Central to rough path theory is the signature transform of a path, an infinite series of tensors given by the iterated integrals of the underlying path. The signature poses an effective way to capture sequentially ordered information, thanks both to its rich analytic and algebraic properties as well as its universality when used as a basis to approximate functions on path space. Whilst a truncated version of the signature can be efficiently computed using Chen's identity, there is a lack of efficient methods for computing a sparse collection of iterated integrals contained in high levels of the signature. We address this problem by leveraging signature kernels, defined as the inner product of two signatures, and computable efficiently by means of PDE-based methods. By forming a filter in signature space with which to take kernels, one can effectively isolate specific groups of signature coefficients and, in particular, a singular coefficient at any depth of the transform. We show that such a filter can be expressed as a linear combination of suitable signature transforms and demonstrate empirically the effectiveness of our approach. To conclude, we give an example use case for sparse collections of signature coefficients based on the construction of N-step Euler schemes for sparse CDEs.

翻译：粗糙路径理论的核心在于路径的签名变换，即由路径的迭代积分给出的无限维张量级数。签名变换凭借其丰富的解析与代数特性，以及作为路径空间函数逼近基的普适性，成为捕获序列化信息的有效工具。虽然签名的截断版本可通过陈氏恒等式高效计算，但目前缺乏有效方法计算签名高层级中包含的稀疏迭代积分集合。本文通过利用签名核——定义为两个签名变换的内积，并借助基于偏微分方程的方法高效计算——来解决该问题。通过在签名空间中构建滤波器进行核运算，可有效分离特定组的签名系数，特别是变换任意深度处的单一系数。我们证明此类滤波器可表示为适当签名变换的线性组合，并通过实验验证了方法的有效性。最后，我们基于稀疏CDE的N步欧拉格式构建，给出了稀疏签名系数集合的应用示例。