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Deep Residual Learning for Image Recognition

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Mathematics of Identity Shortcuts and Dimension Matching

Residual Block with Identity Shortcut

Residual Block with Identity Shortcut

Source equation

y=F(x,{Wi})+x\mathbf{y}=\mathcal{F}(\mathbf{x},\{W_{i}\})+\mathbf{x}

This equation defines the fundamental formulation of a residual learning block. Instead of directly fitting an underlying mapping with a stack of layers, the network is designed to approximate a residual mapping F(x,{Wi})=y−x\mathcal{F}(\mathbf{x}, \{W_i\}) = \mathbf{y} - \mathbf{x}. The original mapping is recast into F(x)+x\mathcal{F}(\mathbf{x}) + \mathbf{x}, which is realized by a feedforward neural network with shortcut connections.

Sources

S3.E1

𝐲=ℱ​(𝐱,{Wi})+𝐱.𝐲ℱ𝐱subscript𝑊𝑖𝐱\mathbf{y}=\mathcal{F}(\mathbf{x},\{W_{i}\})+\mathbf{x}. (1)
\mathbf{y}=\mathcal{F}(\mathbf{x},\{W_{i}\})+\mathbf{x}

y=F(x,{Wi})+x\mathbf{y}=\mathcal{F}(\mathbf{x},\{W_{i}\})+\mathbf{x}

Sources

S3.E1

𝐲=ℱ​(𝐱,{Wi})+𝐱.𝐲ℱ𝐱subscript𝑊𝑖𝐱\mathbf{y}=\mathcal{F}(\mathbf{x},\{W_{i}\})+\mathbf{x}. (1)
\mathbf{y}=\mathcal{F}(\mathbf{x},\{W_{i}\})+\mathbf{x}

The formulation adds the input vector x\mathbf{x} directly to the output of the residual function F\mathcal{F}. This identity shortcut connection introduces neither extra parameters nor computational complexity, allowing the network to be trained end-to-end via backpropagation.

Sources

S3.E1

𝐲=ℱ​(𝐱,{Wi})+𝐱.𝐲ℱ𝐱subscript𝑊𝑖𝐱\mathbf{y}=\mathcal{F}(\mathbf{x},\{W_{i}\})+\mathbf{x}. (1)
\mathbf{y}=\mathcal{F}(\mathbf{x},\{W_{i}\})+\mathbf{x}
Implementation detail

Illustrative Numerical Example

Let us compute a forward pass for a single-channel 1D feature vector:

  • Input vector: x=[1.02.0]\mathbf{x} = \begin{bmatrix} 1.0 \\ 2.0 \end{bmatrix}
  • Residual function: F(x,{Wi})=[−0.50.3]\mathcal{F}(\mathbf{x}, \{W_i\}) = \begin{bmatrix} -0.5 \\ 0.3 \end{bmatrix}

Applying the residual formula: y=F(x,{Wi})+x=[−0.50.3]+[1.02.0]=[0.52.3]\mathbf{y} = \mathcal{F}(\mathbf{x}, \{W_i\}) + \mathbf{x} = \begin{bmatrix} -0.5 \\ 0.3 \end{bmatrix} + \begin{bmatrix} 1.0 \\ 2.0 \end{bmatrix} = \begin{bmatrix} 0.5 \\ 2.3 \end{bmatrix}

Sources

S3.E1

𝐲=ℱ​(𝐱,{Wi})+𝐱.𝐲ℱ𝐱subscript𝑊𝑖𝐱\mathbf{y}=\mathcal{F}(\mathbf{x},\{W_{i}\})+\mathbf{x}. (1)
\mathbf{y}=\mathcal{F}(\mathbf{x},\{W_{i}\})+\mathbf{x}
y\mathbf{y}
Output vector of the residual block · D-dimensional vector
x\mathbf{x}
Input vector to the residual block · D-dimensional vector
F\mathcal{F}
Residual mapping function to be learned · Function mapping from D-dimensional space to D-dimensional space
{Wi}\{W_{i}\}
Set of weights associated with the layers in the residual block · Set of weight matrices