JL Lemma: Proof Sketch

Key Steps

1. Choose a random $m \times d$ matrix $R$ with entries $R_{ij} \sim N(0, 1/m)$
2. For any fixed vector $x$, show $E[||Rx||^2] = ||x||^2$
3. By concentration: $\Pr(|||Rx||^2 - ||x||^2| \geq \epsilon ||x||^2) \leq 2\exp(-m(\epsilon^2/2 - \epsilon^3/3))$
4. Union bound over $\binom{n}{2}$ pairs gives the result when $m = O(\log n / \epsilon^2)$ Join Discord PreviousJohnson-LindenstraussNextLocality-Sensitive Hashing