# Drill 32 Convolution with a negative coefficient

Subject: Deep Learning

Exam: MT2

Reader: https://deep-learning-study-room.vercel.app/#mt2/read/17/mt2-c17-s32

Canonical: https://deep-learning-study-room.vercel.app/read/dl/mt2/section/mt2-c17-s32/

## Question

**Question.** Use the input A and kernel K shown below, stride 1, padding 0. Compute all outputs.

Given matrices

Input A

Input A

|  |  |  |  |
| --- | --- | --- | --- |
| 1 | 2 | 0 | 1 |
| 0 | 1 | 3 | 2 |
| 2 | 1 | 0 | 1 |
| 1 | 0 | 2 | 3 |

Kernel K

Kernel K

|  |  |
| --- | --- |
| 1 | −1 |
| 0 | 1 |

Stride 1; P = 0 (no padding).

## Answer — step by step

### Given

Input A and kernel K are printed with the question.

Stride S = 1; padding P = 0; bias = 0.

Use the kernel without flipping it.

### To find

Calculate all nine output cells.

### 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

### Formulas and the operation we are performing

A filter window produces one output value. Multiply entries at matching positions and add the products. In this deep-learning convention, keep the filter in its printed orientation.



$$
Y_{ij}=\sum_{u=0}^{K_h-1}\sum_{v=0}^{K_w-1}X_{iS+u,jS+v}K_{uv}+b
$$



Here i and j identify an output position; u and v count positions inside its window. S is stride, and b is the filter bias, zero in this question. The equation uses a padded input if padding is present and zero-based indices; the working below labels output positions from 1 for readability.



$$
N_{\mathrm{out}}=\left\lfloor\frac{N+2P-K}{S}\right\rfloor+1
$$



N is input width/height, P is padding per side, and K is filter width/height. Floor means take the integer below the result, then add one.

Follow each selected input window below. Matching row colors connect its input cells, kernel coefficients and products; the outlined output cell shows where their sum belongs.

**Step 1 shape:**   $\left\lfloor\frac{4+2 \times  0-2}{1}\right\rfloor+1$  = 3 along each axis. Use the deep-learning cross-correlation convention: do not flip the printed kernel.

### Output position (1, 1)

Use this position's input window. Multiply corresponding entries row by row; then add all row totals.

Follow window (1, 1) into one output cell

Input rows **1–2**, columns **1–2**. The window moves **1** cell per output step.

1. Select the input window

1. Select the input window

|  |  |  |  |
| --- | --- | --- | --- |
| 1 | 2 | 0 | 1 |
| 0 | 1 | 3 | 2 |
| 2 | 1 | 0 | 1 |
| 1 | 0 | 2 | 3 |

Only the outlined cells belong to this output.

2. Match the kernel

2. Match the kernel

|  |  |
| --- | --- |
| 1 | −1 |
| 0 | 1 |

Same row and column positions; no matrix multiplication.

3. Multiply cell by cell

3. Multiply cell by cell

|  |  |
| --- | --- |
| 1 | −2 |
| 0 | 1 |

Each cell is input × matching kernel coefficient.

4. Add the products row by row

Row 1**1 + (−2) = −1**

Row 2**0 + 1 = 1**

**(−1) + 1 = 0**

5. Place it in the output

5. Place it in the output

|  |  |  |
| --- | --- | --- |
| 0 | 5 | 1 |
| 0 | −2 | 2 |
| 1 | 3 | 2 |

Output row 1, column 1.

**Window row 1.**



$$
R_1=(1)\times(1)+(2)\times(-1)=-1
$$



**Window row 2.**



$$
R_2=(0)\times(0)+(1)\times(1)=1
$$



**Add the rows to obtain one output cell.**



$$
Y_{1,1}=(-1)+(1)=0
$$



### Output position (1, 2)

Use this position's input window. Multiply corresponding entries row by row; then add all row totals.

Follow window (1, 2) into one output cell

Input rows **1–2**, columns **2–3**. The window moves **1** cell per output step.

1. Select the input window

1. Select the input window

|  |  |  |  |
| --- | --- | --- | --- |
| 1 | 2 | 0 | 1 |
| 0 | 1 | 3 | 2 |
| 2 | 1 | 0 | 1 |
| 1 | 0 | 2 | 3 |

Only the outlined cells belong to this output.

