K-Means是否存在“各簇观测值数量大致相等”的假设？

Hey there, let’s untangle this confusing mix-up about what "size" refers to in K-means—this is such a common point of confusion, so you’re definitely not alone in scratching your head over conflicting takes online.

First, let’s set the record straight on the core point your lecturer brought up: K-means does NOT require clusters to have roughly equal numbers of observations as a hard rule. That’s a common misinterpretation of one of its implicit assumptions. Here’s why the two "size" definitions are floating around:

• When people say "size" means cluster cardinality (number of points):
  This is the most casual, everyday usage. If someone mentions "the size of a K-means cluster" without extra context, they’re almost certainly talking about how many data points are grouped into that cluster. This comes up when discussing K-means limitations—for example, if your true data has clusters with wildly uneven sample counts, K-means might split large clusters or merge small ones because its objective (minimizing squared error) prioritizes balancing the "mass" around centroids, not raw point counts.

• When people say "size" means spatial range/area:
  This ties directly to K-means’ key implicit assumption: it expects clusters to be spherical and have similar variance (i.e., similar "spread" in the feature space). In this context, "size" refers to how much space the cluster occupies, not how many points are in it. If your data has clusters with drastically different spatial sizes (one tight cluster, one spread-out cloud), K-means will struggle to group them correctly because it’s optimized for clusters with comparable variance.

So where did your lecturer’s statement come from? It’s likely a mix-up of these two ideas. K-means doesn’t care about equal point counts, but its underlying math makes it perform best when clusters have similar spatial sizes (variance). When people simplify this to "equal size clusters," they sometimes accidentally swap "spatial size" with "point count."

To wrap it up:

• Default, casual use of "cluster size" = number of points in the cluster
• When discussing K-means assumptions, "equal size" almost always refers to similar spatial variance/distribution range

内容的提问来源于stack exchange，提问作者xji