Neural Signature Kernels

近年来，微分方程和深度学习之间的共生关系已经成为一个活跃的研究领域，特别是通过引入名为神经微分方程的混合模型。事实上，许多标准的神经网络架构可以被解释为对某些微分方程的近似。这种近似已经被严格地处理为有限宽度的ResNets，其在无限深度极限下按分布收敛于零漂移神经随机微分方程（神经系统随机微分方程）的扩散取决于激活函数的选择。在序列数据的深度学习背景下，有限宽度的RNN已经被非正式地确定为神经控制微分方程的近似（神经CDE）的灵感来自于粗糙分析中研究的动力系统的homonymous类。我的研究项目的主要目标是提供一个严格的数学分析的随机初始化控制ResNets在大宽度和深度制度的行为。

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.