2. Match the kernel

2. Match the kernel

|  |  |
| --- | --- |
| 1 | −1 |
| 0 | 1 |

Same row and column positions; no matrix multiplication.

3. Multiply cell by cell

3. Multiply cell by cell

|  |  |
| --- | --- |
| 2 | 0 |
| 0 | 3 |

Each cell is input × matching kernel coefficient.

4. Add the products row by row

Row 1**2 + 0 = 2**

Row 2**0 + 3 = 3**

**2 + 3 = 5**

5. Place it in the output

5. Place it in the output

|  |  |  |
| --- | --- | --- |
| 0 | 5 | 1 |
| 0 | −2 | 2 |
| 1 | 3 | 2 |

Output row 1, column 2.

**Window row 1.**



$$
R_1=(2)\times(1)+(0)\times(-1)=2
$$



**Window row 2.**



$$
R_2=(1)\times(0)+(3)\times(1)=3
$$



**Add the rows to obtain one output cell.**



$$
Y_{1,2}=(2)+(3)=5
$$



### Output position (1, 3)

Use this position's input window. Multiply corresponding entries row by row; then add all row totals.

Follow window (1, 3) into one output cell

Input rows **1–2**, columns **3–4**. The window moves **1** cell per output step.

1. Select the input window

1. Select the input window

|  |  |  |  |
| --- | --- | --- | --- |
| 1 | 2 | 0 | 1 |
| 0 | 1 | 3 | 2 |
| 2 | 1 | 0 | 1 |
| 1 | 0 | 2 | 3 |

Only the outlined cells belong to this output.

2. Match the kernel

2. Match the kernel

|  |  |
| --- | --- |
| 1 | −1 |
| 0 | 1 |

Same row and column positions; no matrix multiplication.

3. Multiply cell by cell

3. Multiply cell by cell

|  |  |
| --- | --- |
| 0 | −1 |
| 0 | 2 |

Each cell is input × matching kernel coefficient.

4. Add the products row by row

Row 1**0 + (−1) = −1**

Row 2**0 + 2 = 2**

**(−1) + 2 = 1**

5. Place it in the output

5. Place it in the output

|  |  |  |
| --- | --- | --- |
| 0 | 5 | 1 |
| 0 | −2 | 2 |
| 1 | 3 | 2 |

Output row 1, column 3.

**Window row 1.**



$$
R_1=(0)\times(1)+(1)\times(-1)=-1
$$



**Window row 2.**



$$
R_2=(3)\times(0)+(2)\times(1)=2
$$



**Add the rows to obtain one output cell.**



$$
Y_{1,3}=(-1)+(2)=1
$$



### Output position (2, 1)

Use this position's input window. Multiply corresponding entries row by row; then add all row totals.

Follow window (2, 1) into one output cell

Input rows **2–3**, columns **1–2**. The window moves **1** cell per output step.

1. Select the input window

1. Select the input window

|  |  |  |  |
| --- | --- | --- | --- |
| 1 | 2 | 0 | 1 |
| 0 | 1 | 3 | 2 |
| 2 | 1 | 0 | 1 |
| 1 | 0 | 2 | 3 |

Only the outlined cells belong to this output.

2. Match the kernel

2. Match the kernel

|  |  |
| --- | --- |
| 1 | −1 |
| 0 | 1 |

Same row and column positions; no matrix multiplication.

3. Multiply cell by cell

3. Multiply cell by cell

|  |  |
| --- | --- |
| 0 | −1 |
| 0 | 1 |

Each cell is input × matching kernel coefficient.

4. Add the products row by row

Row 1**0 + (−1) = −1**

Row 2**0 + 1 = 1**

**(−1) + 1 = 0**

5. Place it in the output

5. Place it in the output

|  |  |  |
| --- | --- | --- |
| 0 | 5 | 1 |
| 0 | −2 | 2 |
| 1 | 3 | 2 |

Output row 2, column 1.

