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