236779: Foundations of Algorithms for Massive Datasets Lecture 4 the Johnson-lindenstrauss Lemma
چکیده
The Johnson-Lindenstrauss lemma and its proof This lecture aims to prove the Johnson–Lindenstrauss lemma. Since the lemma is proved easily with another interesting lemma, a part of this lecture is focused on the proof of this second lemma. At the end, the optimality of the Johnson–Lindenstrauss lemma is discussed. Lemma 1 (Johnson-Lindenstrauss). Given the initial space X ⊆ R n s.t. |X| = N , < 1/4 the distortion parameter, δ the probability of failure, and K < n the target dimension. ∃Φ ∈ R K×n a random matrix such that, with probability 1 − δ : and K = O log N δ 2 The following lemma is relevant to prove the Johnson-Lindenstrauss lemma.