## What is a metric map?

A metric map is a very simple idea: a function between two metric spaces is considered a metric map if for every set of two points in the original space, the two points in the image space are as close or closer together. In mathematical terms, a function $$f:X\rightarrow Y$$ is a metric map if $$d_Y\left(f\left(x\right),f\left(y\right)\right) \le d_X\left(x,y\right)$$.

As we will begin to see later, these maps are very useful for image mapping and serve a key role in what is called the gradient descent method for minimization problems.

But first, we need to develop a better understanding of what the problem is before we can attempt to solve it.