**Window row 1.**



$$
R_1=(0)\times(1)+(1)\times(-1)=-1
$$



**Window row 2.**



$$
R_2=(2)\times(0)+(1)\times(1)=1
$$



**Add the rows to obtain one output cell.**



$$
Y_{2,1}=(-1)+(1)=0
$$



### Output position (2, 2)

Use this position's input window. Multiply corresponding entries row by row; then add all row totals.

Follow window (2, 2) into one output cell

Input rows **2–3**, columns **2–3**. The window moves **1** cell per output step.

1. Select the input window

1. Select the input window

|  |  |  |  |
| --- | --- | --- | --- |
| 1 | 2 | 0 | 1 |
| 0 | 1 | 3 | 2 |
| 2 | 1 | 0 | 1 |
| 1 | 0 | 2 | 3 |

Only the outlined cells belong to this output.

2. Match the kernel

2. Match the kernel

|  |  |
| --- | --- |
| 1 | −1 |
| 0 | 1 |

Same row and column positions; no matrix multiplication.

3. Multiply cell by cell

3. Multiply cell by cell

|  |  |
| --- | --- |
| 1 | −3 |
| 0 | 0 |

Each cell is input × matching kernel coefficient.

4. Add the products row by row

Row 1**1 + (−3) = −2**

Row 2**0 + 0 = 0**

**(−2) + 0 = −2**

5. Place it in the output

5. Place it in the output

|  |  |  |
| --- | --- | --- |
| 0 | 5 | 1 |
| 0 | −2 | 2 |
| 1 | 3 | 2 |

Output row 2, column 2.

**Window row 1.**



$$
R_1=(1)\times(1)+(3)\times(-1)=-2
$$



**Window row 2.**



$$
R_2=(1)\times(0)+(0)\times(1)=0
$$



**Add the rows to obtain one output cell.**



$$
Y_{2,2}=(-2)+(0)=-2
$$



### Output position (2, 3)

Use this position's input window. Multiply corresponding entries row by row; then add all row totals.

Follow window (2, 3) into one output cell

Input rows **2–3**, columns **3–4**. The window moves **1** cell per output step.

1. Select the input window

1. Select the input window

|  |  |  |  |
| --- | --- | --- | --- |
| 1 | 2 | 0 | 1 |
| 0 | 1 | 3 | 2 |
| 2 | 1 | 0 | 1 |
| 1 | 0 | 2 | 3 |

Only the outlined cells belong to this output.

2. Match the kernel

2. Match the kernel

|  |  |
| --- | --- |
| 1 | −1 |
| 0 | 1 |

Same row and column positions; no matrix multiplication.

3. Multiply cell by cell

3. Multiply cell by cell

|  |  |
| --- | --- |
| 3 | −2 |
| 0 | 1 |

Each cell is input × matching kernel coefficient.

4. Add the products row by row

Row 1**3 + (−2) = 1**

Row 2**0 + 1 = 1**

**1 + 1 = 2**

5. Place it in the output

5. Place it in the output

|  |  |  |
| --- | --- | --- |
| 0 | 5 | 1 |
| 0 | −2 | 2 |
| 1 | 3 | 2 |

Output row 2, column 3.

**Window row 1.**



$$
R_1=(3)\times(1)+(2)\times(-1)=1
$$



**Window row 2.**



$$
R_2=(0)\times(0)+(1)\times(1)=1
$$



**Add the rows to obtain one output cell.**



$$
Y_{2,3}=(1)+(1)=2
$$



### Output position (3, 1)

Use this position's input window. Multiply corresponding entries row by row; then add all row totals.

Follow window (3, 1) into one output cell

Input rows **3–4**, columns **1–2**. The window moves **1** cell per output step.

1. Select the input window

1. Select the input window

|  |  |  |  |
| --- | --- | --- | --- |
| 1 | 2 | 0 | 1 |
| 0 | 1 | 3 | 2 |
| 2 | 1 | 0 | 1 |
| 1 | 0 | 2 | 3 |

Only the outlined cells belong to this output.

2. Match the kernel

2. Match the kernel

|  |  |
| --- | --- |
| 1 | −1 |
| 0 | 1 |

Same row and column positions; no matrix multiplication.

