We present a novel framework for kernel learning with sequential data of any kind, such as time series, sequences of graphs, or strings. Our approach is based on signature features which can be seen as an ordered variant of sample (cross-)moments; it allows to obtain a "sequentialized" version of any static kernel. The sequential kernels are efficiently computable for discrete sequences and are shown to approximate a continuous moment form in a sampling sense. A number of known kernels for sequences arise as "sequentializations" of suitable static kernels: string kernels may be obtained as a special case, and alignment kernels are closely related up to a modification that resolves their open non-definiteness issue. Our experiments indicate that our signature-based sequential kernel framework may be a promising approach to learning with sequential data, such as time series, that allows to avoid extensive manual pre-processing.
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