Locality-sensitive Hashing

SUMMARY

Locality-sensitive hashing (LSH) is a probabilistic technique for finding approximate nearest neighbors in high-dimensional spaces. It works by hashing similar items into the same "buckets" with high probability, while ensuring dissimilar items are unlikely to collide.

How locality-sensitive hashing works

LSH uses special hash functions that preserve similarity relationships - items that are close in the original space remain close in the hashed space. The key property is:

$P(h(x) = h(y)) \propto \text{similarity}(x,y)$

Where:

$h(x)$ is the hash function
$P(h(x) = h(y))$ is the probability of collision
$\text{similarity}(x,y)$ is a domain-specific similarity measure

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Common LSH families

MinHash

Used for Jaccard similarity between sets:

$P(\text{MinHash}(A) = \text{MinHash}(B)) = J(A,B)$

Where $J(A,B)$ is the Jaccard similarity between sets A and B.

SimHash

Preserves cosine similarity between vectors:

$P(\text{SimHash}(x) = \text{SimHash}(y)) = 1 - \frac{\cos^{-1}(\frac{x \cdot y}{||x|| ||y||})}{\pi}$

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Applications in time-series data

LSH enables efficient similarity search in time-series databases through:

Dimensionality reduction: Converting time-series segments into compact hash signatures
Fast retrieval: Only comparing sequences that hash to the same bucket
Approximate matching: Finding similar patterns without exhaustive search

This makes it valuable for:

Pattern detection
Anomaly identification
Nearest neighbor search
Duplicate detection

Performance considerations

The effectiveness of LSH depends on:

Hash function selection: Must preserve relevant similarity metrics
Number of hash functions: Trades accuracy vs computational cost
Bucket size: Affects collision probability and search efficiency

The optimal configuration balances:

$\text{Precision} = \frac{\text{True Positives}}{\text{True Positives + False Positives}}$

Against:

$\text{Recall} = \frac{\text{True Positives}}{\text{True Positives + False Negatives}}$

Implementation strategies

Multi-probe LSH: Queries multiple nearby buckets to improve recall
LSH Forest: Uses prefix trees to enable dynamic parameter adjustment
Distributed LSH: Parallelizes computation across multiple nodes

LSH provides a powerful framework for approximate similarity search in high-dimensional spaces, making it particularly valuable for time-series analysis and pattern matching at scale.