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The problem of efficient approximation of a linear operator induced by theGaussian or softmax kernel is often addressed using random features (RFs) whichyield an unbiased approximation of the operator's result. Such operators emergein important applications ranging from kernel methods to efficientTransformers. We propose parameterized, positive, non-trigonometric RFs whichapproximate Gaussian and softmax-kernels. In contrast to traditional RFapproximations, parameters of these new methods can be optimized to reduce thevariance of the approximation, and the optimum can be expressed in closed form.We show that our methods lead to variance reduction in practice ($e^{10}$-timessmaller variance and beyond) and outperform previous methods in a kernelregression task. Using our proposed mechanism, we also present FAVOR#, a methodfor self-attention approximation in Transformers. We show that FAVOR#outperforms other random feature methods in speech modelling and naturallanguage processing.

FAVOR#: Sharp Attention Kernel Approximations via New Classes of Positive Random Features