2nd/10_Wiki/Topics/AI_and_ML/Dimensionality-Reduction.md

---
id: wiki-2026-0508-dimensionality-reduction
title: Dimensionality Reduction
category: 10_Wiki/Topics
status: verified
canonical_id: self
aliases: [PCA, t-SNE, UMAP, autoencoder, curse of dimensionality, feature extraction]
duplicate_of: none
source_trust_level: A
confidence_score: 0.93
verification_status: applied
tags: [dimensionality-reduction, pca, tsne, umap, autoencoder, visualization, manifold-learning]
raw_sources: []
last_reinforced: 2026-05-10
github_commit: pending
tech_stack:
  language: Python
  framework: scikit-learn / umap-learn / PyTorch
---

# Dimensionality Reduction

## 매 한 줄
> **"매 high-dim 의 essence 의 low-dim"**. 매 PCA (linear) → 매 t-SNE / UMAP (nonlinear, 시각화) → 매 Autoencoder / VAE (deep). 매 modern: 매 embedding (CLIP, sentence-transformers) 의 implicit dim reduction.

## 매 핵심 method

### Linear

#### PCA (Principal Component Analysis)
- 매 variance 의 maximum direction.
- 매 orthogonal axis.
- 매 SVD.
- 매 fast + interpretable.

#### LDA (Linear Discriminant Analysis)
- 매 class separation 의 maximize.
- 매 supervised.

#### Factor Analysis
- 매 latent factor 의 explain variance.

### Nonlinear (manifold)

#### t-SNE (Maaten 2008)
- 매 local neighborhood 의 preserve.
- 매 visualization 강.
- 매 global structure 의 weak.
- 매 stochastic.

#### UMAP (McInnes 2018)
- 매 t-SNE 의 successor.
- 매 faster + 매 global structure 도 better.
- 매 default for high-dim viz.

#### Isomap
- 매 geodesic distance 의 preserve.

#### LLE (Locally Linear Embedding).

### Neural

#### Autoencoder
- 매 bottleneck 의 dim reduce.

#### VAE (Variational AE)
- 매 probabilistic.

#### Self-Supervised Embedding
- 매 CLIP, BERT, sentence-transformers.
- 매 implicit dim reduction.

### 매 PaCMAP / TriMap (recent)
- 매 UMAP 의 variant.
- 매 better global structure.

### 매 응용
1. **Visualization** (2D / 3D): 매 t-SNE, UMAP.
2. **Speed** (preprocess): 매 PCA.
3. **Anomaly detection**: 매 autoencoder.
4. **Feature extraction**: 매 embedding.
5. **Compression**: 매 quantization + 매 embed.
6. **Clustering preprocessing**.
7. **RAG** (vector DB): 매 PCA / quantization.

### 매 curse of dimensionality
- 매 distance 의 meaningless.
- 매 sparsity in 매 high-dim.
- 매 sample requirement 의 exponential.

## 💻 패턴

### PCA (sklearn)
```python
from sklearn.decomposition import PCA
from sklearn.preprocessing import StandardScaler

X_scaled = StandardScaler().fit_transform(X)

pca = PCA(n_components=0.95)  # 매 95% variance 의 keep
X_reduced = pca.fit_transform(X_scaled)

print(f'Original: {X.shape[1]}, reduced: {pca.n_components_}')
print(f'Explained variance: {pca.explained_variance_ratio_.cumsum()}')
```

### t-SNE
```python
from sklearn.manifold import TSNE

tsne = TSNE(
    n_components=2,
    perplexity=30,
    n_iter=1000,
    random_state=42,
)
X_2d = tsne.fit_transform(X[:5000])  # 매 t-SNE 의 slow → 매 sample
```

### UMAP (modern)
```python
import umap

reducer = umap.UMAP(
    n_components=2,
    n_neighbors=15,
    min_dist=0.1,
    metric='cosine',  # 매 embedding 에 좋음
    random_state=42,
)
X_2d = reducer.fit_transform(X)
```

### Autoencoder (PyTorch)
```python
import torch.nn as nn

class AE(nn.Module):
    def __init__(self, input_dim, latent_dim=32):
        super().__init__()
        self.encoder = nn.Sequential(
            nn.Linear(input_dim, 128), nn.ReLU(),
            nn.Linear(128, 64), nn.ReLU(),
            nn.Linear(64, latent_dim),
        )
        self.decoder = nn.Sequential(
            nn.Linear(latent_dim, 64), nn.ReLU(),
            nn.Linear(64, 128), nn.ReLU(),
            nn.Linear(128, input_dim),
        )
    
    def forward(self, x):
        z = self.encoder(x)
        return self.decoder(z), z

# 매 latent 의 use
model = AE(input_dim=784)
# ... train ...
_, latent = model(X_test)
```

