仿射变换（affine transformations）不变性及simplicial median相关技术问询

Hey there! Let's unpack your confusion about this property step by step — it's a common point of mix-up, so you're not alone.

What "Affine Invariance" Actually Means #

When we say the simplicial median is invariant under affine transformations, we don't mean the median value stays exactly the same no matter what transformation you apply. Instead, it means the transformation and the median calculation "play nice" together:

If you apply an affine transformation ( T ) to every point in your dataset, the simplicial median of the transformed dataset is exactly the result of applying ( T ) to the original simplicial median.

In formula terms, for a dataset ( S ) and affine transformation ( T ):
T(med_simp(S)) = med_simp(T(S))

Let's use a super simple 2D example to make this concrete:

• Suppose your original dataset is ( S = {(0,0), (2,0), (0,2)} ), whose simplicial median is ( m = (1,1) ).
• Apply a shear transformation ( T(x,y) = (x + y, y) ) to every point in ( S ): the transformed dataset becomes ( T(S) = {(0,0), (2,0), (2,2)} ).
• The simplicial median of ( T(S) ) is ( (2,1) ), which is exactly ( T(m) = T(1,1) = (2,1) ). That's the invariance in action!

It's Not "Staying the Same" — It's "Commuting with Transformations" #

Your question about whether it "means always keeping the same" hits on a key distinction. The simplicial median doesn't stay fixed when you apply an affine transformation — it moves exactly the same way every other point in your dataset moves. The invariance here refers to the relationship between the transformation and the median, not the median's absolute value.

For contrast: if you used the standard Euclidean median (which minimizes the sum of squared distances), this property wouldn't hold. Applying a shear transformation to your dataset would result in a Euclidean median that isn't just the transformed original median.

Simple Related Context #

Here are a few quick bullet points to ground this further:

• Affine transformations include translations, scalings, rotations, shears, and combinations of these — all the transformations that preserve straight lines and distance ratios (as you noted).
• The simplicial median's affine invariance makes it useful for datasets that might be distorted by affine transformations (e.g., in computer vision, where images can be sheared or scaled). You don't need to "undo" the transformation first to compute a meaningful median.
• Unlike the Euclidean median, the simplicial median can be computed in a way that doesn't depend on the coordinate system you're using — which ties directly to its affine invariance.

内容的提问来源于stack exchange，提问作者Koala