Residual block with shortcut

Background

A residual block is the building block of ResNets. Instead of asking a stack of layers to learn a full mapping $H(x)$, it learns a residual $F(x)$ and adds back the input: $H(x)=F(x)+x$. This shortcut (skip) connection lets gradients flow directly to earlier layers, making very deep networks trainable and largely curing the degradation problem.

Problem statement

Implement residual_block(x, w1, w2) for a two-layer block with a ReLU between the weight layers, an additive shortcut, and a final ReLU:

$$F(x) = W_2\,\operatorname{ReLU}(W_1 x), \qquad y = \operatorname{ReLU}\big(F(x) + x\big)$$

Input

• x — input vector, np.ndarray of shape (n,).
• w1 — first weight matrix, shape (m, n).
• w2 — second weight matrix, shape (n, m) (so $F(x)$ matches x's shape).

Output

An np.ndarray of shape (n,): the block output $y$.

Examples

Example 1

Input:  x = [1.0, 2.0], w1 = [[1,0],[0,1]], w2 = [[0.5,0],[0,0.5]]
Output: [1.5, 3.0]

Explanation: $W_1 x=[1,2]$, ReLU keeps it; $W_2[1,2]=[0.5,1.0]$; add the shortcut $x$ to get $[1.5,3.0]$; the final ReLU leaves it unchanged.

Constraints

• ReLU is $\max(0,\cdot)$, applied after the first weight layer and again after the addition.
• The shortcut adds the original x (identity mapping), so $F(x)$ must have the same shape as x.
• Use matrix-vector products (np.dot / @).

Notes

• When the optimal mapping is close to identity, the block only has to drive $F(x)\to 0$ — far easier than learning identity through nonlinear layers.
• If the block changes the channel/spatial dimensions, a real ResNet projects the shortcut (e.g. a 1×1 conv) so the shapes line up before adding.