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+---
+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 |