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This is a tutorial and survey paper on the Johnson-Lindenstrauss (JL) lemmaand linear and nonlinear random projections. We start with linear randomprojection and then justify its correctness by JL lemma and its proof. Then,sparse random projections with $\ell_1$ norm and interpolation norm areintroduced. Two main applications of random projection, which are low-rankmatrix approximation and approximate nearest neighbor search by randomprojection onto hypercube, are explained. Random Fourier Features (RFF) andRandom Kitchen Sinks (RKS) are explained as methods for nonlinear randomprojection. Some other methods for nonlinear random projection, includingextreme learning machine, randomly weighted neural networks, and ensemble ofrandom projections, are also introduced.

Johnson-Lindenstrauss Lemma, Linear and Nonlinear Random Projections, Random Fourier Features, and Random Kitchen Sinks: Tutorial and Survey