3. Multiply cell by cell

3. Multiply cell by cell

|  |  |
| --- | --- |
| 2 | −1 |
| 0 | 0 |

Each cell is input × matching kernel coefficient.

4. Add the products row by row

Row 1**2 + (−1) = 1**

Row 2**0 + 0 = 0**

**1 + 0 = 1**

5. Place it in the output

5. Place it in the output

|  |  |  |
| --- | --- | --- |
| 0 | 5 | 1 |
| 0 | −2 | 2 |
| 1 | 3 | 2 |

Output row 3, column 1.

**Window row 1.**



$$
R_1=(2)\times(1)+(1)\times(-1)=1
$$



**Window row 2.**



$$
R_2=(1)\times(0)+(0)\times(1)=0
$$



**Add the rows to obtain one output cell.**



$$
Y_{3,1}=(1)+(0)=1
$$



### Output position (3, 2)

Use this position's input window. Multiply corresponding entries row by row; then add all row totals.

Follow window (3, 2) into one output cell

Input rows **3–4**, columns **2–3**. The window moves **1** cell per output step.

1. Select the input window

1. Select the input window

|  |  |  |  |
| --- | --- | --- | --- |
| 1 | 2 | 0 | 1 |
| 0 | 1 | 3 | 2 |
| 2 | 1 | 0 | 1 |
| 1 | 0 | 2 | 3 |

Only the outlined cells belong to this output.

2. Match the kernel

2. Match the kernel

|  |  |
| --- | --- |
| 1 | −1 |
| 0 | 1 |

Same row and column positions; no matrix multiplication.

3. Multiply cell by cell

3. Multiply cell by cell

|  |  |
| --- | --- |
| 1 | 0 |
| 0 | 2 |

Each cell is input × matching kernel coefficient.

4. Add the products row by row

Row 1**1 + 0 = 1**

Row 2**0 + 2 = 2**

**1 + 2 = 3**

5. Place it in the output

5. Place it in the output

|  |  |  |
| --- | --- | --- |
| 0 | 5 | 1 |
| 0 | −2 | 2 |
| 1 | 3 | 2 |

Output row 3, column 2.

**Window row 1.**



$$
R_1=(1)\times(1)+(0)\times(-1)=1
$$



**Window row 2.**



$$
R_2=(0)\times(0)+(2)\times(1)=2
$$



**Add the rows to obtain one output cell.**



$$
Y_{3,2}=(1)+(2)=3
$$



### Output position (3, 3)

Use this position's input window. Multiply corresponding entries row by row; then add all row totals.

Follow window (3, 3) into one output cell

Input rows **3–4**, columns **3–4**. The window moves **1** cell per output step.

1. Select the input window

1. Select the input window

|  |  |  |  |
| --- | --- | --- | --- |
| 1 | 2 | 0 | 1 |
| 0 | 1 | 3 | 2 |
| 2 | 1 | 0 | 1 |
| 1 | 0 | 2 | 3 |

Only the outlined cells belong to this output.

2. Match the kernel

2. Match the kernel

|  |  |
| --- | --- |
| 1 | −1 |
| 0 | 1 |

Same row and column positions; no matrix multiplication.

3. Multiply cell by cell

3. Multiply cell by cell

|  |  |
| --- | --- |
| 0 | −1 |
| 0 | 3 |

Each cell is input × matching kernel coefficient.

4. Add the products row by row

Row 1**0 + (−1) = −1**

Row 2**0 + 3 = 3**

**(−1) + 3 = 2**

5. Place it in the output

5. Place it in the output

|  |  |  |
| --- | --- | --- |
| 0 | 5 | 1 |
| 0 | −2 | 2 |
| 1 | 3 | 2 |

Output row 3, column 3.

**Window row 1.**



$$
R_1=(0)\times(1)+(1)\times(-1)=-1
$$



**Window row 2.**



$$
R_2=(2)\times(0)+(3)\times(1)=3
$$



**Add the rows to obtain one output cell.**



$$
Y_{3,3}=(-1)+(3)=2
$$



**Final output:**   $\begin{bmatrix}0 & 5 & 1\\0 & -2 & 2\\1 & 3 & 2\end{bmatrix}$ . Each output came from its own complete window; stride changes the next window position, not the multiplication rule.