased towards learning from local textures. Learning local texture does not require invariance of networks. Instead, equivariance is sufficient.  
Geometric information as well as texture is important for object recognition. Therefore, the feature representation schemes have to obtain geometrical invariance. Several papers have applied the polar coordinate system to mitigate variations coming from geometric transforms [17; 18; 19]. This transform converts the rotational variant problem into the translational variant problem so that it can be easily learned by convolutional layers. [PERSON] et al. [20] proposed the localization and grid generator layers to cope with geometric transformations, but it requires mandatory augmentation to train the proposed layers.  
In this paper, we investigate how vulnerable CNNs without pooling or augmentation are to translation in object recognition. In addition, motivated by the use of additional information as input to CNNs [7; 8; 9; 10; 11], we propose a method to explicitly transform an image into a global feature image and then provide it as input to the neural network since CNNs are specialized in learning local textures, but encounter difficulties when learning global geometric information. For this purpose, we firstly adopt the 2D-discrete Fourier transform (2D-DFT) [21] which transforms spatial images into the global invariant magnitude response images. We can use the fast Fourier transform (FFT) [22] algorithm to estimate 2D-DFT. For additional invariance, we can consider a mixture model [23] based on a statistical transform such as independent component analysis (ICA) [24]. Our experimental results clearly show that the convolution layers are extremely vulnerable to a few pixels of translation, and the proposed method is far superior to the baseline network in terms of the performance improvement.  
This paper is organized as follows. Section 2 briefly reviews equivariance and invariance. In Section 3, we describe the properties of frequency images and three network structures for using frequency images as additional input. Section 4 presents an experimental setup to demonstrate the weaknesses of equivariant convolutional networks for translation and shows that the utilization of frequency images improves translational invariance. We give conclusion remarks in Section 5.