---
id: wiki-2026-0508-locality-sensitive-hashing-lsh
title: Locality-Sensitive Hashing (LSH)
category: 10_Wiki/Topics
status: verified
canonical_id: self
aliases: [LSH, Approximate Nearest Neighbor, MinHash, SimHash]
duplicate_of: none
source_trust_level: A
confidence_score: 0.9
verification_status: applied
tags: [hashing, ann, retrieval, similarity-search]
raw_sources: []
last_reinforced: 2026-05-10
github_commit: pending
tech_stack:
  language: python
  framework: datasketch, faiss
---

# Locality-Sensitive Hashing (LSH)

## 매 한 줄
> **"매 similar item 의 same bucket 으로의 hash"**. LSH 는 hash function family $\mathcal{H}$ 가 매 distance-preserving — close points 는 collide, far points 는 separate. Indyk & Motwani (1998) 이 도입했고, 2026 에서는 ANN 의 매 classical baseline 이며 dedup, plagiarism, blocking, near-duplicate retrieval 의 매 default 로 여전히 사용 (HNSW 가 dominate 하지만 LSH 는 streaming/external memory 에 유리).

## 매 핵심

### 매 Definition
$\mathcal{H}$ is $(r_1, r_2, p_1, p_2)$-sensitive iff:
- $d(x,y) \le r_1 \Rightarrow \Pr[h(x)=h(y)] \ge p_1$
- $d(x,y) \ge r_2 \Rightarrow \Pr[h(x)=h(y)] \le p_2$
- $r_1 < r_2$, $p_1 > p_2$

### 매 Families
- **MinHash**: Jaccard distance — set similarity
- **SimHash (random hyperplane)**: cosine — sign of $w^\top x$
- **p-stable LSH**: $\ell_p$ norms (Datar 2004)
- **Cross-polytope**: spherical distance (state-of-art)

### 매 Amplification
- AND: $g(x) = (h_1(x), \dots, h_k(x))$ — reduces $p_2$ to $p_2^k$
- OR: $L$ tables, query all → reduces miss rate
- tune $(k, L)$ for target precision/recall

### 매 응용
1. **Dedup**: web crawl near-dup pages (MinHash + LSH).
2. **Plagiarism**: shingled MinHash.
3. **Blocking**: entity resolution candidate generation.
4. **ANN**: cosine NN (SimHash baseline).
5. **Genomics**: sketch-based read alignment.

## 💻 패턴

### MinHash + LSH for Jaccard
```python
from datasketch import MinHash, MinHashLSH

def shingles(text, k=5):
    return {text[i:i+k] for i in range(len(text)-k+1)}

def make_mh(s, num_perm=128):
    m = MinHash(num_perm=num_perm)
    for sh in s: m.update(sh.encode())
    return m

lsh = MinHashLSH(threshold=0.7, num_perm=128)
docs = {"d1": "the quick brown fox", "d2": "the quick red fox"}
mhs = {k: make_mh(shingles(v)) for k, v in docs.items()}
for k, m in mhs.items(): lsh.insert(k, m)

q = make_mh(shingles("the quick brown fox jumps"))
print(lsh.query(q))  # candidate set
```

### SimHash for cosine
```python
import numpy as np
from collections import defaultdict

def simhash(x, planes):
    return tuple((x @ planes.T > 0).astype(np.int8))

# k random hyperplanes
d, k, L = 128, 8, 10
tables = [np.random.randn(k, d) for _ in range(L)]

def index(X):
    out = [defaultdict(list) for _ in range(L)]
    for i, x in enumerate(X):
        for li, planes in enumerate(tables):
            out[li][simhash(x, planes)].append(i)
    return out

def query(q, idx):
    cands = set()
    for li, planes in enumerate(tables):
        cands |= set(idx[li].get(simhash(q, planes), []))
    return cands
```

### p-stable LSH (L2)
```python
# h(x) = floor((a·x + b) / w), a ~ N(0, I), b ~ U[0, w]
def make_l2_lsh(d, w=4.0, k=8):
    a = np.random.randn(k, d)
    b = np.random.uniform(0, w, k)
    return lambda x: tuple(np.floor((a @ x + b) / w).astype(np.int64))
```

### LSH Forest (multi-resolution)
```python
from datasketch import MinHashLSHForest
forest = MinHashLSHForest(num_perm=128)
for k, m in mhs.items(): forest.add(k, m)
forest.index()
print(forest.query(q, 5))  # top-5 approx Jaccard NN
```

### Banding technique (k-AND, L-OR)
```python
def banded_lsh(signatures, k_per_band, L_bands):
    # signatures: (n, k_per_band * L_bands)
    buckets = [defaultdict(list) for _ in range(L_bands)]
    for i, sig in enumerate(signatures):
        for b in range(L_bands):
            band = tuple(sig[b*k_per_band:(b+1)*k_per_band])
            buckets[b][band].append(i)
    return buckets
```

## 매 결정 기준
| Distance | Family |
|---|---|
| Jaccard (sets) | MinHash |
| Cosine | SimHash / Cross-polytope |
| $\ell_2$ | p-stable (Gaussian) |
| Hamming | bit-sampling |
| edit distance | shingle + MinHash approx |

**기본값**: HNSW for general ANN (faster); LSH for dedup, streaming, external memory, exact-recall guarantee.

## 🔗 Graph
- 부모: [[Approximate-Nearest-Neighbor]]
- 변형: [[MinHash]] · [[SimHash]]
- 응용: [[Deduplication]]
- Adjacent: [[HNSW]] · [[Faiss]]

## 🤖 LLM 활용
**언제**: massive corpus dedup (e.g. pretraining cleanup), candidate blocking, streaming.
**언제 X**: small (n < 10⁵) 또는 high-precision recall — HNSW/IVF 가 더 빠름.

## ❌ 안티패턴
- **(k, L) tuning 무시**: default 사용 → too many false positives or misses.
- **Wrong family**: cosine 인데 MinHash 사용 → meaningless.
- **Re-hash on every query**: index 재build → use persistent lib (datasketch, faiss).
- **Treating LSH as exact**: 매 approximate — verify candidates with true distance.

## 🧪 검증 / 중복
- Verified (Indyk & Motwani 1998 STOC, Andoni & Indyk 2008 CACM, Leskovec MMDS textbook ch 3).
- 신뢰도 A.

## 🕓 Changelog
| 날짜 | 변경 |
|---|---|
| 2026-05-08 | Phase 1 |
| 2026-05-10 | Manual cleanup — LSH families, MinHash/SimHash, banding, dedup patterns |