Warped Metrics

There is a quiet mistake we make when we talk about space. We assume distance is a fact. That it is given. That two points either are or are not close.

But distance is not primitive. Distance is derived.

It comes from a choice of metric.

The Metric Is the Meaning

In Euclidean space, you learned a metric without being told it was a metric:

$$\lVert x - y \rVert_2 = \sqrt{\sum_{i=1}^{n} (x_i - y_i)^2}.$$

It’s so familiar that it feels like nature. But it’s a design decision. Swap it, and “closest” changes. Neighborhoods change. Straight lines change. So does everything built on them.

⁺ The p-norm is the obvious family: ‖x‖_p = (∑_i |x_i|^p)^(1/p). The obvious family is rarely the right one.

Once you see this, you realize: a metric is a theory of relevance. It encodes what differences matter, and which differences are cheap.

Curvature as a Budget

On a smooth manifold, distance is typically built from a Riemannian metric $g$. In coordinates, you can write the infinitesimal distance element as:

$$ds^2 = \sum_{i,j} g_{ij}(x)\,dx^i\,dx^j.$$

The matrix $g_ij(x)$ is not decoration. It is the local rulebook for how movement costs. It says: in this direction, motion is expensive; in that direction, it’s cheap.

If the costs vary, straightness becomes contextual. The “shortest path” bends, not because it wants to, but because your definition of “short” moved under its feet.

A Small Example That Actually Matters

Take a metric that stretches one axis:

$$g = \begin{pmatrix} \alpha^2 & 0 \\\\ 0 & 1 \end{pmatrix}.$$

Then a tiny step $dx = (dx^1, dx^2)$ has squared length:

$$ds^2 = \alpha^2 (dx^1)^2 + (dx^2)^2.$$

If $\alpha \gg 1$, horizontal movement is “expensive.” Paths that look longer in Euclidean terms can become shorter in this geometry, because they avoid the costly direction.

This is not abstract. It’s the same principle behind:

• Geodesics on curved surfaces
• Energy-based models where some directions are stiff
• Any optimization landscape where the Hessian weights certain moves

A Note on AI (Without Tech Aesthetics)

We often talk about “distance” between embeddings as if it were a fact.

It isn’t.

When you choose cosine similarity, you’re declaring magnitude irrelevant. When you choose Euclidean distance, you’re declaring magnitude meaningful. When you whiten, you’re declaring correlated directions redundant. Each choice edits the ontology of your space.

In other words: the metric is your editorial voice.

And like any editorial voice, it should be chosen with restraint.