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+---
+id: wiki-2026-0508-spectral-clustering
+title: Spectral Clustering
+category: 10_Wiki/Topics
+status: verified
+canonical_id: self
+aliases: [Graph Spectral Clustering, Laplacian Clustering, Normalized Cuts]
+duplicate_of: none
+source_trust_level: A
+confidence_score: 0.93
+verification_status: applied
+tags: [clustering, graph, unsupervised, laplacian, eigendecomposition]
+raw_sources: []
+last_reinforced: 2026-05-10
+github_commit: pending
+tech_stack:
+  language: Python
+  framework: scikit-learn/scipy/networkx
+---
+
+# Spectral Clustering
+
+## 매 한 줄
+> **"매 graph Laplacian 의 eigenvector 의 lower-dim embed → k-means"**. Spectral clustering 매 affinity-graph 매 cluster 의 detect, 매 non-convex / manifold 의 흐름 의 break (concentric circle, moons). 매 von Luxburg 2007 tutorial 의 canonical reference; 매 modern 매 Nyström approx + GPU eigen 의 large-scale.
+
+## 매 핵심
+
+### 매 3-step recipe
+1. **Affinity matrix** $W$: $w_{ij} = \exp(-\|x_i - x_j\|^2 / 2\sigma^2)$ 또는 k-NN graph.
+2. **Laplacian**:
+   - Unnormalized: $L = D - W$
+   - Symmetric normalized (Ng-Jordan-Weiss): $L_{sym} = I - D^{-1/2} W D^{-1/2}$
+   - Random-walk: $L_{rw} = I - D^{-1} W$
+3. **Eigendecompose** → take k smallest eigenvectors → row-normalize → k-means on rows.
+
+### 매 why eigenvectors?
+- 매 graph cut (RatioCut / NCut) 매 NP-hard.
+- 매 spectral relaxation 매 continuous: 매 2nd-smallest eigenvector (Fiedler) 의 sign 매 binary cut 의 approximate.
+- 매 k cluster 매 k smallest eigenvectors 의 use.
+
+### 매 variant
+- **Ng-Jordan-Weiss (2002)**: $L_{sym}$ + row-normalize.
+- **Shi-Malik (2000)**: Normalized Cuts, $L_{rw}$, image segmentation.
+- **Self-tuning** (Zelnik-Manor 2004): per-point sigma.
+- **Power Iteration Clustering** (Lin-Cohen 2010): 매 cheap approx.
+
+### 매 응용
+1. Image segmentation (NCut on pixel graph).
+2. Community detection (small social nets).
+3. Manifold-aware clustering (Swiss-roll, moons).
+4. Speaker diarization (utterance affinity).
+5. Document clustering (TF-IDF cosine graph).
+
+## 💻 패턴
+
+### scikit-learn
+```python
+from sklearn.cluster import SpectralClustering
+from sklearn.datasets import make_moons
+
+X, _ = make_moons(n_samples=400, noise=0.05)
+sc = SpectralClustering(
+    n_clusters=2,
+    affinity="nearest_neighbors",  # k-NN graph
+    n_neighbors=10,
+    assign_labels="kmeans",
+    random_state=42,
+)
+labels = sc.fit_predict(X)
+```
+
+### From scratch (numpy + scipy)
+```python
+import numpy as np
+from scipy.sparse import csgraph
+from scipy.sparse.linalg import eigsh
+from sklearn.cluster import KMeans
+from sklearn.neighbors import kneighbors_graph
+
+def spectral_cluster(X, k, n_neighbors=10):
+    # 1. k-NN affinity
+    W = kneighbors_graph(X, n_neighbors=n_neighbors, mode='connectivity')
+    W = 0.5 * (W + W.T)  # symmetrize
+    # 2. Symmetric normalized Laplacian
+    L = csgraph.laplacian(W, normed=True)
+    # 3. k smallest eigenvectors
+    vals, vecs = eigsh(L, k=k, which='SM')
+    # 4. Row-normalize
+    norm = np.linalg.norm(vecs, axis=1, keepdims=True)
+    vecs = vecs / np.clip(norm, 1e-10, None)
