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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.

Random Fourier features approximate a continuous translation-invariant positive-semidefinite kernel $k(x,y)=\kappa (x-y)$ on ${\mathbb{R}}^d$ with $\kappa (0)=1$. Its spectral probability distribution gives frequencies $\omega$ satisfying $\kappa (x-y)=\mathbb{E}\cos (\omega \cdot (x-y))$.

Draw $m$ independent and identically distributed random variables ${\omega }_j$ from this distribution and independent $b_j$ from the continuous uniform distribution on $[0,2\pi ]$. Define the feature mapAveraging the product of two cosines over $b_j$ leaves one half of $\cos ({\omega }_j\cdot (x-y))$, so $\mathbb{E}[z(x{)}^{T}z(y)]=k(x,y)$. Every realized approximate kernel matrix is positive semidefinite because it is a Gram matrix. Increasing $m$ averages more sampled features; it does not make the finite approximation identically equal to the original positive-semidefinite kernel.^[ref]

For a positive-semidefinite kernel with positive diagonal constant $\kappa (0)=c$, normalize by $c$ and multiply $z$ by $\sqrt{c}$. The identically zero case needs no random features.