# The convolution calculation

Subject: Deep Learning

Exam: MT2

Reader: https://deep-learning-study-room.vercel.app/#mt2/read/7/mt2-c7-s2

Canonical: https://deep-learning-study-room.vercel.app/read/dl/mt2/section/mt2-c7-s2/

**Definition.** A convolutional layer calculates local weighted sums with a spatially shared kernel and optional bias. Standard deep-learning implementations ordinarily use cross-correlation, without flipping the kernel; examination calculations must follow the stated convention.

### Explanation and working

In deep-learning software, the operation normally called convolution is cross-correlation: use the kernel as written, without flipping it. This is the convention in this course’s CNN numericals and the supplied latest answer key. Mathematical convolution would rotate the kernel; state that convention if a question explicitly requests it.

**Question.** Apply a 2 × 2 kernel Matrix 1 shown below to the 3 × 3 image Matrix 2 shown below, with stride 1, no padding and zero bias.

### Given

Input X =  $\begin{bmatrix}1 & 2 & 3\\4 & 5 & 6\\7 & 8 & 9\end{bmatrix}$ .

Kernel K =  $\begin{bmatrix}1 & 0\\0 & -1\end{bmatrix}$ .

Stride = 1; padding = 0; bias = 0; no kernel flip.

### To find

Find the 2 × 2 output map.

### Formula — which one and why

**Convolution values.** At each valid location, multiply corresponding window and kernel entries, add all products and add the filter bias once. Move by the stated stride and repeat.

X = input; K = kernel; S = stride; b = filter bias; u,v locate a kernel entry. Use the printed kernel without flipping under the stated deep-learning convention.



$$
Y_{ij}=\sum_{u,v}X_{iS+u,jS+v}K_{uv}+b
$$



**Yaad rakhne ka cue:** Matching cells multiply; sab products add; bias sirf ek baar.

### Solution — step by step

![Given matrices for this question](https://deep-learning-study-room.vercel.app/assets/figures/given_cd39cabe48.svg)

Diagram: Given matrices for this question. Labels: GIVEN VALUES — read each matrix row from left to right | Matrix 1 | 1 | 0 | 0 | −1 | Matrix 2 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9

Given matrices for this question Tap the diagram to enlarge.



$$
X=\begin{bmatrix}1&2&3\\4&5&6\\7&8&9\end{bmatrix},\quad K=\begin{bmatrix}1&0\\0&-1\end{bmatrix}
$$



**Step 1. Top-left window.** 1(1) + 2(0) + 4(0) + 5(−1) = 1 − 5 = −4.

**Step 2. Move right by one.** 2(1) + 3(0) + 5(0) + 6(−1) = 2 − 6 = −4.

**Step 3. Move down and return left.** 4(1) + 5(0) + 7(0) + 8(−1) = 4 − 8 = −4.

**Step 4. Bottom-right window.** 5(1) + 6(0) + 8(0) + 9(−1) = 5 − 9 = −4. Output is  $\begin{bmatrix}-4 & -4\\-4 & -4\end{bmatrix}$ . If ReLU follows, that next layer changes all four entries to zero; do not silently apply it when it was not requested.

![Four windows produce four output cells](https://deep-learning-study-room.vercel.app/assets/figures/convolution.svg)

Diagram: Four windows produce four output cells. Labels: Step 1: start (1,1) | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | K = [1, 0] | [0, −1] | 1×1 + 2×0 + 4×0 | + 5×(−1) = −4 | Step 2: start (1,2) | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | K = [1, 0] | [0, −1] | 2×1 + 3×0 + 5×0 | + 6×(−1) = −4 | Step 3: start (2,1) | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | K = [1, 0] | [0, −1] | 4×1 + 5×0 + 7×0 | + 8×(−1) = −4 | Step 4: start (2,2) | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | K = [1, 0] | [0, −1] | 5×1 + 6×0 + 8×0 | + 9×(−1) = −4 | Output: [−4, −4; −4, −4]. Kernel is not flipped.

Four windows produce four output cells Tap the diagram to enlarge.