  We provide a theoretical, numerical, and experimental investigation of the
Kerr nonlinearity impact on the performance of a frequency-multiplexed Extreme
Learning Machine (ELM). In such ELM, the neuron signals are encoded in the
lines of a frequency comb. The Kerr nonlinearity facilitates the randomized
neuron connections allowing for efficient information mixing. A programmable
spectral filter applies the output weights. The system operates in a
continuous-wave regime. Even at low input peak powers, the resulting weak Kerr
nonlinearity is sufficient to significantly boost the performance on several
tasks. This boost already arises when one uses only the very small Kerr
nonlinearity present in a 20-meter long erbium-doped fiber amplifier. In
contrast, a subsequent propagation in 540 meters of a single-mode fiber
improves the performance only slightly, whereas additional information mixing
with a phase modulator does not result in a further improvement at all. We
introduce a model to show that, in frequency-multiplexed ELMs, the Kerr
nonlinearity mixes information via four-wave mixing, rather than via self- or
cross-phase modulation. At low powers, this effect is quartic in the comb-line
amplitudes. Numerical simulations validate our experimental results and
interpretation.
