Vector Convolution is a special operation for vector. In this tutorial, we will use some simple examples to illustrate it for deep learning beginners.

## What is vector convolution?

As to n dimension vector **A** and **B**

A = [a_{0}, a_{1}, a_{2}, …, a_{n-1}]

B = [b_{0}, b_{1}, b_{2}, …, b_{n-1}]

The convolution of f(A,B) is **C**:

C = [c_{0}, c_{1}, …, c_{2n-2}]

where

From the equation above, we can find:

- The dimension of vector
**A**and**B**must be the same. - The dimension of result
**C**is**2n-1**

We can understand the vector C as following:

Here is an example.

A = [1,2,3,4] B = [2,3,4,5]

Here vector A and B is 4 dimension. It means the dimension of convolutional C is 2 * 4 – 1 = 7.

A * B^{T} =

[2, 3, 4, 5 4, 6, 8, 10 6, 9, 12, 15 8, 12, 16, 20]

c_{0} = 2

c_{1} = 3 + 4 = 7

c_{2} = 4 + 6 + 6 = 16

c_{3} = 5 + 8 + 9 + 8 = 30

c_{4} = 10 + 12 + 12 = 34

c_{5} = 15 + 16 = 31

c_{6} = 20

It menas **C** is:

[2,7,16,30,34,31,20]