UMAP: Still Have To Be Smart With Compressing

(Typing on a physical keyboard nearby) You're drooling on the headset.

"Topology doesn't care about units?"

You are absolutely wrong.

Shez, how does UMAP build this beautiful graph?

Stop. You said *finds the nearest neighbors.*

How does it measure *Nearest*?

Which distance?

Euclidean Distance. The Pythagorean Theorem. $a^2 + b^2 = c^2$.

To UMAP, Point A is **1000 times closer** to Point B than Point C.

The algorithm thinks the Voltage Spike is "The Same State," but the tiny movement is "Far Away."

The neighbors are determined entirely by the variable with the biggest numbers.

You haven't escaped the Unit Problem. You just hid it inside the Neighbor Search.

It adapts to the *density* of the neighbors it found.

But if it found the *wrong* neighbors because your units were messed up, it's just adapting to garbage.

UMAP is like a camera that focuses automatically.

But if you leave the lens cap on (Wrong Units), it just focuses perfectly on the back of the lens cap.

Look at your "Islands."

Island 1 is "High Meters." Island 2 is "Low Meters."

Where is the Voltage info?

Yes. You always have to normalize if your units are different. There is no "God Algorithm" that fixes bad physics.

Welcome to **Stochasticity**.

PCA is Linear Algebra. It solves for Eigenvectors. It is the same every time.

UMAP is an Optimization Process. It uses random initialization and stochastic gradient descent.

Neither. Or both.

They are just 2D shadows of a high-dimensional object. The "Global Structure" in UMAP is not guaranteed to be preserved.

**Global Structure vs. Local Structure.**

UMAP cares about *Neighbors*. It will tear the global map apart to keep neighbors together.

Imagine the Earth.

PCA tries to flatten it like a Mercator projection. It preserves the general "North is Up."

UMAP ensures that New York is next to New Jersey.

But it might put the USA next to China just because they fit together nicely on the table.

Yes.

Exactly. The distance *between* clusters is often meaningless in UMAP.

It's a tool for **Visualization**, not for rigorous engineering.

If you need features for a model: Use **Autoencoders** (or PCA). They are parametric functions. They are stable.

If you need to make a pretty chart for the CEO: Use UMAP.

And if you *must* use UMAP for analysis... change the metric.

Mahalanobis distance. It accounts for the variance and covariance of the data.

It essentially "learns" the units for you.

Because you used a metric that understands how the variables relate to each other.

Don't be seduced by the "Manifold Learning". At the bottom of every fancy algorithm is a simple function calculating `Distance(A, B)`.

If you feed that function garbage units, it will build you a garbage castle.