Positive-definite kernel

Imagine you have a magical box that can tell you if two things are similar or not. Let's say you want to compare two apples. You put one apple in the box and close it. Then, you put the second apple in the box and close it again. The magic box will give you a number that describes how similar or dissimilar the two apples are. If the number is high, it means the apples are very similar. If it's low, it means they are very different.

Now, let's extend this idea to more than just apples. Let's say you want to compare pictures of different dogs. You can put each picture in the magic box and get a number that tells you how similar or different they are. This magic box is called a kernel.

A positive-definite kernel is a type of kernel that always gives positive numbers for every input (like apples, or dog pictures). It also satisfies some special mathematical properties. These special properties ensure that the numbers you get from the kernel always make sense and can be used to do useful things like categorizing similar objects.

So, you can think of a positive-definite kernel as a very smart and helpful box that can tell you how similar or different things are, but always gives positive numbers and follows special rules to make sure it's accurate and reliable.