---
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) |