+    # 5. k-means
+    return KMeans(n_clusters=k, n_init=10).fit_predict(vecs)
+```
+
+### RBF affinity
+```python
+from sklearn.metrics.pairwise import rbf_kernel
+
+def rbf_affinity(X, sigma=1.0):
+    gamma = 1.0 / (2.0 * sigma**2)
+    return rbf_kernel(X, gamma=gamma)
+```
+
+### Sigma auto-tuning (k-th NN distance)
+```python
+from sklearn.neighbors import NearestNeighbors
+
+def auto_sigma(X, k=7):
+    nn = NearestNeighbors(n_neighbors=k+1).fit(X)
+    d, _ = nn.kneighbors(X)
+    return np.median(d[:, k])
+```
+
+### Eigengap heuristic (choose k)
+```python
+def eigengap_k(L, max_k=15):
+    vals, _ = eigsh(L, k=max_k, which='SM')
+    vals = np.sort(vals)
+    gaps = np.diff(vals)
+    return int(np.argmax(gaps)) + 1
+```
+
+### Large-scale Nyström approximation
+```python
+from sklearn.kernel_approximation import Nystroem
+from sklearn.cluster import KMeans
+
+# For N >> 10k
+nys = Nystroem(kernel='rbf', gamma=0.1, n_components=300, random_state=0)
+X_low = nys.fit_transform(X)
+labels = KMeans(n_clusters=k, n_init=10).fit_predict(X_low)
+```
+
+### Image segmentation (NCut)
+```python
+from skimage import data, segmentation, color
+from skimage.future import graph
+
+img = data.coffee()
+labels1 = segmentation.slic(img, compactness=30, n_segments=400)
+g = graph.rag_mean_color(img, labels1, mode='similarity')
+labels2 = graph.cut_normalized(labels1, g)
+out = color.label2rgb(labels2, img, kind='avg')
+```
+
+### Diarization affinity (cosine)
+```python
+def speaker_affinity(embeddings):
+    # (N, D) speaker embeddings, L2-normalized
+    sim = embeddings @ embeddings.T
+    sim = (sim + 1) / 2  # [0,1]
+    return sim
+```
+
+## 매 결정 기준
+| 상황 | Approach |
+|---|---|
+| Convex blob clusters | k-means (faster) |
+| Non-convex / manifold | Spectral (k-NN affinity) |
+| N < 5k | Full eigendecomp |
+| 5k < N < 50k | k-NN sparse + eigsh |
+| N > 50k | Nyström / mini-batch |
+| Image seg | NCut + SLIC superpixels |
+| Speaker diar | Cosine affinity + spectral |
+
+**기본값**: sklearn `SpectralClustering(affinity='nearest_neighbors', n_neighbors=10)`.
+
+## 🔗 Graph
+- 부모: [[Clustering]]
+- 변형: [[K-Means]]
+- 응용: [[Image-Segmentation]]
+- Adjacent: [[Normalized-Cuts]]
+
+## 🤖 LLM 활용
+**언제**: 매 affinity choice rationale, 매 eigengap interpretation, 매 sklearn pipeline scaffolding.
+**언제 X**: 매 numerical eigendecomp (use scipy/PyTorch), 매 cluster validation 매 ground-truth needed.
+
+## ❌ 안티패턴
+- **Dense N×N for N>10k**: 매 OOM. 매 k-NN sparse 의 use.
+- **Sigma 의 untuned**: 매 RBF kernel 매 useless. 매 median distance heuristic.
+- **k 매 hand-pick**: 매 eigengap heuristic 의 first try.
+- **No symmetrization**: 매 k-NN graph 의 directed → 매 complex eigenvalues.
+- **Wrong Laplacian for unbalanced**: 매 unnormalized 매 cluster size 의 sensitive. 매 $L_{sym}$ default.
+
+## 🧪 검증 / 중복
+- Verified (von Luxburg "A Tutorial on Spectral Clustering" 2007; Ng-Jordan-Weiss NIPS 2002; sklearn docs 1.5).
+- 신뢰도 A.
+
+## 🕓 Changelog
+| 날짜 | 변경 |
+|---|---|
+| 2026-05-08 | Phase 1 |
+| 2026-05-10 | Manual cleanup — full content (Laplacian variants + sklearn/scipy + Nyström patterns) |