### Visualization combo (UMAP + scatter)
```python
import matplotlib.pyplot as plt

X_2d = umap.UMAP().fit_transform(X)
plt.figure(figsize=(10, 8))
plt.scatter(X_2d[:, 0], X_2d[:, 1], c=labels, cmap='tab10', alpha=0.5, s=10)
plt.colorbar()
plt.title('UMAP projection')
plt.show()
```

### PCA for speed (vector DB preprocessing)
```python
from sklearn.decomposition import PCA
import faiss

# 매 매 768 의 OpenAI embedding → 매 256
embeddings = get_embeddings(documents)
pca = PCA(n_components=256)
reduced = pca.fit_transform(embeddings).astype('float32')

# 매 Faiss
index = faiss.IndexFlatIP(256)
index.add(reduced)
```

### Quantization (vector DB modern)
```python
import faiss

dim = 768
quantizer = faiss.IndexFlatIP(dim)
index = faiss.IndexIVFPQ(quantizer, dim, nlist=100, m=8, nbits=8)
# 매 8 byte 의 768-dim 의 represent — 매 매 100× compression.

index.train(embeddings_np)
index.add(embeddings_np)
```

### Word2Vec / CLIP-style (implicit reduction)
```python
from sentence_transformers import SentenceTransformer

model = SentenceTransformer('all-MiniLM-L6-v2')  # 매 384-dim
embeddings = model.encode(sentences)
# 매 매 sentence (potentially infinite words) → 매 384-dim.
```

### Reconstruction error (anomaly)
```python
def detect_anomaly(model, X, threshold):
    X_recon, _ = model(X)
    error = ((X_recon - X) ** 2).mean(dim=1)
    return error > threshold
```

### Choose dimension (elbow / cumvar)
```python
import numpy as np
import matplotlib.pyplot as plt

pca = PCA().fit(X_scaled)
cumvar = np.cumsum(pca.explained_variance_ratio_)

plt.plot(cumvar)
plt.xlabel('Component')
plt.ylabel('Cumulative variance')
plt.axhline(0.95, color='r', linestyle='--')
plt.show()

n_components = np.argmax(cumvar >= 0.95) + 1
```

### Manifold visualization comparison
```python
def viz_compare(X, labels):
    fig, axes = plt.subplots(1, 3, figsize=(20, 6))
    
    for ax, (name, reducer) in zip(axes, [
        ('PCA', PCA(n_components=2)),
        ('t-SNE', TSNE(n_components=2, random_state=42)),
        ('UMAP', umap.UMAP(n_components=2, random_state=42)),
    ]):
        proj = reducer.fit_transform(X)
        ax.scatter(proj[:, 0], proj[:, 1], c=labels, cmap='tab10', s=5)
        ax.set_title(name)
```

## 매 결정 기준
| 상황 | Method |
|---|---|
| Speed (preprocess) | PCA |
| Visualization | UMAP |
| Cluster preserve | UMAP |
| Variance interpret | PCA |
| Class-aware | LDA |
| Text → embedding | Sentence-transformer |
| Image → embedding | CLIP |
| Vector DB compress | PCA / PQ quantization |
| Anomaly | Autoencoder |
| Generative | VAE |

**기본값**: PCA (preprocess) + UMAP (viz) + embedding (semantic).

## 🔗 Graph
- 부모: [[Feature Engineering|Feature-Engineering]]
- 변형: [[PCA]] · [[t-SNE]] · [[UMAP]] · [[Autoencoder]] · [[VAE]]
- 응용: [[CLIP]] · [[Sentence-Transformers]] · [[Faiss]] · [[Anomaly-Detection]]
- Adjacent: [[Auto-Encoding]] · [[Bag of Words (BoW)]] · [[Bias-vs-Variance]]

## 🤖 LLM 활용
**언제**: 매 visualization. 매 vector DB. 매 cluster preprocessing. 매 anomaly detection.
**언제 X**: 매 already low-dim. 매 lossless 필수.

## ❌ 안티패턴
- **PCA without standardize**: 매 wrong principal component.
- **t-SNE 의 cluster size 의 interpret**: 매 not preserved.
- **UMAP 의 distance 의 absolute interpret**: 매 local 만.
- **Too aggressive reduction**: 매 information loss.
- **Forget train-test split**: 매 leakage in PCA.

## 🧪 검증 / 중복
- Verified (Jolliffe PCA, van der Maaten t-SNE, McInnes UMAP).
- 신뢰도 A.
- Related: [[Auto-Encoding]] · [[Bag of Words (BoW)]] · [[CLIP]] · [[Sentence-Transformers]] · [[Bias-vs-Variance]].

## 🕓 Changelog
| 날짜 | 변경 |
|---|---|
| 2026-05-08 | Phase 1 |
| 2026-05-10 | Manual cleanup — methods + 매 PCA / t-SNE / UMAP / AE / Faiss / quantization code |