Johnson-Lindenstrauss Lemma
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# Johnson-Lindenstrauss Lemma ## 4.1 Statement For any $\epsilon \in (0, 1/2)$ and any set of $n$ points in $\mathbb{R}^d$, there exists a map $f: \mathbb{R}^d \to \mathbb{R}^m$ with $m = O(\log n / \epsilon^2)$ such that for all $u, v$: $$ (1-\epsilon)||u-v||^2 \leq ||f(u)-f(v)||^2 \leq (1+\epsilon)||u-v||^2 $$ ##...
Johnson-Lindenstrauss Lemma
4.1 Statement
For any and any set of points in , there exists a map with such that for all :
4.2 Random Projection
pythonimport numpy as np def random_projection(X, epsilon=0.1): n, d = X.shape m = int(4 * np.log(n) / (epsilon**2 / 2 - epsilon**3 / 3)) R = np.random.randn(d, m) / np.sqrt(m) return X @ R # Example X = np.random.randn(1000, 10000) X_proj = random_projection(X, epsilon=0.1) print(f"Reduced from {X.shape[1]} to {X_proj.shape[1]} dimensions")
4.3 Key Insight
The target dimension depends only on (number of points), not on (original dimension). This makes JL extremely powerful for high-dimensional data.
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