Neural Signature Kernels

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The symbiosis between differential equations and deep learning has become an active research area in recent years, notably through the introduction of hybrid models named neural differential equations. In fact, many standard neural network architectures may be interpreted as approximations to some differential equations. This approximation has been treated rigorously for finite-width ResNets, which in the infinite-depth limit converge in distribution to zero-drift neural stochastic differential equations (Neural SDEs) with diffusion depending on the choice of activation function.In the context of deep learning for sequential data, finite-width RNNs have been informally identified as approximations to neural controlled differential equations (Neural CDEs) inspired from the homonymous class of dynamical systems studied in rough analysis. The main goal of my research project is to provide a rigorous mathematical analysis of the behaviour of randomly initialised controlled ResNets in the large width and depth regimes.