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### Lecture 9 — Sept 29, 2017 1 Fast JL transform - Sketching

The approach we cover here was investigated by Ailon and Chazelle [AC09]. This approach gives a running time to compute Πx of roughly O(dlog d). They called their transformation the Fast. Johnson-Lindenstrauss Transform (FJLT). A construction similar to theirs, which we will analyze here, is the m × n matrix Π defined

### The Fast Johnson–Lindenstrauss Transform and Approximate

We introduce a new low-distortion embedding of $\ell_2^d$ into $\ell_p^{O(\log n)}$ ($p=1,2$) called the fast Johnson–Lindenstrauss transform (FJLT). The FJLT is faster than standard random projections and just as easy to implement. It is based upon the preconditioning of a sparse projection matrix with a randomized