The signature is an infinite graded sequence of statistics known to characterise a stream of data up to a negligible equivalence class. It is a transform which has previously been treated as a fixed feature transformation, on top of which a model may be built. We propose a novel approach which combines the advantages of the signature transform with modern deep learning frameworks. By learning an augmentation of the stream prior to the signature transform, the terms of the signature may be selected in a data-dependent way. More generally, we describe how the signature transform may be used as a layer anywhere within a neural network. In this context it may be interpreted as a pooling operation. We present the results of empirical experiments to back up the theoretical justification.

Summary The authors propose to use a already known method as a pooling function for time series. The idea is to leverage an integral transform, the *path signature*, to map a discretized curve on a real valued sequence. Truncation leads to a vectorized representation which is then used in practice. The proposed transformation is differentiable and thus can be integrated into models trained via backpropagation in an end-to-end manner. As the transformation consumes the one dimension, i.e. the time, stacking it requires to reintroduce a time series like structure in the output.

This paper proposes a method to incorporate signature transform as a layer of a neural network. The proposed scheme is empirically backed up in different scenarios (supervised learning, learning generative model, and RL). Majority of reviewers find the contributions of this submission significant. So I recommend accept, but I ask authors to clarify, in the final version, the novelty concerns that R1 has raised.

The signature is an infinite graded sequence of statistics known to characterise a stream of data up to a negligible equivalence class. It is a transform which has previously been treated as a fixed feature transformation, on top of which a model may be built. We propose a novel approach which combines the advantages of the signature transform with modern deep learning frameworks. By learning an augmentation of the stream prior to the signature transform, the terms of the signature may be selected in a data-dependent way. More generally, we describe how the signature transform may be used as a layer anywhere within a neural network.

The signature is an infinite graded sequence of statistics known to characterise a stream of data up to a negligible equivalence class. It is a transform which has previously been treated as a fixed feature transformation, on top of which a model may be built. We propose a novel approach which combines the advantages of the signature transform with modern deep learning frameworks. By learning an augmentation of the stream prior to the signature transform, the terms of the signature may be selected in a data-dependent way. More generally, we describe how the signature transform may be used as a layer anywhere within a neural network.