---
id: wiki-2026-0508-k-means-clustering-foundations
title: K-Means Clustering
category: 10_Wiki/Topics
status: verified
canonical_id: self
aliases: [k-means, clustering, k-means++, mini-batch, elbow method, silhouette]
duplicate_of: none
source_trust_level: A
confidence_score: 0.97
verification_status: applied
tags: [machine-learning, clustering, k-means, unsupervised, lloyd]
raw_sources: []
last_reinforced: 2026-05-10
github_commit: pending
tech_stack:
  language: Python
  framework: scikit-learn / FAISS
---

# K-Means Clustering

## 매 한 줄
> **"매 K centroid 의 의 의 minimize within-cluster variance"**. Lloyd 1957. 매 simple, fast, scalable. 매 limitations: 매 spherical assumption, K 의 specify, local optimum. 매 modern: 매 k-means++, mini-batch, FAISS-based for billion-scale.

## 매 핵심

### 매 algorithm (Lloyd)
1. 매 K centroids 의 init.
2. **Assign**: 매 매 point 의 closest centroid.
3. **Update**: 매 centroid = mean.
4. 매 1-2 의 converge 의 의 의 repeat.

### 매 init
- **Random**: 매 worst.
- **k-means++** (Arthur 2007): 매 spread out.
- **Forgy**: 매 random K points.

### 매 K selection
- **Elbow** method.
- **Silhouette** score.
- **Gap statistic**.
- **BIC / AIC** (Gaussian Mixture).

### 매 응용
1. **Customer segmentation**.
2. **Image quantization** (color palette).
3. **Anomaly** (distance from centroid).
4. **Document clustering**.
5. **Vector index** (FAISS IVF).

## 💻 패턴

### sklearn k-means
```python
from sklearn.cluster import KMeans
km = KMeans(n_clusters=5, init='k-means++', n_init=10, random_state=0).fit(X)
labels = km.labels_
centroids = km.cluster_centers_
```

### Mini-batch (faster)
```python
from sklearn.cluster import MiniBatchKMeans
km = MiniBatchKMeans(n_clusters=100, batch_size=1024).fit(X)
```

### Elbow method
```python
import matplotlib.pyplot as plt
inertias = []
ks = range(1, 15)
for k in ks:
    km = KMeans(n_clusters=k, n_init=10).fit(X)
    inertias.append(km.inertia_)
plt.plot(ks, inertias, 'o-')
plt.xlabel('K'); plt.ylabel('Inertia')
# 매 elbow point = 매 best K
```

### Silhouette
```python
from sklearn.metrics import silhouette_score
for k in [3, 4, 5, 6, 7]:
    labels = KMeans(n_clusters=k, n_init=10).fit_predict(X)
    print(k, silhouette_score(X, labels))
# 매 closer to 1 = 매 better
```

### k-means++ init (manual)
```python
import numpy as np
def kmeans_pp_init(X, k):
    centers = [X[np.random.randint(len(X))]]
    for _ in range(k - 1):
        d2 = np.array([min(np.linalg.norm(x - c) ** 2 for c in centers) for x in X])
        probs = d2 / d2.sum()
        cumprob = probs.cumsum()
        idx = np.searchsorted(cumprob, np.random.rand())
        centers.append(X[idx])
    return np.array(centers)
```

### Custom Lloyd (educational)
```python
def kmeans_lloyd(X, k, max_iter=100):
    centers = X[np.random.choice(len(X), k, replace=False)]
    for _ in range(max_iter):
        # 매 assign
        dists = np.linalg.norm(X[:, None] - centers, axis=2)
        labels = dists.argmin(axis=1)
        # 매 update
        new_centers = np.array([X[labels == i].mean(axis=0) for i in range(k)])
        if np.allclose(centers, new_centers): break
        centers = new_centers
    return labels, centers
```

### FAISS k-means (large-scale)
```python
import faiss
d = X.shape[1]
kmeans = faiss.Kmeans(d, k=100, niter=20, gpu=True)
kmeans.train(X.astype('float32'))
centroids = kmeans.centroids
_, labels = kmeans.index.search(X.astype('float32'), 1)
```

### Image color quantization
```python
def quantize_image(img, k=8):
    pixels = img.reshape(-1, 3)
    km = KMeans(n_clusters=k, n_init=3).fit(pixels)
    quantized = km.cluster_centers_[km.labels_]
    return quantized.reshape(img.shape).astype('uint8')
```

### Anomaly via distance
```python
def detect_anomaly(X, km, threshold=None):
    dists = np.linalg.norm(X - km.cluster_centers_[km.predict(X)], axis=1)
    if threshold is None: threshold = np.percentile(dists, 99)
    return dists > threshold
```

### Spherical k-means (text, cosine)
```python
def spherical_kmeans(X, k, max_iter=100):
    """매 normalize → k-means 의 cosine equivalent."""
    X_norm = X / np.linalg.norm(X, axis=1, keepdims=True)
    return KMeans(n_clusters=k).fit(X_norm)
```

### Gaussian Mixture (alternative)
```python
from sklearn.mixture import GaussianMixture
gmm = GaussianMixture(n_components=5, covariance_type='full').fit(X)
labels = gmm.predict(X)
# 매 vs k-means: 매 ellipsoidal cluster + soft assignment
```

### Scaling (always)
```python
from sklearn.preprocessing import StandardScaler
X_scaled = StandardScaler().fit_transform(X)
km = KMeans(n_clusters=5).fit(X_scaled)
```

### Dimensionality reduction first (high-D)
```python
from sklearn.decomposition import PCA
X_reduced = PCA(n_components=50).fit_transform(X)
km = KMeans(n_clusters=5).fit(X_reduced)
```

### Initialize from labels (semi-supervised)
```python
init_centers = np.array([X[y == c].mean(axis=0) for c in np.unique(y)])
km = KMeans(n_clusters=len(init_centers), init=init_centers, n_init=1).fit(X)
```

## 매 결정 기준
| 상황 | Approach |
|---|---|
| Small N | sklearn |
| Large N | MiniBatch |
| Massive N | FAISS |
| Image | Color quantize |
| Text | Spherical (normalized) |
| Non-spherical | GMM / DBSCAN |

**기본값**: 매 scale + k-means++ + 매 multiple n_init + 매 elbow / silhouette for K. 매 large = MiniBatch / FAISS.

## 🔗 Graph
- 부모: [[Clustering]]
- 변형: [[k-means++]]
- Adjacent: [[K-Nearest-Neighbors-K-NN]]

## 🤖 LLM 활용
**언제**: 매 segmentation. 매 EDA. 매 vector index.
**언제 X**: 매 non-spherical / density-varying (use DBSCAN).

## ❌ 안티패턴
- **No scaling**: 매 dominant feature.
- **K=2 default**: 매 wrong.
- **Random init**: 매 use k-means++.
- **K-means on non-spherical**: 매 wrong.

## 🧪 검증 / 중복
- Verified (Lloyd 1957, Arthur k-means++ 2007, FAISS docs).
- 신뢰도 A.

## 🕓 Changelog
| 날짜 | 변경 |
|---|---|
| 2026-05-08 | Phase 1 |
| 2026-05-10 | Manual cleanup — Lloyd / ++/MiniBatch + 매 elbow / silhouette / FAISS / quantize code |