[dnautics]

"inner products/gaussians" - the absolute value (and also cuberoot of absolute cubes, fourth root of fourth powers) also define inner products. Likewise, there are "gaussian-like formulas" which take these powers instead of squared.

However: if you look at the shape of the squareroot of sum squares, it's a circle, so you can rotate it. If you take the absolute, it's a square, so that cannot be rotated; the cuberoot of cubes and fourthroot of fourths, etc. look like rounded edge squares, and that cannot be rotated either, so if you have a change of vector basis, you're out of luck.

With the gaussian forms of other powers, none of them have the central limit property.

[grodeni]

What kind of inner products are defined by the absolute value, cuberoot of absolute cubes, fourth root of fourth powers? I never heard of that and would be glad to learn about it.

[ska]

You may find it interesting to read about Lp norms, and their relationship to inner products on vector spaces. I think the OP is mixing up norm and inner product terminology. This happens often because you derive an norm from any inner product, but the other way may not exist.

If you plot on the plane the distance = 1 line, then L_1 gives you a diamond, L_2 a circle, L_inf a square. [More precisely, the unit circle under the related metric (distance function) looks like those euclidean shapes]

[Chinjut]

They don't give inner products, but they do give norms. But inner products are, in some ways, more convenient than general norms, hence squared error as opposed to other things. It's not that squared error is necessarily what you fundamentally care about; it just happens to be so conveniently analyzed, because the mathematics of inner products is convenient.

[lordnacho]

It's possible he means Lp norms?

https://en.wikipedia.org/wiki/Norm_(mathematics)

[dnautics]

hah whoops! I did confuse inner products with norms. But it is true that the L2 norm is the only one that survives transformations to arbitrary unit basis vectors.