exercise:D4f9286df0

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Let [math]n \gt k[/math] be integers.

• Implement a random projection [math]T\colon\mathbb{R}^d\rightarrow\mathbb{R}^k[/math] and test it for small [math]d[/math] and [math]k[/math].
• Put [math]d=1000[/math], generate 10\,000 points in [math]\mathbb{R}^d[/math] at random, project them via [math]T[/math] to [math]\mathbb{R}^k[/math] and compute, for different values of [math]k[/math], the worst distorsion of a pairwise distance
  [[math]] \epsilon:=\max\Bigl(1-\min_{x\not=y}\medfrac{\|Tx-Ty\|}{\sqrt{k}\|x-y\|}, \max_{x\not=y}\medfrac{\|Tx-Ty\|}{\sqrt{k}\|x-y\|}-1\Bigr) [[/math]]
  that occurs.
• Compare the outcome of your experiment with the relation of [math]\epsilon[/math] and [math]k[/math] that is given in the Johnson-Lindenstrauss Lemma, by replicating Figure based on the data from (ii).
• Explain how you would pick [math]\epsilon[/math] and [math]k[/math] if you are given a dataset in [math]\mathbb{R}^d[/math] and want to do a dimensionality reduction that does not corrupt a classifier based on nearest neighbors.