## 2 Proposed Network
### Lightweight Model  
To build a compact network and improve the detection speed for practical applications, we present a Lightweight architecture. We embed factorization of standard convolutions inspired by MobileNetv2 [32]. In our lightweight module, a convolutional operation comprises a combination of pointwise and depthwise separable convolutional layer. Pointwise is a standard \(1\times 1\) convolution operation that performs linear transformation of the input and changes the channel dimensionality. Depthwise convolution applies a single filter per each channel to filter the features. Our network uses Batchnorm and ReLU activation operation after the depthwise convolutional layer. The same pattern of layers is followed for skip connection layers. Spatial dimension is handled by the max-pooling operation. This design is used with the LogoNet and Dual-Attention LogoNet architecture. We convert each standard residual convolution block of our architecture into a depthwise convolution block that follows the approach of MobileNetv2 block. We employ the approach only for hourglass module layers, feature transformations of other layers including the attention modules is performed using a standard convolution operation. This approach reduces network complexity and computation compared to standard convolution. Depthwise computation can be expressed as:  
\[\hat{O}_{l,m,c}=\sum_{i,j}\hat{K}_{i,j,c}\cdot F_{l+i-1,m+j-1,c} \tag{8}\]  
where \(F\) and \(\hat{O}\) are input and output feature maps with \(C\) number of channels. \(\hat{K}\) is a depthwise convolution kernel of size \(D_{K}\times D_{K}\times D_{C}\) where \(D_{K}\) is the size of kernel, which is 3 in our case. For a feature map of \(D_{F}\) height and width, the total computation cost of depthwise and pointwise convolution operation can be computed as:  
\[C_{in}\cdot C_{out}\cdot D_{F}\cdot D_{F}+D_{K}\cdot D_{K}\cdot C_{out}\cdot D _{F}\cdot D_{F} \tag{9}\]  
exploits where \(C_{in}\) and \(C_{out}\) are the input and output channels.  
To compare the proposed architecture, we also demonstrate lightweight models, exploring the CP-Decomposition (CPD) [33]. The CPD method is the typical method for reducing complexity, which factorizes a tensor into a sum of outer products of vectors. For a given tensor of 3-dimensional space, the CP decomposition can be explained as:  
\[T\approx\sum_{r=1}^{R}l_{r}\circ m_{r}\circ n_{r} \tag{10}\]where \(R>0\), and \(l_{r},m_{r},n_{r}\) are vectors of relevant dimension, and '\(\circ\)' denotes the outer product of two tensors, i.e.,  
\[t_{i,j,k}\approx\sum_{r=1}^{R}l_{ri}\circ m_{rj}\circ n_{rk} \tag{11}\]  
In case of rank one assumption of CPD (i.e., \(R=1\)), the 4D kernel \(\hat{C}\in R^{X\times Y\times Z\times S}\) will be separated into cross-products of four 1D filters as follows:  
\[\hat{C}=\alpha\times\beta\times\gamma\times\eta \tag{12}\]  
where \(\alpha,\beta,\gamma\) are 1D convolution vectors convolving across the dimensions and the fourth corresponds to channels.  
Here, we converted a standard convolution to two 1D convolutions within each residual block of proposed feature extractor. We use 1D convolution from two axes \((X\times 1,\)\(1\times Y)\) to convolve the feature maps. First, we convolve the features using single filter each channel (depthwise) by a kernel size of \((3\times 1)\). Then a kernel of size \((1\times 3)\) is applied to map the number of feature channels. Same approach is applied with skip connection layer to transform the feature maps. Block structures of feature extractor with depthwise convolution and CPD methods are shown in Figure 6.