What are approximation in high dimensions?

Approximation in high dimensions is like trying to guess what shape a cookie will be before it’s baked, but there are way more ingredients than you can count.

Imagine you're baking cookies, and each ingredient is a different kind of flavor. If you only use sugar and chocolate chips, guessing the shape is easy, maybe round or square. But if you add hundreds of flavors like cinnamon, vanilla, peanut butter, almond extract, and more, it's much harder to tell what the cookie will look like before it comes out of the oven.

High dimensions mean there are a lot of these ingredients (or variables), making it hard to see the whole picture. It’s like trying to find your way through a maze with 100 turns instead of just 2, you need more clues!

But here's the fun part: sometimes, even though there are a lot of flavors, we can approximate what the cookie will be by looking at just a few key ingredients. It’s like tasting a tiny bit and guessing the whole recipe, not perfect, but close enough to enjoy!