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Ali Rahimi coauthored Random Features for Large-Scale Kernel Machines by Ali Rahimi and Benjamin Recht in 2007. With Benjamin Recht, he proposed explicit randomized maps whose inner products approximate a chosen kernel, enabling familiar fast linear methods in the transformed feature space.^[ref]

The Random Fourier feature mechanism samples frequencies from a shift-invariant kernel's spectral representation. This separates the cost of constructing a feature map from the cost of repeatedly evaluating every pairwise kernel value. Accuracy and computation depend on the feature count; randomness does not eliminate approximation error or the need to validate the learned predictor.

Benjamin Recht coauthored the 2007 Random Fourier feature construction with Ali Rahimi. Randomly sampled features approximate a shift-invariant positive-definite kernel by ordinary inner products, permitting a linear learning method to approximate a kernel method without storing the whole pairwise kernel matrix.^[ref]

For suitable normalized kernels, features $z_j(x)=\sqrt{2/D}\cos ({\omega }_j^{T}x+b_j)$ use frequencies sampled from the kernel's spectral distribution and uniform phases. Their expected inner product recovers the kernel; finitely many features introduce an approximation error. The number of features controls a computational and statistical tradeoff rather than an exact identity for every finite sample.

By Ali Rahimi and Benjamin Recht. Advances in Neural Information Processing Systems 20, 2007.

Official proceedings: proceedings.neurips.cc/paper_files/paper/2007/hash/013a006f03dbc5392effeb8f18fda755-Abstract.html

Author-hosted PDF: people.eecs.berkeley.edu/~brecht/papers/07.rah.rec.nips.pdf

Section 3, random Fourier features, gives the spectral sampling construction and random-offset cosine feature representation.